diff b1f9bc4..56bc728 restricted_geometric_langlands_pos_v2.txt
diff --git a/restricted_geometric_langlands_pos_v2.txt b/restricted_geometric_langlands_pos_v2.txt
index b94b315..3ba507c 100644
--- a/restricted_geometric_langlands_pos_v2.txt
+++ b/restricted_geometric_langlands_pos_v2.txt
@@ -65,14 +65,14 @@ allowdisplaybreaks[2]
maketitle
Abstract:
-We prove the $mathbb Qbar_bbQbar_ell$-linear restricted geometric Langlands
+We prove the mathbb Qbar_bbQbar_ell-linear restricted geometric Langlands
equivalence for a smooth projective connected curve and a connected reductive
-group over a field of characteristic $p>0$, with $ellne p$, in two regimes.
+group over a field of characteristic p>0, with ellne p, in two regimes.
-Over $mathbb Fbar_q$,bbFbar_q$, we assume only the four characteristic
+Over mathbb Fbar_q,bbFbar_q, we assume only the four characteristic
conditions of the restricted theory (Hypotheses~hyp:common
and~hyp:A). Over an arbitrary algebraically closed field, we assume
-these conditions plus that $p$ is very good and does not divide the order
+these conditions plus that p is very good and does not divide the order
of the Weyl group (Hypothesis~hyp:B). In both regimes we prove
Gaitsgory--Raskin's full-support conjecture
cite[Conj.~1.3.10]{GR}: the primed equivalence is already an equivalence
@@ -95,9 +95,9 @@ tableofcontents
## The question
Geometric Langlands predicts that automorphic objects on the moduli
-$Bun_G(X)$ of $G$-bundles on a smooth projective curve $X$ are matched
-with spectral objects depending on local systems on $X$ for the dual
-group $check G$. The modern formulation compares whole sheaf categories,
+Bun_G(X) of G-bundles on a smooth projective curve X are matched
+with spectral objects depending on local systems on X for the dual
+group check G. The modern formulation compares whole sheaf categories,
and this comparison is organised by the stack of local systems itself:
each point of the spectral stack should determine a summand of the
automorphic category, and this pinning-down should be coherent in families.
@@ -105,7 +105,7 @@ automorphic category, and this pinning-down should be coherent in families.
Over characteristic zero this is a theorem, established in the five-part
project of Arinkin, Beraldo, Campbell, Chen, Faergeman, Gaitsgory, Lin,
Raskin, and Rozenblyum [cite]. Its input is a comparison between
-$ell$-adic sheaves on $Bun_G$ and ind-coherent sheaves on the
+ell-adic sheaves on Bun_G and ind-coherent sheaves on the
stack of local systems with restricted variation introduced by
Arinkin--Gaitsgory--Kazhdan--Raskin--Rozenblyum--Varshavsky [cite].
Positive characteristic is different: the theory has additional
@@ -114,112 +114,112 @@ only reaches an open-and-closed union of spectral components. The
question left open by [cite] is whether any component can be missing.
We prove that none is, in the two regimes stated in the abstract.
-Our proof is geometric. A missing component $C$ would give a nonzero
-constructible object $F$ on $Bun_G$ over the geometric generic fibre
-of a lift $mathscr X/S$scrX/S$ of $X$ to the Witt trait, satisfying $Psi(F)=0$
-for geometric nearby cycles $Psi$. We then construct a small, finite
-slice transverse to the support of $F$ on which $Psi(F)$ cannot vanish,
+Our proof is geometric. A missing component C would give a nonzero
+constructible object F on Bun_G over the geometric generic fibre
+of a lift mathscr X/SscrX/S of X to the Witt trait, satisfying Psi(F)=0
+for geometric nearby cycles Psi. We then construct a small, finite
+slice transverse to the support of F on which Psi(F) cannot vanish,
provided the slice suitably avoids the nilpotent cone. That contradiction
finishes the proof.
## The spectral stack
-Let $k$ be an algebraically closed field of characteristic $p>0$,
-let $ellne p$ be a prime, and set $E=mathbb Qbar_bbQbar_ell$. Let
-$X/k$ be a smooth projective connected curve and $G/k$ a connected
-reductive group; such a group is split over $k$. Write
-$check G/E$ for the Langlands dual group and
-$mathfrak g=frakg=Lie(G)$.
+Let k be an algebraically closed field of characteristic p>0,
+let ellne p be a prime, and set E=mathbb Qbar_bbQbar_ell. Let
+X/k be a smooth projective connected curve and G/k a connected
+reductive group; such a group is split over k. Write
+check G/E for the Langlands dual group and
+mathfrak g=frakg=Lie(G).
-We work with the $E$-linear ind-constructible sheaf categories of
-[cite]. In particular $Shv(X)$ is ind-constructible, and
-$QLisse_E(X)subsetShv(X)$ is the full subcategory whose ordinary
+We work with the E-linear ind-constructible sheaf categories of
+[cite]. In particular Shv(X) is ind-constructible, and
+QLisse_E(X)subsetShv(X) is the full subcategory whose ordinary
cohomology sheaves are filtered colimits of finite-rank lisse
-$E$-sheaves cite[Def.~1.2.6]{AGKRRV}. The restricted spectral prestack
-$Y=LS^{restr}_{check G}(X)$ assigns to a derived affine
-$E$-scheme $T$ the groupoid
+E-sheaves cite[Def.~1.2.6]{AGKRRV}. The restricted spectral prestack
+Y=LS^{restr}_{check G}(X) assigns to a derived affine
+E-scheme T the groupoid
Y(T)=Fun^{ ⊗ ,r.t.ex}!(Rep_E(check G),,
QCoh(T) ⊗ _EQLisse_E(X)),
-of right-$t$-exact symmetric monoidal functors. An $E$-parameter
-$sigmain Y(E)$ is such a functor with $T=Spec E$: concretely, an
-assignment of a lisse sheaf $sigma(V)$ to each finite-dimensional
-representation of $check G$, monoidal in $V$.
+of right-t-exact symmetric monoidal functors. An E-parameter
+sigmain Y(E) is such a functor with T=Spec E: concretely, an
+assignment of a lisse sheaf sigma(V) to each finite-dimensional
+representation of check G, monoidal in V.
The reason for the restricted variation in (eq:Y-def) is the same
as in characteristic zero. Ind-coherent sheaves alone with no support
condition do not capture the Arthur-type data that automorphic sheaves
must carry. On the automorphic side one restricts to sheaves with
nilpotent singular support, i.e. those whose wavefront set is contained
-in the global nilpotent cone $mathcal NcalNsubset T^*Bun_G$; write this
-subcategory $Shv_{Nilp}(Bun_G(X))$ (see Ssec:intro-cat below
+in the global nilpotent cone mathcal NcalNsubset T^*Bun_G; write this
+subcategory Shv_{Nilp}(Bun_G(X)) (see Ssec:intro-cat below
for the precise definition). On the spectral side one imposes an
-analogous condition and uses $IndCoh_{Nilp}(Y)$. These categories
+analogous condition and uses IndCoh_{Nilp}(Y). These categories
already exist in [cite]; our new assertion is about the
spectral components on which they are supported.
-Both sides carry a coherent action of $QCoh(Y)$, coming from the
-universal family of local systems on $Ytimes X$ and the construction
+Both sides carry a coherent action of QCoh(Y), coming from the
+universal family of local systems on Ytimes X and the construction
of Hecke operators. This makes the comparison functor
-$QCoh(Y)$-linear, and lets us localise to individual components.
+QCoh(Y)-linear, and lets us localise to individual components.
## Two characteristic regimes and the four standing hypotheses
All four hypotheses below are already required by the construction of
-$Shv_{Nilp}$ in [cite]. They hold for $G=GL_n$ in
+Shv_{Nilp} in [cite]. They hold for G=GL_n in
any characteristic and for many other groups; each captures a way in
which positive characteristic can complicate Lie theory.
[Common characteristic assumptions]
-The following hold for $G/k$.
+The following hold for G/k.
[label=(roman*)]
-item There is a nondegenerate $G$-invariant symmetric bilinear form
- $kappa$ on $mathfrak g$frakg$ whose restriction to $Lie Z(M)$ is
- nondegenerate for every Levi subgroup $M$.
-item Chevalley restriction $k[Lie M]^Mto k[Lie T_M]^{W_M}$ is an
- isomorphism for every Levi $M$ and every maximal torus $T_Msubset M$.
-item The scheme-theoretic centraliser in $G$ of every semisimple
- element of $mathfrak g$frakg$ is a Levi subgroup.
-item Every nilpotent element of $mathfrak g$frakg$ lies in $Lie R_u(P)$ for
- some parabolic $Psubset G$, after any field extension.
+ There is a nondegenerate G-invariant symmetric bilinear form
+ kappa on mathfrak gfrakg whose restriction to Lie Z(M) is
+ nondegenerate for every Levi subgroup M.
+ Chevalley restriction k[Lie M]^Mto k[Lie T_M]^{W_M} is an
+ isomorphism for every Levi M and every maximal torus T_Msubset M.
+ The scheme-theoretic centraliser in G of every semisimple
+ element of mathfrak gfrakg is a Levi subgroup.
+ Every nilpotent element of mathfrak gfrakg lies in Lie R_u(P) for
+ some parabolic Psubset G, after any field extension.
Conditions (i)--(iii) are cite[S14.4.1]{AGKRRV}; condition (iv) is
cite[S D.1.1]{AGKRRV}. Both hypotheses below keep the notion of
semisimple and nilpotent as in [cite]: semisimple elements of
-$mathfrak g$frakg$ are exactly the ones geometrically conjugate into a
+mathfrak gfrakg are exactly the ones geometrically conjugate into a
Cartan subalgebra, nilpotent ones the ones whose adjoint orbit closure
-contains $0$.
+contains 0.
[Regime A: finite-field descent]
-There are $q=p^r$, a smooth projective geometrically connected curve
-$X_0/mathbb F_q$,bbF_q$, and a split connected reductive group
-$G_0/mathbb F_q$bbF_q$ with $k=mathbb Fbar_q$,bbFbar_q$,
-$X=X_0 ⊗ _{mathbb F_q}bbF_qk$, $G=G_0 ⊗ _{mathbb F_q}bbF_qk$.
-Hypothesis~hyp:common holds. No restriction on $|W_G|$ is imposed.
+There are q=p^r, a smooth projective geometrically connected curve
+X_0/mathbb F_q,bbF_q, and a split connected reductive group
+G_0/mathbb F_qbbF_q with k=mathbb Fbar_q,bbFbar_q,
+X=X_0 ⊗ _{mathbb F_q}bbF_qk, G=G_0 ⊗ _{mathbb F_q}bbF_qk.
+Hypothesis~hyp:common holds. No restriction on |W_G| is imposed.
[Regime B: arbitrary algebraically closed base]
-$k$ is any algebraically closed field of characteristic $p>0$.
-Hypothesis~hyp:common holds, $p$ is very good for $G$, and
-$pnmid|W_G|$. No finite-field descent is assumed.
+k is any algebraically closed field of characteristic p>0.
+Hypothesis~hyp:common holds, p is very good for G, and
+pnmid|W_G|. No finite-field descent is assumed.
-``Very good'' is the usual root-system condition. For type $A_n$ it
-means $pnmid n+1$; for $B_n,C_n,D_n$ it means $pne2$; for
-$G_2,F_4,E_6,E_7$ that $pne2,3$; for $E_8$ that $pne2,3,5$; applied on
+``Very good'' is the usual root-system condition. For type A_n it
+means pnmid n+1; for B_n,C_n,D_n it means pne2; for
+G_2,F_4,E_6,E_7 that pne2,3; for E_8 that pne2,3,5; applied on
each irreducible factor. A torus has no such restriction, and every
prime is very good for a torus.
The two regimes differ only in one Lie-theoretic input
(Section~sec:lie): how to produce a central cocharacter with a
nonvanishing pairing with the constant term of a Higgs field. Regime A
-solves this with a lattice argument that uses only $kappa$ and
+solves this with a lattice argument that uses only kappa and
Hypothesis~hyp:common(i)-(iii). Regime B solves it with the
-separately assumed invertibility of $|W_G|$. Neither is used as a
+separately assumed invertibility of |W_G|. Neither is used as a
substitute for the other; both produce the identical statement of
Proposition~lie:input, and everything downstream is common to
the two regimes.
@@ -232,19 +232,19 @@ it is worth pinning them down once, because several comparisons later
depend on the precise definition.
On the automorphic side, the global nilpotent cone
-$mathcal NcalNsubset T^*Bun_G(X)$ consists of Higgs pairs $(P,A)$ with
-$Ain H^0(X,Ad P ⊗ omega_X)$ pointwise nilpotent. A bounded
-constructible $E$-sheaf on $Bun_G(X)$ has nilpotent singular support
-if its wavefront set lies in $mathcal N$;calN$; the full subcategory
-$Shv_{Nilp}(Bun_G(X))subsetShv(Bun_G(X))$ consists of the
+mathcal NcalNsubset T^*Bun_G(X) consists of Higgs pairs (P,A) with
+Ain H^0(X,Ad P ⊗ omega_X) pointwise nilpotent. A bounded
+constructible E-sheaf on Bun_G(X) has nilpotent singular support
+if its wavefront set lies in mathcal N;calN; the full subcategory
+Shv_{Nilp}(Bun_G(X))subsetShv(Bun_G(X)) consists of the
ind-objects whose every cohomology sheaf has this property. This
definition uses Beilinson's theory of singular support for constructible
sheaves in arbitrary characteristic [cite];
it is not the coherent singular-support theory of [cite],
which lives on the spectral side.
-On the spectral side, ind-coherent sheaves on $Y$ with nilpotent singular
-support, denoted $IndCoh_{Nilp}(Y)$, are defined in cite[SS21.2.1--21.2.5]{AGKRRV}.
+On the spectral side, ind-coherent sheaves on Y with nilpotent singular
+support, denoted IndCoh_{Nilp}(Y), are defined in cite[SS21.2.1--21.2.5]{AGKRRV}.
The nilpotent condition there is a condition on a horizontal section of
the adjoint local system, matching Laumon's prediction. This condition,
its correspondence to Laumon's conjecture for Hecke eigensheaves, and the
@@ -257,37 +257,37 @@ correspondence we prove matches them.
## The primed equivalence and full support
By cite[Main Thm.~1.3.9]{GR}, in either regime there is a
-$QCoh(Y)$-linear equivalence of the form
+QCoh(Y)-linear equivalence of the form
mathbb L_G^bbL_G^{restr}colon
Shv_{Nilp}(Bun_G(X))
xrightarrow{ sim }IndCoh_{Nilp}(Y')
↪IndCoh_{Nilp}(Y),
-where $Y'subset Y$ is an open-and-closed union of connected components
-of $Y$. Equivalently, it is a $QCoh(Y)$-linear equivalence onto the
-direct summand $IndCoh_{Nilp}(Y')$ of $IndCoh_{Nilp}(Y)$. The
+where Y'subset Y is an open-and-closed union of connected components
+of Y. Equivalently, it is a QCoh(Y)-linear equivalence onto the
+direct summand IndCoh_{Nilp}(Y') of IndCoh_{Nilp}(Y). The
inclusion is the natural fully faithful functor, and its image is a
summand.
-What is missing from (eq:primed) is the assertion $Y'=Y$: all
+What is missing from (eq:primed) is the assertion Y'=Y: all
components, not just some, support a nonzero summand of automorphic
sheaves. This is the content of the following theorem.
[Full support]
Under either Hypothesis~hyp:A or Hypothesis~hyp:B, the
-inclusion $Y'subset Y$ in (eq:primed) is equality of spectral
+inclusion Y'subset Y in (eq:primed) is equality of spectral
prestacks, and the natural functor
[
mathbb L_G^bbL_G^{restr}colon
Shv_{Nilp}(Bun_G(X))xrightarrow{ sim }
IndCoh_{Nilp}(Y)
]
-is a $QCoh(Y)$-linear equivalence.
+is a QCoh(Y)-linear equivalence.
The primed equivalence used as input comes from cite[Main Thm.~1.3.9(i)]{GR};
-the full characteristic-zero equivalence and the case of $GL_n$
+the full characteristic-zero equivalence and the case of GL_n
are also established there. Theorem~thm:full-support upgrades the
positive-characteristic case from a summand to the whole category, and
in so doing proves cite[Conj.~1.3.10]{GR} in the two regimes.
@@ -296,22 +296,22 @@ in so doing proves cite[Conj.~1.3.10]{GR} in the two regimes.
Three consequences are developed in Section~sec:consequences:
[label=(alph*)]
-item localising at any one component $Csubset Y$ gives a nonzero
- automorphic category $mathcal C_calC_{s,C}$ equivalent to
- $IndCoh_{Nilp}(C)$;
-item in regime A, the arithmetic trace formula holds with the full
- derived fixed-point stack $Y^{Frob}$ rather than the primed
- $(Y')^{Frob}$;
-item every $E$-parameter $sigmain Y(E)$ admits a nonzero Hecke
+ localising at any one component Csubset Y gives a nonzero
+ automorphic category mathcal C_calC_{s,C} equivalent to
+ IndCoh_{Nilp}(C);
+ in regime A, the arithmetic trace formula holds with the full
+ derived fixed-point stack Y^{Frob} rather than the primed
+ (Y')^{Frob};
+ every E-parameter sigmain Y(E) admits a nonzero Hecke
eigenobject with coherent tensor and fusion structure.
None of these requires any additional hypothesis beyond
Theorem~thm:full-support.
Section~sec:coefficients proves the same support statement for
-rational coefficients $mathbb Q_bbQ_ell$. The theorem there is conditional
+rational coefficients mathbb Q_bbQ_ell. The theorem there is conditional
on a single input: the existence of the primed trace formula over
-$mathbb Q_bbQ_ell$ itself, intertwining the excursion and spherical Hecke
+mathbb Q_bbQ_ell itself, intertwining the excursion and spherical Hecke
actions with the Frobenius and Satake conventions. We do not construct
that map. What we prove is that once it is given, the primed formula
unprimes without any further hypothesis, at the level of the whole
@@ -323,29 +323,29 @@ The rest of the paper is organised so that each section proves one point.
smallskipnoindent
Section~sec:specialization (specialisation). We lift
-$(X,G)$ to $(mathscr X/S,G/S)$scrX/S,G/S)$ over the Witt trait, write
-$mathcal B=calB=Bun_G(mathscr X/S)$scrX/S)$ and set up geometric specialisation
-$Psi$ from the geometric generic fibre to the closed fibre. We show
-that a missing component $C$ produces a nonzero compact $F$ on
-$mathcal B_K$calB_K$ with $Psi(F)=0$, and that this vanishing persists
-after pulling back along smooth maps $acolon Ttomathcal B$calB$ and after
-applying any Hecke operator $mathrm H_V$.rmH_V$. Crucially, the leg direction
-of the Hecke map is arbitrary; only the projection to $mathcal B$calB$ must
+(X,G) to (mathscr X/S,G/S)scrX/S,G/S) over the Witt trait, write
+mathcal B=calB=Bun_G(mathscr X/S)scrX/S) and set up geometric specialisation
+Psi from the geometric generic fibre to the closed fibre. We show
+that a missing component C produces a nonzero compact F on
+mathcal B_KcalB_K with Psi(F)=0, and that this vanishing persists
+after pulling back along smooth maps acolon Ttomathcal BcalB and after
+applying any Hecke operator mathrm H_V.rmH_V. Crucially, the leg direction
+of the Hecke map is arbitrary; only the projection to mathcal BcalB must
be smooth. This is the freedom later needed to cut Hecke correspondences
by a hypersurface.
smallskipnoindent
Section~sec:lie (Lie-theoretic reduction). Starting from a
-non-nilpotent Higgs field $A$ at some point $x$ on the curve, we find a
-Levi subgroup $Msubset G$, a cocharacter $lambda$ of its centre, and a
-change of formal frame, such that $A$ lies in $mathfrak m[[t]]$,frakm[[t]]$,
-$kappa(A,dlambda)ne0$, and $ad_{A}$ is invertible on
-$mathfrak g/frakg/mathfrak m$.frakm$. This is the input for constructing a Hecke
+non-nilpotent Higgs field A at some point x on the curve, we find a
+Levi subgroup Msubset G, a cocharacter lambda of its centre, and a
+change of formal frame, such that A lies in mathfrak m[[t]],frakm[[t]],
+kappa(A,dlambda)ne0, and ad_{A} is invertible on
+mathfrak g/frakg/mathfrak m.frakm. This is the input for constructing a Hecke
modification whose moving-point covector does not vanish.
smallskipnoindent
Section~sec:hecke (Hecke modification). Applying the above
-modification to a fixed bundle, we get a nonzero leg covector $xi$ and
+modification to a fixed bundle, we get a nonzero leg covector xi and
two relative cotangent sections (fixed-input and fixed-output) whose
derivatives are isomorphisms on the two Hecke fibres. This is the
tranversality input.
@@ -353,9 +353,9 @@ tranversality input.
smallskipnoindent
Section~sec:twohecke (two Hecke correspondences). Composing
two Hecke correspondences with common output and common leg, and cutting
-by a hypersurface dual to $xi$, we isolate one stalk of $Psi(F)$ and
+by a hypersurface dual to xi, we isolate one stalk of Psi(F) and
identify its local neighbourhood with a split quadratic equation
-$sum_{i=1}^m u_iv_i+f=0$.
+sum_{i=1}^m u_iv_i+f=0.
smallskipnoindent
Section~sec:slices (from quadric to slices). A proper
@@ -366,16 +366,16 @@ incidence construction then handles higher-codimension transverse slices.
smallskipnoindent
Section~sec:support (contradiction). The support of
-$Psi(F)$ can be cut down to a finite scheme over the trait, on which
+Psi(F) can be cut down to a finite scheme over the trait, on which
nearby cycles are a finite direct sum of stalks. Since the summands
cannot all vanish (the object is nonzero), we get a slice on which
-$Psi$ does not vanish, contradicting the transverse test. Since the
-nilpotent cone has dimension at most $dimmathcal B_s$,calB_s$, a slice of
+Psi does not vanish, contradicting the transverse test. Since the
+nilpotent cone has dimension at most dimmathcal B_s,calB_s, a slice of
sufficiently high codimension always exists. This finishes
Theorem~thm:full-support.
smallskipnoindent
-Sections~ref{sec:consequences--sec:coefficients} apply the
+Sections~ apply the
theorem to the previously listed consequences and prove the rational
coefficient statement by coefficient comparison.
@@ -386,11 +386,11 @@ coefficient statement by coefficient comparison.
Throughout this section we assume Hypothesis~hyp:A or
Hypothesis~hyp:B; we never assume Theorem~thm:full-support
-itself. The goal is a single statement: if $Y'ne Y$, then there exists
-a nonzero sheaf $F$ on the geometric generic fibre for which all the
+itself. The goal is a single statement: if Y'ne Y, then there exists
+a nonzero sheaf F on the geometric generic fibre for which all the
nearby-cycle vanishings we need later do actually hold. The
construction has three parts. First we place everything over the Witt
-trait. Second we extract such an $F$ from the missing component.
+trait. Second we extract such an F from the missing component.
Third we propagate its vanishing along smooth pullbacks and Hecke
transforms.
@@ -398,64 +398,64 @@ transforms.
Let
[
- R=W(k),qquad S=Spec(R),qquad K=Frac(R)bar,
+ R=W(k), S=Spec(R), K=Frac(R)bar,
]
-where $W(k)$ is the ring of Witt vectors of $k$. Since $k$ is perfect,
-$R$ is a complete Noetherian discrete valuation ring with uniformizer
-$p$ and residue field $k$. It is strictly henselian, and it is
+where W(k) is the ring of Witt vectors of k. Since k is perfect,
+R is a complete Noetherian discrete valuation ring with uniformizer
+p and residue field k. It is strictly henselian, and it is
excellent because it is a complete Noetherian local ring
cite[Tag~07QS]{Stacks}.
-The curve $X/k$ admits a smooth projective lift $mathscr X/S$.scrX/S$.
+The curve X/k admits a smooth projective lift mathscr X/S.scrX/S.
Concretely, smooth-lifting theory gives a smooth formal lift over
-$SpfR$, since the obstruction to lifting a smooth curve
-lies in $H^2(X,T_X)=0$; the ample line bundle $O_X(1)$ lifts for the
-same reason ($H^2(X,mathcal O_X)=0$);calO_X)=0$); Grothendieck's existence theorem
+SpfR, since the obstruction to lifting a smooth curve
+lies in H^2(X,T_X)=0; the ample line bundle O_X(1) lifts for the
+same reason (H^2(X,mathcal O_X)=0);calO_X)=0); Grothendieck's existence theorem
algebraises the resulting formal data to a projective lift. The lift
is smooth near the closed fibre, and the nonsmooth locus is a closed
-subset of a proper $S$-scheme; it cannot meet the closed fibre without
+subset of a proper S-scheme; it cannot meet the closed fibre without
meeting it as a horizontal divisor, so it is empty. See
cite[Tags~0DZQ,~0E7R]{Stacks}; the same lift is used in
-cite[S3.1.1]{GR}. This construction does not require that $X$ descends
+cite[S3.1.1]{GR}. This construction does not require that X descends
to a finite field.
-The root datum of the split $G$ gives a split reductive group over
-$mathbb Z$,bbZ$, and base change to $S$ gives a lift, which we also denote
-$G$ cite[Thms.~6.1.16(2) and~6.1.17]{Conrad}. Write
+The root datum of the split G gives a split reductive group over
+mathbb Z,bbZ, and base change to S gives a lift, which we also denote
+G cite[Thms.~6.1.16(2) and~6.1.17]{Conrad}. Write
[
- mathcal B=calB=Bun_G(mathscr X/S),scrX/S),qquad
- s=Spec(k),qquad eta=Spec(K).
+ mathcal B=calB=Bun_G(mathscr X/S),scrX/S),
+ s=Spec(k), eta=Spec(K).
]
-Then $mathcal B=calB=Hom_S(mathscr X,B_SG)$scrX,B_SG)$ is an algebraic stack locally
-of finite type over $S$ with affine diagonal
-cite[Thm.~1.2(i)--(iii)]{HallRydh}: $mathscr X/S$scrX/S$ is proper, flat and
-of finite presentation, and $B_SG$ is locally of finite presentation
+Then mathcal B=calB=Hom_S(mathscr X,B_SG)scrX,B_SG) is an algebraic stack locally
+of finite type over S with affine diagonal
+cite[Thm.~1.2(i)--(iii)]{HallRydh}: mathscr X/SscrX/S is proper, flat and
+of finite presentation, and B_SG is locally of finite presentation
and quasi-separated, with affine stabilisers and affine diagonal. The
-obstruction to deforming a bundle on a curve lies in $H^2(X,Ad P)=0$,
-so $mathcal B$calB$ is smooth over $S$.
+obstruction to deforming a bundle on a curve lies in H^2(X,Ad P)=0,
+so mathcal BcalB is smooth over S.
-Later we will replace $R$ by its integral closure in a finite extension
-of $Frac(R)$ inside $K$. Such an integral closure is a
-finite complete discrete valuation ring with residue field $k$, still
+Later we will replace R by its integral closure in a finite extension
+of Frac(R) inside K. Such an integral closure is a
+finite complete discrete valuation ring with residue field k, still
excellent and strictly henselian cite[Tags~0EXQ,~0323]{Stacks}. All
subsequent models are understood to have been replaced in this way,
-and we continue to use $R,S,mathscr X,scrX,mathcal B$.calB$. The point is that
+and we continue to use R,S,mathscr X,scrX,mathcal B.calB. The point is that
every dimension argument below takes place over a Noetherian trait
-which is not the integral closure in the whole of $K$.
+which is not the integral closure in the whole of K.
-For a finite-type $S$-scheme or algebraic space $T$, geometric
+For a finite-type S-scheme or algebraic space T, geometric
specialisation with monodromy forgotten is a functor
[
Psi_Tcolon Shv(T_K)→Shv(T_s).
]
-Here $Shv$ always means the ind-completion of bounded constructible
-$E$-adic complexes, using ordinary $*$-pullback, ordinary tensor and
+Here Shv always means the ind-completion of bounded constructible
+E-adic complexes, using ordinary *-pullback, ordinary tensor and
ordinary stalks. It extends to locally finite-type algebraic stacks by
-smooth descent, and $Psi_T$ is defined on the whole sheaf category.
+smooth descent, and Psi_T is defined on the whole sheaf category.
The construction is as in cite[SS4.1.1--4.1.5]{GR}: at finite
coefficients and finite extensions of the base field one takes
geometric nearby cycles; one passes through the compatible coefficient
-limits, inverts $ell$, extends coefficients to $E$, and extends
+limits, inverts ell, extends coefficients to E, and extends
continuously from constructibles. The result is exact, continuous,
preserves constructibility, and commutes with smooth pullback and
representable proper pushforward cite[S4.1.6]{GR}. These properties
@@ -463,9 +463,9 @@ are unchanged by the finite trait replacements above; the coefficient
functors and the models at successive stages agree on the identified
geometric-fibre coefficients.
-We will use one tensor identity repeatedly. If $acolon Ttomathcal B$calB$
-is smooth and $mathcal L$calL$ is a finite-rank lisse sheaf on $T$, then
-for every $DinShv(mathcal B_K)$calB_K)$
+We will use one tensor identity repeatedly. If acolon Ttomathcal BcalB
+is smooth and mathcal LcalL is a finite-rank lisse sheaf on T, then
+for every DinShv(mathcal B_K)calB_K)
Psi_T(a_K^*D ⊗ mathcal L_K)calL_K)
simeq a_s^*Psi_{mathcal B}calB(D) ⊗ mathcal L_s.calL_s.
@@ -474,19 +474,19 @@ A lisse sheaf on the source of a smooth morphism is universally locally
acyclic over the target; (eq:spec-lisse-tensor) is exactly the
universally-locally-acyclic tensor-pullback formula
cite[Lemma~4.1.9]{GR}. Continuity makes it applicable to arbitrary
-$D$ in the indicated ind-category.
+D in the indicated ind-category.
## The spectral summand attached to a missing component
Set
[
- mathcal C_s=calC_s=Shv_{Nilp}(mathcal B_s),calB_s),qquad
- mathcal C_K=calC_K=Shv_{Nilp}(mathcal B_K),calB_K),qquad
+ mathcal C_s=calC_s=Shv_{Nilp}(mathcal B_s),calB_s),
+ mathcal C_K=calC_K=Shv_{Nilp}(mathcal B_K),calB_K),
Y_K=LS^{restr}_{check G}(mathscr X_K).scrX_K).
]
Restriction to the closed fibre gives an identification
-$QLisse(mathscr X)scrX)simeqQLisse(X)$ cite[S3.1.3]{GR}. Its inverse,
+QLisse(mathscr X)scrX)simeqQLisse(X) cite[S3.1.3]{GR}. Its inverse,
composed with restriction to the geometric generic fibre, gives a
symmetric monoidal functor
[
@@ -496,21 +496,21 @@ The corresponding map
[
icolon Y↪ Y_K
]
-identifies $Y$ with an open-and-closed union of components of $Y_K$
+identifies Y with an open-and-closed union of components of Y_K
cite[S3.1.3]{GR}. In particular the tensor product
mathcal D_Y:=calD_Y:=QCoh(Y)underset{QCoh(Y_K)}{ ⊗ }mathcal C_KcalC_K
-is a direct summand of $mathcal C_K$.calC_K$.
+is a direct summand of mathcal C_K.calC_K.
We need the actual specialisation functor to commute with this summand,
-not merely a functor obtained by transporting $mathbb L_G$bbL_G$ across the
+not merely a functor obtained by transporting mathbb L_GbbL_G across the
specialisation. Corollary~4.3.6 and S4.3.9 of [cite] show that the
chartwise geometric nearby-cycle functor we built above restricts to
[
Psi_{mathcal B}calBcolon mathcal D_Y→calD_Y→ mathcal C_s,calC_s,
]
-and it is $QCoh(Y)$-linear by cite[Thm.~3.1.6(B) and S4.3.10]{GR}.
+and it is QCoh(Y)-linear by cite[Thm.~3.1.6(B) and S4.3.10]{GR}.
The characteristic-zero equivalence of cite[Main Thm.~1.3.9(ii)]{GR},
together with its spectral linearity, identifies
@@ -518,51 +518,51 @@ together with its spectral linearity, identifies
mathcal D_YcalD_YsimeqIndCoh_{Nilp}(Y).
In positive characteristic we use only the primed version:
-$mathcal C_scalC_ssimeqIndCoh_{Nilp}(Y')$ for some open-and-closed
-$Y'subset Y$.
+mathcal C_scalC_ssimeqIndCoh_{Nilp}(Y') for some open-and-closed
+Y'subset Y.
-If a connected component $Csubset Y$ is disjoint from $Y'$, then the
+If a connected component Csubset Y is disjoint from Y', then the
summand
[
mathcal D_C:=calD_C:=QCoh(C)underset{QCoh(Y)}{ ⊗ }mathcal D_YcalD_Y
]
-of $mathcal D_Y$calD_Y$ is nonzero; $Psi_{mathcal B}calB(D)=0$ for every
-$Dinmathcal D_C$;calD_C$; and there exists a nonzero $Finmathcal D_C$calD_C$
-whose image under $mathcal D_YcalD_Ysubsetmathcal C_KcalC_Ksubset
-Shv(mathcal B_K)$calB_K)$ is compact, and which restricts to a bounded
-constructible complex on every finite-type smooth chart $Utomathcal B$.calB$.
+of mathcal D_YcalD_Y is nonzero; Psi_{mathcal B}calB(D)=0 for every
+Dinmathcal D_C;calD_C; and there exists a nonzero Finmathcal D_CcalD_C
+whose image under mathcal D_YcalD_Ysubsetmathcal C_KcalC_Ksubset
+Shv(mathcal B_K)calB_K) is compact, and which restricts to a bounded
+constructible complex on every finite-type smooth chart Utomathcal B.calB.
Proof.
-Let $e_CinQCoh(Y)$ be the structure sheaf of $C$ extended by zero.
-Its action is the projection onto $mathcal D_C$.calD_C$. On $mathcal C_s$calC_s$
-the same object acts by zero, because $Ccap Y'=∅$ and the
+Let e_CinQCoh(Y) be the structure sheaf of C extended by zero.
+Its action is the projection onto mathcal D_C.calD_C. On mathcal C_scalC_s
+the same object acts by zero, because Ccap Y'=∅ and the
primed equivalence is spectral-linear. Therefore for every
-$Dinmathcal D_C$,calD_C$,
+Dinmathcal D_C,calD_C,
[
Psi_{mathcal B}calB(D)
simeqPsi_{mathcal B}calB(e_Ccdot D)
simeq e_CcdotPsi_{mathcal B}calB(D)=0.
]
-Under (eq:spec-generic-equivalence), $mathcal D_C$calD_C$ is
-$IndCoh_{Nilp}(C)$, which is nonzero: the fully faithful
-$QCoh(C)↪IndCoh_{Nilp}(C)$ sends $mathcal O_CcalO_Cne0$ to
+Under (eq:spec-generic-equivalence), mathcal D_CcalD_C is
+IndCoh_{Nilp}(C), which is nonzero: the fully faithful
+QCoh(C)↪IndCoh_{Nilp}(C) sends mathcal O_CcalO_Cne0 to
a nonzero object cite[SS1.3.1--1.3.4]{GR}.
-The category $mathcal D_Y$calD_Y$ is compactly generated
+The category mathcal D_YcalD_Y is compactly generated
cite[S3.1.7]{GR}. In the binary decomposition
-$mathcal D_YcalD_Ysimeqmathcal D_CcalD_Ctimesmathcal D_calD_{Y∖ C}$ both
+mathcal D_YcalD_Ysimeqmathcal D_CcalD_Ctimesmathcal D_calD_{Y∖ C} both
projections preserve compactness, because both their right adjoints
(the inclusions of the summands) are continuous. Project a set of
-compact generators of $mathcal D_Y$calD_Y$ under $e_C$; at least one
+compact generators of mathcal D_YcalD_Y under e_C; at least one
projection is nonzero, since otherwise the colimit closure would give
-$mathcal D_C=0$.calD_C=0$. Choose such an $F$.
+mathcal D_C=0.calD_C=0. Choose such an F.
-Compactness in $mathcal C_K$calC_K$ is the same argument: $F$ is compact in
-$mathcal D_Y$,calD_Y$, which is a summand of $mathcal C_K$,calC_K$, and the inclusion
+Compactness in mathcal C_KcalC_K is the same argument: F is compact in
+mathcal D_Y,calD_Y, which is a summand of mathcal C_K,calC_K, and the inclusion
of a summand preserves compactness. The inclusion
-$mathcal C_K↪calC_K↪Shv(mathcal B_K)$calB_K)$ preserves compactness
+mathcal C_K↪calC_K↪Shv(mathcal B_K)calB_K) preserves compactness
by cite[Thm.~1.1.7]{GR}. Finally, a compact sheaf on an algebraic
stack restricts to a bounded constructible complex on every smooth
finite-type chart cite[S F.2.2]{AGKRRV}.
@@ -570,19 +570,19 @@ finite-type chart cite[S F.2.2]{AGKRRV}.
The argument uses only one missing component and binary decomposition.
-It does not require finitely many components of $Y$ or compactness of
-$e_C$. If $Y'ne Y$ then such a $C$ exists because $Y'$ is a union of
-open-and-closed components of $Y$.
+It does not require finitely many components of Y or compactness of
+e_C. If Y'ne Y then such a C exists because Y' is a union of
+open-and-closed components of Y.
For the remainder of the geometric argument we fix a missing component
-$C$ and the associated $F$ as in Proposition~spec:missing. If
-$Y'=Y$ there is nothing to prove.
+C and the associated F as in Proposition~spec:missing. If
+Y'=Y there is nothing to prove.
## Hecke transforms of a specialising object
-Let $V$ be a finite-dimensional representation of $check G$ and let
+Let V be a finite-dimensional representation of check G and let
[
mathrm H_VrmH_Vcolonmathcal C_K→calC_K→
Shv(mathcal B_KcalB_Ktimes_Kmathscr X_K)scrX_K)
@@ -594,94 +594,94 @@ facts about the universal family.
The universal objects are
[
- mathcal E_calE_{V,s}inQCoh(Y) ⊗ QLisse(X),qquad
+ mathcal E_calE_{V,s}inQCoh(Y) ⊗ QLisse(X),
mathcal E_calE_{V,K}inQCoh(Y_K) ⊗ QLisse(mathscr X_K).scrX_K).
]
-On an affine test $Tto Y$ the restriction of $mathcal E_V$calE_V$ is the
+On an affine test Tto Y the restriction of mathcal E_VcalE_V is the
family of local systems obtained by applying the corresponding tensor
-functor to $V$. Acting by its first tensor factor and tensoring with
-the exterior product gives $mathrm H_V$rmH_V$ on the spectral side
+functor to V. Acting by its first tensor factor and tensoring with
+the exterior product gives mathrm H_VrmH_V on the spectral side
cite[S14.3.3]{AGKRRV}.
-By definition of $i$,
+By definition of i,
(i^* ⊗ id)(mathcal E_calE_{V,K})
simeq(id ⊗ rho)(mathcal E_calE_{V,s}).
On every affine test the two sides are the same local system; the
passage from these identities to (eq:spec-universal-family) uses
-dualizability of $QLisse(X)$. Concretely, $QLisse(X)$ for a smooth
+dualizability of QLisse(X). Concretely, QLisse(X) for a smooth
projective curve is compactly generated
cite[S1.2.11,~SS E.2.6--E.2.7]{AGKRRV}, hence dualizable as a
presentable dg category, and tensoring with a dualizable category
-commutes with the descent limit defining $QCoh(Y)$.
+commutes with the descent limit defining QCoh(Y).
The second, distinct input is cite[Rem.~14.3.9]{AGKRRV}:
mathcal E_calE_{V,s}inQCoh(Y) ⊗ IndLisse(X),
-where $IndLisse(X)=Ind(Lisse(X))$ and
-$Lisse(X)$ consists of bounded constructible complexes
+where IndLisse(X)=Ind(Lisse(X)) and
+Lisse(X) consists of bounded constructible complexes
with finite-rank lisse cohomology sheaves. We do not identify
-$IndLisse(X)$ with $QLisse(X)$; that identification fails for
-$X=mathbb P^1$.bbP^1$. The two properties used above (dualizability for
-descent, membership in $IndLisse$ for propagation) are separate.
-
-Now apply the $QCoh(Y)$-action in (eq:spec-IndLisse) to $F$. The
-result is an object of $mathcal D_CcalD_C ⊗ IndLisse(X)$. After applying
-$id ⊗ rho$ and the exterior-product functor, it is precisely
-$mathrm H_V(F)$rmH_V(F)$ by (eq:spec-universal-family). Elementary tensors
-generate $mathcal D_CcalD_C ⊗ IndLisse(X)$ under colimits, and bounded
-truncations build every object of $Lisse(X)$ from
-finite-rank lisse sheaves. Therefore $mathrm H_V(F)$rmH_V(F)$ lies in the stable
+IndLisse(X) with QLisse(X); that identification fails for
+X=mathbb P^1.bbP^1. The two properties used above (dualizability for
+descent, membership in IndLisse for propagation) are separate.
+
+Now apply the QCoh(Y)-action in (eq:spec-IndLisse) to F. The
+result is an object of mathcal D_CcalD_C ⊗ IndLisse(X). After applying
+id ⊗ rho and the exterior-product functor, it is precisely
+mathrm H_V(F)rmH_V(F) by (eq:spec-universal-family). Elementary tensors
+generate mathcal D_CcalD_C ⊗ IndLisse(X) under colimits, and bounded
+truncations build every object of Lisse(X) from
+finite-rank lisse sheaves. Therefore mathrm H_V(F)rmH_V(F) lies in the stable
colimit closure of objects
- Dboxtimesmathcal L_K,calL_K,qquad Dinmathcal D_C,calD_C,
+ Dboxtimesmathcal L_K,calL_K, Dinmathcal D_C,calD_C,
-where $mathcal L_K$calL_K$ is the restriction of a finite-rank lisse sheaf
-$mathcal L_S$calL_S$ on $mathscr X$.scrX$.
+where mathcal L_KcalL_K is the restriction of a finite-rank lisse sheaf
+mathcal L_ScalL_S on mathscr X.scrX.
-Let $acolon Ttomathcal B$calB$ be smooth, with $T$ a finite-type $S$-scheme
-or algebraic space, and let $ell_Tcolon Ttomathscr X$scrX$ be any
-$S$-morphism. Then for $F$ as in Proposition~spec:missing and
-every finite-dimensional $V$,
+Let acolon Ttomathcal BcalB be smooth, with T a finite-type S-scheme
+or algebraic space, and let ell_Tcolon Ttomathscr XscrX be any
+S-morphism. Then for F as in Proposition~spec:missing and
+every finite-dimensional V,
[
Psi_T(a_K^*F)=0,
- qquad
+
Psi_T((a,ell_T)_K^*mathrm H_V(F))=0.rmH_V(F))=0.
]
-The same statements hold on the smooth locus of $a$, and after any of
+The same statements hold on the smooth locus of a, and after any of
the finite trait replacements described above.
Proof.
-The first equality follows from $Psi_{mathcal B}calB(F)=0$ and smooth base
-change along $a$. For an exterior generator in
+The first equality follows from Psi_{mathcal B}calB(F)=0 and smooth base
+change along a. For an exterior generator in
(eq:spec-exterior-generators), ordinary pullback gives
[
(a,ell_T)_K^*(Dboxtimesmathcal L_K)calL_K)
=a_K^*D ⊗ ell_{T,K}^*mathcal L_K.calL_K.
]
-The sheaf $ell_T^*mathcal L_S$calL_S$ is lisse on $T$, regardless of whether
-$ell_T$ is smooth. Formula~(eq:spec-lisse-tensor) and
-$Psi_{mathcal B}calB(D)=0$ therefore give
+The sheaf ell_T^*mathcal L_ScalL_S is lisse on T, regardless of whether
+ell_T is smooth. Formula~(eq:spec-lisse-tensor) and
+Psi_{mathcal B}calB(D)=0 therefore give
[
Psi_T(a_K^*D ⊗ ell_{T,K}^*mathcal L_K)calL_K)
simeq a_s^*Psi_{mathcal B}calB(D) ⊗ ell_{T,s}^*mathcal L_s=0.calL_s=0.
]
Both pullback and specialisation are continuous, so the same vanishing
holds for the stable colimit closure of these exterior generators. That
-closure contains $mathrm H_V(F)$.rmH_V(F)$. Finite trait replacement preserves
+closure contains mathrm H_V(F).rmH_V(F). Finite trait replacement preserves
geometric specialisation, so no new hypothesis is needed there.
[What is and is not used]
-The proof uses smoothness only of $acolon Ttomathcal B$;calB$; the leg
-map $ell_T$ may be arbitrary. It makes no smooth-base-change assertion
-for the combined map $(a,ell_T)$. This is exactly the freedom needed
+The proof uses smoothness only of acolon Ttomathcal B;calB; the leg
+map ell_T may be arbitrary. It makes no smooth-base-change assertion
+for the combined map (a,ell_T). This is exactly the freedom needed
in Section~sec:twohecke, where we cut the Hecke correspondence
-by a hypersurface in $T$: after the cut, the leg coordinate floats
+by a hypersurface in T: after the cut, the leg coordinate floats
freely over the legacy of the modification, and we do not control
whether the cut is smooth in the leg direction.
@@ -695,43 +695,43 @@ nonzero incoming covector and both transverse derivatives.
% =========================================================================
Section~sec:specialization reduced the main theorem to the following
-local problem. Suppose $Pinmathcal B_s(k)$calB_s(k)$ is a bundle and
-$Ain T^*_Pmathcal B_s$calB_s$ is a non-nilpotent Higgs field: a section
-of $Ad P ⊗ omega_X$ whose values are not all nilpotent. We want to
-find a Hecke modification of $P$ whose moving-point covector is nonzero,
+local problem. Suppose Pinmathcal B_s(k)calB_s(k) is a bundle and
+Ain T^*_Pmathcal B_scalB_s is a non-nilpotent Higgs field: a section
+of Ad P ⊗ omega_X whose values are not all nilpotent. We want to
+find a Hecke modification of P whose moving-point covector is nonzero,
and whose fixed-input / fixed-output cotangent sections have invertible
derivatives. Both properties will come from a central cocharacter of a
Levi subgroup.
-The Lie-theoretic input is the following. Choose a point $xin X(k)$
-where $A$ is non-nilpotent. In a formal coordinate $t$ and bundle frame
-at $x$, write $A=a(t),dt$ with $a(t)inmathfrak g[[t]]$.frakg[[t]]$. Chevalley
-restriction identifies the nilpotent cone in $mathfrak g$frakg$ with
-$chi^{-1}(chi(0))$, where $chicolonmathfrak gfrakgtomathfrak g/frakg/!/G$ is
-the adjoint quotient; so the requirement that $A$ be non-nilpotent at $x$
-becomes $chi(a(0))nechi(0)$.
+The Lie-theoretic input is the following. Choose a point xin X(k)
+where A is non-nilpotent. In a formal coordinate t and bundle frame
+at x, write A=a(t),dt with a(t)inmathfrak g[[t]].frakg[[t]]. Chevalley
+restriction identifies the nilpotent cone in mathfrak gfrakg with
+chi^{-1}(chi(0)), where chicolonmathfrak gfrakgtomathfrak g/frakg/!/G is
+the adjoint quotient; so the requirement that A be non-nilpotent at x
+becomes chi(a(0))nechi(0).
[Lie-theoretic reduction]
-Let $k$ and $G$ satisfy Hypothesis~hyp:common and either
-Hypothesis~hyp:A or Hypothesis~hyp:B, and let $kappa$ be the
+Let k and G satisfy Hypothesis~hyp:common and either
+Hypothesis~hyp:A or Hypothesis~hyp:B, and let kappa be the
invariant symmetric form of Hypothesis~hyp:common(i). Let
-$a(t)inmathfrak g[[t]]$frakg[[t]]$ satisfy $chi(a(0))nechi(0)$. Then after a
-change of frame by an element of $G(k[[t]])$, there are a Levi subgroup
-$Msubset G$ and a cocharacter $lambdacolonmathbb G_mbbG_mto Z(M)$ such
+a(t)inmathfrak g[[t]]frakg[[t]] satisfy chi(a(0))nechi(0). Then after a
+change of frame by an element of G(k[[t]]), there are a Levi subgroup
+Msubset G and a cocharacter lambdacolonmathbb G_mbbG_mto Z(M) such
that
- a(t)inmathfrak m[[t]],frakm[[t]],qquad kappa(a(0),dlambda)ne0,qquad
+ a(t)inmathfrak m[[t]],frakm[[t]], kappa(a(0),dlambda)ne0,
ad_{a(0)}colonmathfrak g/frakg/mathfrak mfrakm
xrightarrow{ sim }mathfrak g/frakg/mathfrak m,frakm,
-where $mathfrak m=frakm=Lie M$ and $dlambda=dlambda(1)inmathfrak g$frakg$ is the
-image of $1inLie(mathbb G_m)=k$.bbG_m)=k$. The subgroup $M$ may equal $G$; the
+where mathfrak m=frakm=Lie M and dlambda=dlambda(1)inmathfrak gfrakg is the
+image of 1inLie(mathbb G_m)=k.bbG_m)=k. The subgroup M may equal G; the
invertibility includes the zero vector space.
The proof has three layers. First, root-pairing and Jordan facts
-(Ssec:lie-basic): $kappa$ pairs root vectors with their opposite
-partners, $mathfrak t$frakt$ is nondegenerate, and every element is conjugate
+(Ssec:lie-basic): kappa pairs root vectors with their opposite
+partners, mathfrak tfrakt is nondegenerate, and every element is conjugate
into the Lie algebra of a Borel subgroup. Second, one of two
regime-specific arguments produces a central cocharacter
(Ssec:lie-A and~Ssec:lie-B). Third, a formal smoothing
@@ -741,321 +741,321 @@ argument lifts the constant-term reduction to the full formal arc
## Root pairings and constant terms
-Fix a maximal torus $Tsubset G$. Set
-$Λ=X_*(T)$, $Λ^vee=X^*(T)$, $mathfrak t=frakt=Lie T=Λ ⊗ k$.
-For a root $alpha$ write $dalphacolonmathfrak tfraktto k$ for the
-differential of the character $alpha$ and $dalpha^veeinmathfrak t$frakt$
+Fix a maximal torus Tsubset G. Set
+Λ=X_*(T), Λ^vee=X^*(T), mathfrak t=frakt=Lie T=Λ ⊗ k.
+For a root alpha write dalphacolonmathfrak tfraktto k for the
+differential of the character alpha and dalpha^veeinmathfrak tfrakt
for the differential of the coroot.
-The form $kappa$ is nondegenerate on $mathfrak t$.frakt$. For each root $alpha$
-there is a scalar $c_alphain k^times$ such that
+The form kappa is nondegenerate on mathfrak t.frakt. For each root alpha
+there is a scalar c_alphain k^times such that
- kappa(dalpha^vee,h)=c_alpha,dalpha(h)qquad(hinmathfrak t).frakt).
+ kappa(dalpha^vee,h)=c_alpha,dalpha(h)(hinmathfrak t).frakt).
-In particular neither $dalpha$ nor $dalpha^vee$ vanishes. Every element
-of $mathfrak g$frakg$ is conjugate into the Lie algebra of a Borel subgroup
-containing $T$, and every semisimple element is conjugate into
-$mathfrak t$.frakt$.
+In particular neither dalpha nor dalpha^vee vanishes. Every element
+of mathfrak gfrakg is conjugate into the Lie algebra of a Borel subgroup
+containing T, and every semisimple element is conjugate into
+mathfrak t.frakt.
Proof.
-Under the $T$-weight decomposition
-$mathfrak g=frakg=mathfrak tfrakt ⊕ ⊕_alphamathfrak g_frakg_alpha$,
-invariance of $kappa$ makes $mathfrak t$frakt$ orthogonal to each root space
-and makes $mathfrak g_frakg_alpha$ pair only with $mathfrak g_frakg_{-alpha}$.
-Nondegeneracy of $kappa$ on $mathfrak g$frakg$ then gives nondegeneracy on
-$mathfrak t$frakt$ and on each root pairing. Choose compatible root vectors
-$e_alpha,e_{-alpha}$ with $[e_alpha,e_{-alpha}]=dalpha^vee$; this
-follows from the $SL_2$-root homomorphism and holds for every
+Under the T-weight decomposition
+mathfrak g=frakg=mathfrak tfrakt ⊕ ⊕_alphamathfrak g_frakg_alpha,
+invariance of kappa makes mathfrak tfrakt orthogonal to each root space
+and makes mathfrak g_frakg_alpha pair only with mathfrak g_frakg_{-alpha}.
+Nondegeneracy of kappa on mathfrak gfrakg then gives nondegeneracy on
+mathfrak tfrakt and on each root pairing. Choose compatible root vectors
+e_alpha,e_{-alpha} with [e_alpha,e_{-alpha}]=dalpha^vee; this
+follows from the SL_2-root homomorphism and holds for every
isogeny type cite[SS8.1.1--8.1.4]{Springer}, cite[Cor.~5.1.12, Eq.~(5.1.4)]{Conrad}.
-Invariance of $kappa$ gives (eq:root-coroot-pairing) with
-$c_alpha=kappa(e_alpha,e_{-alpha})ne0$. Hence one of
-$dalpha,dalpha^vee$ vanishes iff the other does. Simultaneous
-vanishing would put $alphain pΛ^vee$ and $alpha^veein pΛ$,
-so $<alpha,alpha^vee>=2$ would be divisible by $p^2$,
+Invariance of kappa gives (eq:root-coroot-pairing) with
+c_alpha=kappa(e_alpha,e_{-alpha})ne0. Hence one of
+dalpha,dalpha^vee vanishes iff the other does. Simultaneous
+vanishing would put alphain pΛ^vee and alpha^veein pΛ,
+so <alpha,alpha^vee>=2 would be divisible by p^2,
impossible. This argument uses characters, not just their differentials,
and works in characteristic two.
For the Borel-conjugation statement: consider the incidence morphism
[
- Gtimes^Bmathfrak b→frakb→mathfrak g,frakg,qquad
+ Gtimes^Bmathfrak b→frakb→mathfrak g,frakg,
[g,x]mapstoAd(g)x,
]
-where $Bsupset T$ is a Borel subgroup and $mathfrak b=frakb=Lie B$. Its
-source is closed in $(G/B)timesmathfrak g$,frakg$, so the morphism is proper.
-Choose $cinmathfrak t$frakt$ outside the root differential hyperplanes.
-At $[1,c]$, the differential of the incidence morphism is onto (the
+where Bsupset T is a Borel subgroup and mathfrak b=frakb=Lie B. Its
+source is closed in (G/B)timesmathfrak g,frakg, so the morphism is proper.
+Choose cinmathfrak tfrakt outside the root differential hyperplanes.
+At [1,c], the differential of the incidence morphism is onto (the
negative-root tangent directions map isomorphically to the negative root
-spaces, and $mathfrak b$frakb$ supplies the rest), so the morphism is dominant,
+spaces, and mathfrak bfrakb supplies the rest), so the morphism is dominant,
hence surjective.
For the semisimple-conjugation statement: write a semisimple element of
-$mathfrak b$frakb$ as $c+u$ with $cinmathfrak t$frakt$ and $u$ in the positive
+mathfrak bfrakb as c+u with cinmathfrak tfrakt and u in the positive
root Lie algebra. Conjugate by positive-root groups in increasing height.
-If $dalpha(c)ne0$, conjugation by the $alpha$-root group changes the
-$alpha$ coefficient by $-z,dalpha(c)$ and only touch higher-height
+If dalpha(c)ne0, conjugation by the alpha-root group changes the
+alpha coefficient by -z,dalpha(c) and only touch higher-height
coefficients; it therefore kills that coefficient. After all steps we
-are at $c+u_0$ with $[c,u_0]=0$. In a faithful representation $c$ is
-semisimple, $u_0$ is nilpotent, and their sum is semisimple, so the
-$(semisimple+nilpotent)$ uniqueness forces $u_0=0$.
+are at c+u_0 with [c,u_0]=0. In a faithful representation c is
+semisimple, u_0 is nilpotent, and their sum is semisimple, so the
+(semisimple+nilpotent) uniqueness forces u_0=0.
-Choose a closed faithful representation $G↪GL(V)$.
-Its differential realises $mathfrak g$frakg$ as a restricted Lie subalgebra of
-$End(V)$, with $xmapsto x^{[p]}$ equal to matrix $p$th power. We use
+Choose a closed faithful representation G↪GL(V).
+Its differential realises mathfrak gfrakg as a restricted Lie subalgebra of
+End(V), with xmapsto x^{[p]} equal to matrix pth power. We use
the ordinary matrix Jordan decomposition; the two lemmas below prove
-its two parts lie in $mathfrak g$frakg$ in each regime; compare
+its two parts lie in mathfrak gfrakg in each regime; compare
cite[S4.4, Thm.~4.4.20]{Springer}.
-Suppose $a_0=s+ninmathfrak g$frakg$ where $s,ninmathfrak g$,frakg$, $s$ is
-semisimple, $n$ is nilpotent in a faithful representation, and
-$[s,n]=0$. Suppose also $chi(a_0)nechi(0)$. After conjugation, $s$ is
-a nonzero element of $mathfrak t$,frakt$, $M=Z_G(s)$ is a Levi subgroup,
-$a_0inmathfrak m$,frakm$, and $ad_{a_0}$ is invertible on
-$mathfrak g/frakg/mathfrak m$.frakm$. Moreover for every $lambdacolonmathbb G_mbbG_m
-to Z(M)$,
+Suppose a_0=s+ninmathfrak gfrakg where s,ninmathfrak g,frakg, s is
+semisimple, n is nilpotent in a faithful representation, and
+[s,n]=0. Suppose also chi(a_0)nechi(0). After conjugation, s is
+a nonzero element of mathfrak t,frakt, M=Z_G(s) is a Levi subgroup,
+a_0inmathfrak m,frakm, and ad_{a_0} is invertible on
+mathfrak g/frakg/mathfrak m.frakm. Moreover for every lambdacolonmathbb G_mbbG_m
+to Z(M),
kappa(a_0,dlambda)=kappa(s,dlambda).
Proof.
-The root-pairing lemma conjugates $s$ into $mathfrak t$.frakt$. If $s=0$ then
-$n=a_0$ is nilpotent. The projection $mathfrak bfrakbtomathfrak t$frakt$ preserves
-restricted $p$-operations; since $n$ is killed by some iterated
-$p$-operation and that operation is injective on $mathfrak t$,frakt$, its
+The root-pairing lemma conjugates s into mathfrak t.frakt. If s=0 then
+n=a_0 is nilpotent. The projection mathfrak bfrakbtomathfrak tfrakt preserves
+restricted p-operations; since n is killed by some iterated
+p-operation and that operation is injective on mathfrak t,frakt, its
toral projection vanishes. A strictly dominant cocharacter contracts
the positive-root Lie algebra to zero. Therefore every invariant
-polynomial has the same value on $n$ as on $0$, contradiction with
-$chi(a_0)nechi(0)$. Thus $sne0$.
-
-By Hypothesis~hyp:common(iii), $M=Z_G(s)$ is a Levi subgroup. Its
-Lie algebra is $kerad_s$, which contains $a_0$. On
-$mathfrak g/frakg/mathfrak m$frakm$ the operator $ad_s$ is diagonalisable with
-nonzero eigenvalues, while $ad_n$ is nilpotent (indeed
-$ad_n^{p^e}=ad_{n^{[p^e]}}=0$ for large $e$) and commutes with
-$ad_s$. Therefore on each $ad_s$-eigenspace the sum
-$ad_{a_0}=ad_s+ad_n$ is invertible.
-
-For (eq:nilpotent-orthogonality): conjugate $n$ inside $M$ into
-the Lie algebra of a Borel of $M$ containing $T$; the incidence
-argument of the previous lemma applies to $M$ too, and its roots have
-nonzero differentials and $kappa$ is nondegenerate on $mathfrak m$frakm$ by
+polynomial has the same value on n as on 0, contradiction with
+chi(a_0)nechi(0). Thus sne0.
+
+By Hypothesis~hyp:common(iii), M=Z_G(s) is a Levi subgroup. Its
+Lie algebra is kerad_s, which contains a_0. On
+mathfrak g/frakg/mathfrak mfrakm the operator ad_s is diagonalisable with
+nonzero eigenvalues, while ad_n is nilpotent (indeed
+ad_n^{p^e}=ad_{n^{[p^e]}}=0 for large e) and commutes with
+ad_s. Therefore on each ad_s-eigenspace the sum
+ad_{a_0}=ad_s+ad_n is invertible.
+
+For (eq:nilpotent-orthogonality): conjugate n inside M into
+the Lie algebra of a Borel of M containing T; the incidence
+argument of the previous lemma applies to M too, and its roots have
+nonzero differentials and kappa is nondegenerate on mathfrak mfrakm by
the same weight decomposition. Restricted nilpotence again kills the
-toral projection, so the conjugated $n$ lies in the positive-root Lie
-algebra of $M$. That Lie algebra is orthogonal to $mathfrak t$,frakt$, so
-$kappa(n,dlambda)=0$ for $dlambdainmathfrak t$frakt$ fixed by $M$; by
+toral projection, so the conjugated n lies in the positive-root Lie
+algebra of M. That Lie algebra is orthogonal to mathfrak t,frakt, so
+kappa(n,dlambda)=0 for dlambdainmathfrak tfrakt fixed by M; by
invariance the same holds before conjugation.
The remaining work is choosing the central cocharacter with
-nonvanishing pairing against $s$. This is where the two regimes
+nonvanishing pairing against s. This is where the two regimes
diverge; everything else is common.
## Regime A: a lattice argument for actual central cocharacters
-Assume Hypothesis~hyp:A, so $k=mathbb Fbar_q$.bbFbar_q$. Two facts
-are needed. First, the Jordan parts $s,n$ of $a_0$ constructed in
-Lemma~lie:constant-parts actually lie in $mathfrak g$;frakg$; they start
-life only in an ambient $End(V)$. Second, some $lambdain
-X_*(Z(M))$ has $kappa(s,dlambda)ne0$. Under Hypothesis~hyp:A,
+Assume Hypothesis~hyp:A, so k=mathbb Fbar_q.bbFbar_q. Two facts
+are needed. First, the Jordan parts s,n of a_0 constructed in
+Lemma~lie:constant-parts actually lie in mathfrak g;frakg; they start
+life only in an ambient End(V). Second, some lambdain
+X_*(Z(M)) has kappa(s,dlambda)ne0. Under Hypothesis~hyp:A,
both are integral.
-The Jordan parts first. In a faithful representation write $a_0=s+n$ for
-the matrix Jordan decomposition. The eigenvalues of $s$ lie in a finite
-field $mathbb F_bbF_{p^h}$; choose $e$ divisible by $h$ and large enough
-to have $n^{p^e}=0$. Then $a_0^{[p^e]}=a_0^{p^e}=s^{p^e}+n^{p^e}=s^{p^e}$
-so $s=a_0^{[p^e]}$, hence $sinmathfrak g$frakg$ as the $p^e$th restricted
-power of an element of $mathfrak g$.frakg$. Consequently $ninmathfrak g$frakg$ too.
-Lemma~lie:constant-parts then supplies a nonzero $sinmathfrak t$,frakt$,
-$M=Z_G(s)$ and the quotient invertibility.
+The Jordan parts first. In a faithful representation write a_0=s+n for
+the matrix Jordan decomposition. The eigenvalues of s lie in a finite
+field mathbb F_bbF_{p^h}; choose e divisible by h and large enough
+to have n^{p^e}=0. Then a_0^{[p^e]}=a_0^{p^e}=s^{p^e}+n^{p^e}=s^{p^e}
+so s=a_0^{[p^e]}, hence sinmathfrak gfrakg as the p^eth restricted
+power of an element of mathfrak g.frakg. Consequently ninmathfrak gfrakg too.
+Lemma~lie:constant-parts then supplies a nonzero sinmathfrak t,frakt,
+M=Z_G(s) and the quotient invertibility.
For the cocharacter, we use the following lattice statement. It uses
only Hypothesis~hyp:common(i) and~(iii); no Weyl-order assumption.
-Assume $kappa$ and Hypothesis~hyp:common(i),~(iii). Then every Levi
-Cartan determinant is a $p$-adic unit. Moreover, for every Levi
-$Msupset T$,
+Assume kappa and Hypothesis~hyp:common(i),~(iii). Then every Levi
+Cartan determinant is a p-adic unit. Moreover, for every Levi
+Msupset T,
span_k{dlambda: lambdain X_*(Z(M))}
=mathfrak z(frakz(mathfrak m),frakm),
-where the $dlambdainmathfrak t$frakt$ are differentials of actual
-central cocharacters of $Z(M)$.
+where the dlambdainmathfrak tfrakt are differentials of actual
+central cocharacters of Z(M).
Proof.
-Fix a positive system and let $I={alpha_1,...,alpha_r}$ be the
-simple roots of $M$. Set
-$J_I=span_k{dalpha_i^vee}subsetmathfrak t$.frakt$. A
-central element of $mathfrak m$frakm$ has no root-space component because all
+Fix a positive system and let I={alpha_1,...,alpha_r} be the
+simple roots of M. Set
+J_I=span_k{dalpha_i^vee}subsetmathfrak t.frakt. A
+central element of mathfrak mfrakm has no root-space component because all
root differentials are nonzero (Lemma~lie:root-pairing), so
(eq:root-coroot-pairing) gives
[
mathfrak z(frakz(mathfrak m)=∩_ifrakm)=∩_iker dalpha_i=J_I^perp.
]
The form on this center is nondegenerate by hypothesis, and on
-$mathfrak t$frakt$ by Lemma~lie:root-pairing; nondegeneracy on $J_I$
+mathfrak tfrakt by Lemma~lie:root-pairing; nondegeneracy on J_I
follows.
-Define integral maps $UcolonΛtomathbb Z^r$,bbZ^r$,
-$ymapsto(<alpha_i,y>)_i$, and $Vcolonmathbb Z^rbbZ^rto
-Λ$, $e_jmapstoalpha_j^vee$. Their composite is the Cartan matrix
-$C_I=UV$. Reduction to $k$ is denoted by a subscript; by
-(eq:root-coroot-pairing) we have $V_k^*kappa=D U_k$ for $D$ a
-diagonal invertible matrix. The radical of $V_k^*kappa V_k$ is exactly
-$ker V_k$, again by nondegeneracy on $J_I=im V_k$. Hence
+Define integral maps UcolonΛtomathbb Z^r,bbZ^r,
+ymapsto(<alpha_i,y>)_i, and Vcolonmathbb Z^rbbZ^rto
+Λ, e_jmapstoalpha_j^vee. Their composite is the Cartan matrix
+C_I=UV. Reduction to k is denoted by a subscript; by
+(eq:root-coroot-pairing) we have V_k^*kappa=D U_k for D a
+diagonal invertible matrix. The radical of V_k^*kappa V_k is exactly
+ker V_k, again by nondegeneracy on J_I=im V_k. Hence
rank(C_{I,k})=rank(V_k)
=rank(U_k).
-If some Levi Cartan determinant were divisible by $p$, the standard
+If some Levi Cartan determinant were divisible by p, the standard
determinant list
cite[Thm.~C.56,~S24(c)]{Milne}
[
{c|ccccccccc}
- type&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2 hline
+ type&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2
det&r+1&2&2&4&3&2&1&1&1
]
-would give a further Levi of type $A_{p-1}$: in $A_r$ take a consecutive
-string of $p-1$ vertices; in determinant-$2$ and determinant-$4$ cases
-take one vertex (forcing $p=2$); in $E_6$ take two adjacent vertices
-(forcing $p=3$). This new Levi is a simple-root subdiagram of $G$, so
+would give a further Levi of type A_{p-1}: in A_r take a consecutive
+string of p-1 vertices; in determinant-2 and determinant-4 cases
+take one vertex (forcing p=2); in E_6 take two adjacent vertices
+(forcing p=3). This new Levi is a simple-root subdiagram of G, so
(eq:cartan-ranks) applies to it.
-For this $A_{p-1}$ Levi, $r=p-1$ and $det C_I=p$. Choose dual bases of
-the full lattices (rank $N_T$); the matrices of $U,V$ are $rtimes N_T$
-and $N_Ttimes r$. By (eq:cartan-ranks) both reductions have rank
-$<r$, so every maximal minor of either matrix is divisible by $p$.
+For this A_{p-1} Levi, r=p-1 and det C_I=p. Choose dual bases of
+the full lattices (rank N_T); the matrices of U,V are rtimes N_T
+and N_Ttimes r. By (eq:cartan-ranks) both reductions have rank
+<r, so every maximal minor of either matrix is divisible by p.
Cauchy--Binet gives
[
det C_I
=sum_{substack{Jsubset{1,...,N_T}|J|=r}}
det(U_{:,J}),det(V_{J,:}),
]
-each summand divisible by $p^2$, contradicting $det C_I=p$. This uses
+each summand divisible by p^2, contradicting det C_I=p. This uses
the full lattices, with no isogeny-type assumption.
-For an arbitrary Levi set $d_I=det C_I$ and define
-$Pi_I=d_Iid_Λ-Vadj(C_I)UinEnd(Λ)$. Then
-$UPi_I=0$; the lattice $ker U$ is exactly the set of integral
-cocharacters into $Z(M)$. For $zinker U_k$, $Pi_{I,k}(z)=d_I z$; since
-$d_I$ is invertible in $k$, the differentials of $ker U$ span
-$ker U_k=mathfrak z(frakz(mathfrak m)$,frakm)$, giving
-(eq:central-span). For $r=0$ take $d_I=1$ and $Pi_I=id$; the
+For an arbitrary Levi set d_I=det C_I and define
+Pi_I=d_Iid_Λ-Vadj(C_I)UinEnd(Λ). Then
+UPi_I=0; the lattice ker U is exactly the set of integral
+cocharacters into Z(M). For zinker U_k, Pi_{I,k}(z)=d_I z; since
+d_I is invertible in k, the differentials of ker U span
+ker U_k=mathfrak z(frakz(mathfrak m),frakm), giving
+(eq:central-span). For r=0 take d_I=1 and Pi_I=id; the
same proof covers tori.
Now return to the regime-A story. Under Hypothesis~hyp:A, the
-Jordan parts $s,ninmathfrak g$frakg$ have been constructed. Lemma~lie:constant-parts
-gives $sinmathfrak z(frakz(mathfrak m)∖frakm)∖{0}$ and the quotient
-invertibility. Since $kappa$ is nondegenerate on $mathfrak z(frakz(mathfrak
-m)$frakm)$ by Hypothesis~hyp:common(i),
-Lemma~lie:lattice produces an actual $lambdain X_*(Z(M))$
-with $kappa(s,dlambda)ne0$. Orthogonality
+Jordan parts s,ninmathfrak gfrakg have been constructed. Lemma~lie:constant-parts
+gives sinmathfrak z(frakz(mathfrak m)∖frakm)∖{0} and the quotient
+invertibility. Since kappa is nondegenerate on mathfrak z(frakz(mathfrak
+m)frakm) by Hypothesis~hyp:common(i),
+Lemma~lie:lattice produces an actual lambdain X_*(Z(M))
+with kappa(s,dlambda)ne0. Orthogonality
(eq:nilpotent-orthogonality) gives the required
-$kappa(a_0,dlambda)ne0$. This used no Weyl-order assumption.
+kappa(a_0,dlambda)ne0. This used no Weyl-order assumption.
## Regime B: Weyl-order averaging
-Now assume Hypothesis~hyp:B. Eigenvalues of a matrix over $k$ need
+Now assume Hypothesis~hyp:B. Eigenvalues of a matrix over k need
not be fixed by any positive power of Frobenius, so the restricted-power
trick of regime A does not apply. Instead we interpolate the inverse of
-Frobenius using an $mathbb F_p$-linearbbF_p$-linear additive polynomial.
-
-In the faithful representation write $a_0=s+n$ and choose $e$ with
-$n^{p^e}=0$, so $a_0^{p^e}=s^{p^e}$. Let $V_0subset k$ be the
-finite-dimensional $mathbb F_p$-spanbbF_p$-span of the eigenvalues of $s$, and let
-$V_e={v^{p^e}:vin V_0}$. The inverse of Frobenius $V_eto V_0$ is
-$mathbb F_p$-linear.bbF_p$-linear. If $V_0=0$ then $s=0$ and $a_0=ninmathfrak g$,frakg$,
-contradicting $chi(a_0)nechi(0)$
-(Lemma~lie:constant-parts). Hence $r=dim_{mathbb F_p}bbF_pV_ege1$ and
-we may assume $V_0ne0$.
-
-Take a basis $beta_1,...,beta_r$ of $V_e$ over $mathbb F_p$.bbF_p$. The
-Moore matrix $(beta_i^{p^j})_{1le ile r, 0le j<r}$ is invertible:
-if not, some nonzero polynomial $sum_{j=0}^{r-1}c_j z^{p^j}$ of degree
-$le p^{r-1}$ would vanish on all $p^r$ elements of $V_e$, contradicting
-$|V_e|=p^r$. Therefore there is an additive polynomial
-$Q(z)=sum_{j=0}^{r-1}c_j z^{p^j}$ agreeing with inverse-Frobenius on
-$V_e$. Applying it to the diagonalisable matrix $s^{p^e}$ gives
+Frobenius using an mathbb F_p-linearbbF_p-linear additive polynomial.
+
+In the faithful representation write a_0=s+n and choose e with
+n^{p^e}=0, so a_0^{p^e}=s^{p^e}. Let V_0subset k be the
+finite-dimensional mathbb F_p-spanbbF_p-span of the eigenvalues of s, and let
+V_e={v^{p^e}:vin V_0}. The inverse of Frobenius V_eto V_0 is
+mathbb F_p-linear.bbF_p-linear. If V_0=0 then s=0 and a_0=ninmathfrak g,frakg,
+contradicting chi(a_0)nechi(0)
+(Lemma~lie:constant-parts). Hence r=dim_{mathbb F_p}bbF_pV_ege1 and
+we may assume V_0ne0.
+
+Take a basis beta_1,...,beta_r of V_e over mathbb F_p.bbF_p. The
+Moore matrix (beta_i^{p^j})_{1le ile r, 0le j<r} is invertible:
+if not, some nonzero polynomial sum_{j=0}^{r-1}c_j z^{p^j} of degree
+le p^{r-1} would vanish on all p^r elements of V_e, contradicting
+|V_e|=p^r. Therefore there is an additive polynomial
+Q(z)=sum_{j=0}^{r-1}c_j z^{p^j} agreeing with inverse-Frobenius on
+V_e. Applying it to the diagonalisable matrix s^{p^e} gives
[
s=Q(a_0^{p^e})=sum_{j=0}^{r-1}c_j a_0^{[p^{e+j}]}inmathfrak g.frakg.
]
-Hence $ninmathfrak g$frakg$ as well. Lemma~lie:constant-parts now
-supplies $sinmathfrak t∖frakt∖{0}$, $M=Z_G(s)$ and the quotient
+Hence ninmathfrak gfrakg as well. Lemma~lie:constant-parts now
+supplies sinmathfrak t∖frakt∖{0}, M=Z_G(s) and the quotient
invertibility.
-To find $lambda$: let $W_M=N_M(T)/T$ be the Weyl group of $M$ and set
-$h_M=|W_M|$. Since $W_Msubset W_G$, Hypothesis~hyp:B implies
-$h_M$ is invertible in $k$. The averaging operator
+To find lambda: let W_M=N_M(T)/T be the Weyl group of M and set
+h_M=|W_M|. Since W_Msubset W_G, Hypothesis~hyp:B implies
+h_M is invertible in k. The averaging operator
[
e_M=frac1{h_M}sum_{win W_M}wcolonmathfrak tfrakttomathfrak tfrakt
]
-is a self-adjoint projection for $kappa$, so $kappa$ is nondegenerate
-on $mathfrak t^frakt^{W_M}=im e_M$. That subspace contains $s$ because $M$
-centralises $s$. In fact $mathfrak t^frakt^{W_M}=mathfrak z(frakz(mathfrak m)$:frakm)$:
-the reflection formula is $s_alpha(h)=h-dalpha(h)dalpha^vee$
-and $dalpha^veene0$ by Lemma~lie:root-pairing. Thus nondegeneracy
-on $mathfrak z(frakz(mathfrak m)$frakm)$ again follows from $kappa$ alone, without
+is a self-adjoint projection for kappa, so kappa is nondegenerate
+on mathfrak t^frakt^{W_M}=im e_M. That subspace contains s because M
+centralises s. In fact mathfrak t^frakt^{W_M}=mathfrak z(frakz(mathfrak m):frakm):
+the reflection formula is s_alpha(h)=h-dalpha(h)dalpha^vee
+and dalpha^veene0 by Lemma~lie:root-pairing. Thus nondegeneracy
+on mathfrak z(frakz(mathfrak m)frakm) again follows from kappa alone, without
needing Hypothesis~hyp:common(i)'s second clause.
-For an integral cocharacter, choose a basis $mu_1,...,mu_{N_T}$ of
-$Λ$ and set $nu_i=sum_{win W_M}wmu_iinΛ^{W_M}$. Their
-differentials span $mathfrak t^frakt^{W_M}$ because $dnu_i=h_M e_M(dmu_i)$.
-The integral reflection formula $s_alpha(lambda)=lambda-
-<alpha,lambda>alpha^vee$ and torsion-freeness of
-$Λ$ show that $Λ^{W_M}=X_*(Z(M))$: a $W_M$-invariant
-cocharacter has zero pairing with every root of $M$, and conversely. No
-division of a cocharacter by $h_M$ is made. Nondegeneracy of $kappa$ on
-$mathfrak t^frakt^{W_M}$ now gives an integral $lambdain X_*(Z(M))$ with
-$kappa(s,dlambda)ne0$, and (eq:nilpotent-orthogonality) gives
-$kappa(a_0,dlambda)ne0$.
-
-This argument applies to any algebraically closed $k$. Its use of
-$pnmid|W_G|$ is specific to Hypothesis~hyp:B.
+For an integral cocharacter, choose a basis mu_1,...,mu_{N_T} of
+Λ and set nu_i=sum_{win W_M}wmu_iinΛ^{W_M}. Their
+differentials span mathfrak t^frakt^{W_M} because dnu_i=h_M e_M(dmu_i).
+The integral reflection formula s_alpha(lambda)=lambda-
+<alpha,lambda>alpha^vee and torsion-freeness of
+Λ show that Λ^{W_M}=X_*(Z(M)): a W_M-invariant
+cocharacter has zero pairing with every root of M, and conversely. No
+division of a cocharacter by h_M is made. Nondegeneracy of kappa on
+mathfrak t^frakt^{W_M} now gives an integral lambdain X_*(Z(M)) with
+kappa(s,dlambda)ne0, and (eq:nilpotent-orthogonality) gives
+kappa(a_0,dlambda)ne0.
+
+This argument applies to any algebraically closed k. Its use of
+pnmid|W_G| is specific to Hypothesis~hyp:B.
## Formal smoothing: from constant term to whole arc
-Both regimes have now produced the same constant-term data: a Levi $M$,
-a central cocharacter $lambda$, and $a_0=a(0)inmathfrak m$frakm$ with
-$kappa(a_0,dlambda)ne0$ and $ad_{a_0}$ invertible on
-$mathfrak g/frakg/mathfrak m$.frakm$. The remaining step moves the entire formal
-arc $a(t)$ into $mathfrak m[[t]]$.frakm[[t]]$.
+Both regimes have now produced the same constant-term data: a Levi M,
+a central cocharacter lambda, and a_0=a(0)inmathfrak mfrakm with
+kappa(a_0,dlambda)ne0 and ad_{a_0} invertible on
+mathfrak g/frakg/mathfrak m.frakm. The remaining step moves the entire formal
+arc a(t) into mathfrak m[[t]].frakm[[t]].
-Let $G$ be a connected reductive group over an algebraically closed
-field $k$, $Msubset G$ a Levi subgroup, $a(t)inmathfrak g[[t]]$frakg[[t]]$ with
-$a_0=a(0)inmathfrak m$.frakm$. If $ad_{a_0}$ is invertible on
-$mathfrak g/frakg/mathfrak m$,frakm$, then there is $g(t)in G(k[[t]])$ with
-$g(0)=1$ and $Ad(g(t))a(t)inmathfrak m[[t]]$.frakm[[t]]$.
+Let G be a connected reductive group over an algebraically closed
+field k, Msubset G a Levi subgroup, a(t)inmathfrak g[[t]]frakg[[t]] with
+a_0=a(0)inmathfrak m.frakm. If ad_{a_0} is invertible on
+mathfrak g/frakg/mathfrak m,frakm, then there is g(t)in G(k[[t]]) with
+g(0)=1 and Ad(g(t))a(t)inmathfrak m[[t]].frakm[[t]].
Proof.
Proceed by successive approximation. Suppose the arc already lies in
-$mathfrak m[[t]]$frakm[[t]]$ modulo $t^r$, $rge1$, and let $c_rin
-mathfrak g/frakg/mathfrak m$frakm$ be the class of its $t^r$-coefficient. Choose
-$Dinmathfrak g$frakg$ with $c_r+[D,a_0]=0$ in the quotient (possible by
-invertibility). Smoothness of $G$ gives $g_r(t)in G(k[[t]])$ with
-$g_r(t)equiv1+t^rDpmod{t^{r+1}}$; formally, this is a lift of a point
-in the tangent kernel of the square-zero ideal $(t^r)/(t^{r+1})$ to
-$G(k[t]/(t^{r+1}))$, obtained by iterating formal smoothness at $G$. Then
-conjugation by $g_r(t)$ changes the $t^r$-coefficient by $[D,a_0]$ and
+mathfrak m[[t]]frakm[[t]] modulo t^r, rge1, and let c_rin
+mathfrak g/frakg/mathfrak mfrakm be the class of its t^r-coefficient. Choose
+Dinmathfrak gfrakg with c_r+[D,a_0]=0 in the quotient (possible by
+invertibility). Smoothness of G gives g_r(t)in G(k[[t]]) with
+g_r(t)equiv1+t^rDpmod{t^{r+1}}; formally, this is a lift of a point
+in the tangent kernel of the square-zero ideal (t^r)/(t^{r+1}) to
+G(k[t]/(t^{r+1})), obtained by iterating formal smoothness at G. Then
+conjugation by g_r(t) changes the t^r-coefficient by [D,a_0] and
leaves all lower coefficients fixed, so it places the arc in
-$mathfrak m[[t]]$frakm[[t]]$ modulo $t^{r+1}$.
+mathfrak m[[t]]frakm[[t]] modulo t^{r+1}.
-Iterating, the products $prod g_r(t)$ converge in the $t$-adic
-topology because $g_requiv1pmod{t^r}$. The affine group $G$ then
-produces a single $g(t)in G(k[[t]])$ with $g(0)=1$ and
-$Ad(g(t))a(t)inmathfrak m[[t]]$.frakm[[t]]$. No exponential series is used.
+Iterating, the products prod g_r(t) converge in the t-adic
+topology because g_requiv1pmod{t^r}. The affine group G then
+produces a single g(t)in G(k[[t]]) with g(0)=1 and
+Ad(g(t))a(t)inmathfrak m[[t]].frakm[[t]]. No exponential series is used.
Proof. [Proof of Proposition~lie:input]
@@ -1063,18 +1063,18 @@ Under Hypothesis~hyp:A, use the restricted-power construction of
Jordan parts together with Lemma~lie:lattice. Under
Hypothesis~hyp:B, use the Moore-matrix construction together with
integral Weyl averaging. In both cases Lemma~lie:constant-parts
-gives the Levi $M$, the quotient invertibility and
-$kappa(a_0,dlambda)ne0$. Apply Lemma~lie:formal after the
-initial constant conjugation; the formal change of frame fixes $a_0$ by
+gives the Levi M, the quotient invertibility and
+kappa(a_0,dlambda)ne0. Apply Lemma~lie:formal after the
+initial constant conjugation; the formal change of frame fixes a_0 by
construction, so all of (eq:lie-output) holds.
-Centrality of $lambda$ in $M$ gives $Ad(lambda(t))a(t)=a(t)$, and
-every nonzero $lambda$-weight space of $mathfrak g$frakg$ injects into
-$mathfrak g/frakg/mathfrak m$,frakm$, so the adjoint action is invertible on the
+Centrality of lambda in M gives Ad(lambda(t))a(t)=a(t), and
+every nonzero lambda-weight space of mathfrak gfrakg injects into
+mathfrak g/frakg/mathfrak m,frakm, so the adjoint action is invertible on the
truncated nonzero-weight modules used below. We do not need
-$Z_G(lambda)=M$. If $lambda$ is central in $G$ the Hecke orbit has
+Z_G(lambda)=M. If lambda is central in G the Hecke orbit has
dimension zero, but the nonzero pairing in (eq:lie-output) still
supplies the moving-point covector. Tori are therefore included in every
allowed characteristic.
@@ -1086,84 +1086,84 @@ allowed characteristic.
% =========================================================================
Section~sec:lie produced Levi data. Now we turn it into a
-modification of a bundle. Let $Pinmathcal B_s(k)$calB_s(k)$ and let
-$Ain T^*_Pmathcal B_s$calB_s$ be non-nilpotent. We will choose a point
-$xin X(k)$, a cocharacter $lambda$, and a modification
-$(P,P',x)in H^lambda_s(k)$ such that:
+modification of a bundle. Let Pinmathcal B_s(k)calB_s(k) and let
+Ain T^*_Pmathcal B_scalB_s be non-nilpotent. We will choose a point
+xin X(k), a cocharacter lambda, and a modification
+(P,P',x)in H^lambda_s(k) such that:
[label=(roman*)]
-item the moving-point displacement $(dlambda/t)|_x$ pairs nonzero
- against $A$ to give a leg covector $xine0$;
-item the fixed-input cotangent section $S_A$ and the fixed-output one
- $S_{A'}$ both vanish at the modification and both have invertible
+ the moving-point displacement (dlambda/t)|_x pairs nonzero
+ against A to give a leg covector xine0;
+ the fixed-input cotangent section S_A and the fixed-output one
+ S_{A'} both vanish at the modification and both have invertible
derivatives on their respective tangent spaces.
These are the "transversality" and "isolation" conditions.
## Correspondence and kernels
-We keep the trait $S$, the curve $mathscr X/S$scrX/S$ and the bundle stack
-$mathcal B$calB$ from Section~sec:specialization. A one-point Hecke
-modification is a quadruple $(P,P',x,alpha)$ where $xinmathscr X$scrX$
-and $alpha$ identifies $P'$ and $P$ away from the graph of $x$. Write
+We keep the trait S, the curve mathscr X/SscrX/S and the bundle stack
+mathcal BcalB from Section~sec:specialization. A one-point Hecke
+modification is a quadruple (P,P',x,alpha) where xinmathscr XscrX
+and alpha identifies P' and P away from the graph of x. Write
[
- p(P,P',x,alpha)=P,qquad o(P,P',x,alpha)=P',qquad
+ p(P,P',x,alpha)=P, o(P,P',x,alpha)=P',
ell_H(P,P',x,alpha)=x.
]
-Convention: in a formal coordinate $t$ at $x$, modification by
-$lambda$ means the output frame is $lambda(t)$ times the input frame.
-Relative position depends only on the Weyl orbit of $lambda$. Let
-$mu=lambda^+$ be its dominant conjugate, $m=< 2rho,mu>$
-where $2rho$ is the sum of positive roots, $mu^vee=-w_0mu$, and
-$mathcal Z=calZ=mathcal BcalBtimes_Smathscr X$.scrX$. Denote the exact-position
-correspondence by $H=H^lambda$, and set
+Convention: in a formal coordinate t at x, modification by
+lambda means the output frame is lambda(t) times the input frame.
+Relative position depends only on the Weyl orbit of lambda. Let
+mu=lambda^+ be its dominant conjugate, m=< 2rho,mu>
+where 2rho is the sum of positive roots, mu^vee=-w_0mu, and
+mathcal Z=calZ=mathcal BcalBtimes_Smathscr X.scrX. Denote the exact-position
+correspondence by H=H^lambda, and set
[
- widetilde p=(p,ell_H)colon Htomathcal Z,calZ,qquad
+ widetilde p=(p,ell_H)colon Htomathcal Z,calZ,
q=(o,ell_H)colon Htomathcal Z.calZ.
]
-Both $widetilde p$ and $q$ forget some of the modification data.
+Both widetilde p and q forget some of the modification data.
[Hecke models and kernel]
-Both $widetilde p$ and $q$ are representable, separated and smooth of
-relative dimension $m$. The correspondence $H$ is open in a bounded
-correspondence $overlineoverline H$H$ for which both projections to $mathcal Z$calZ$
+Both widetilde p and q are representable, separated and smooth of
+relative dimension m. The correspondence H is open in a bounded
+correspondence overlineoverline HH for which both projections to mathcal ZcalZ
are representable and proper. On the output side the bound on
-$overlineoverline H$H$ has dominant label $mu^vee$.
+overlineoverline HH has dominant label mu^vee.
-Let $V$ be the irreducible representation of $check G$ corresponding
+Let V be the irreducible representation of check G corresponding
to this bound, with the indicated input/output convention. Then the
generic Hecke functor has the form
mathrm H_V(F)=rmH_V(F)=overlineoverline q_q_{K,*}mathcal I_K,calI_K,
-with $mathcal I_K$calI_K$ on $H_K$ equal to $p_K^*F$ tensored with a fixed
+with mathcal I_KcalI_K on H_K equal to p_K^*F tensored with a fixed
invertible constant complex (the perverse intersection-cohomology
-normalisation $E[m]$, plus a Tate line if weight-normalised).
+normalisation E[m], plus a Tate line if weight-normalised).
Proof.
-Consider the split integral affine Grassmannian with loops $G(A((t)))$
-and positive loops $G(A[[t]])$. Its integral Schubert bound is
+Consider the split integral affine Grassmannian with loops G(A((t)))
+and positive loops G(A[[t]]). Its integral Schubert bound is
projective and the positive-loop orbit is smooth, open, fibre-dense of
-relative dimension $m$ cite[SS1.2,~2.1]{RicharzScholbach}. We use these
-models by base change from $mathbb Z$.bbZ$.
+relative dimension m cite[SS1.2,~2.1]{RicharzScholbach}. We use these
+models by base change from mathbb Z.bbZ.
A graph coordinate and a formal-disc frame identify modifications of
-$P$ with the corresponding Grassmannian spaces; these descriptions glue
+P with the corresponding Grassmannian spaces; these descriptions glue
by Beauville--Laszlo descent [cite]. More explicitly,
the graph is an effective Cartier divisor. Gluing applies to the
-coordinate algebra, multiplication and coaction of an affine $G$-torsor
+coordinate algebra, multiplication and coaction of an affine G-torsor
(as it does to modules); the torsor identity is checked on the two
pieces. A disc frame exists locally by lifting a frame on the graph
successively through its infinitesimal neighbourhoods using smoothness
-of $G$. Changes of frame and of coordinate preserve the orbit and
+of G. Changes of frame and of coordinate preserve the orbit and
its bound, so properness, smoothness and dimension all descend to the
input-and-leg projection.
-Swapping the two bundles and inverting $alpha$ is an involution of the
-Hecke correspondence. On double cosets it sends $mu$ to $-mu$, whose
-dominant representative is $mu^vee$, and it carries the bound to the
-oppositely-labelled bound. This gives the assertions for $q$, and
-$< 2rho,mu^vee>=< 2rho,mu>=m$. Separatedness
+Swapping the two bundles and inverting alpha is an involution of the
+Hecke correspondence. On double cosets it sends mu to -mu, whose
+dominant representative is mu^vee, and it carries the bound to the
+oppositely-labelled bound. This gives the assertions for q, and
+< 2rho,mu^vee>=< 2rho,mu>=m. Separatedness
of the open correspondence follows from that of the proper bound. This
involution is an involution of the correspondence itself, not of the
right-quotient affine Grassmannian.
@@ -1176,9 +1176,9 @@ For the last assertion, geometric Satake constructs the intersection
complex of the bound. Its restriction to the smooth open orbit is the
shifted constant sheaf
cite[Prop.~4.1]{Richarz}. The Hecke construction tensors this kernel
-with $p_K^*F$ and pushes forward along the proper output-and-leg
-projection cite[SS4.2.1--4.2.4]{GR}. Relabelling by $mu^vee$ if
-needed matches these projections with the chosen $V$. A fixed overall
+with p_K^*F and pushes forward along the proper output-and-leg
+projection cite[SS4.2.1--4.2.4]{GR}. Relabelling by mu^vee if
+needed matches these projections with the chosen V. A fixed overall
shift or Tate line changes neither the description on the open orbit nor
any vanishing assertion below. Only the open-orbit restriction of the
kernel is identified here; on the boundary the kernel remains its
@@ -1187,41 +1187,41 @@ intersection complex.
## The cotangent identity and the transverse section
-On the special fibre a cotangent vector to $mathcal B_s$calB_s$ at $P$ is an
-element of $H^0$ of the cotangent-complex fibre. Deformation theory and
-Serre duality cite[Tag~0BS4]{Stacks}, together with $kappa$, identify
+On the special fibre a cotangent vector to mathcal B_scalB_s at P is an
+element of H^0 of the cotangent-complex fibre. Deformation theory and
+Serre duality cite[Tag~0BS4]{Stacks}, together with kappa, identify
this space as
[
T_P^*mathcal B_s=H^1(X,calB_s=H^1(X,mathfrak g_P)^*=H^0(X,frakg_P)^*=H^0(X,mathfrak g_Pfrakg_P ⊗ omega_X),
]
-where $mathfrak g_P$frakg_P$ is the adjoint bundle. We call such a vector $A$ a
+where mathfrak g_Pfrakg_P is the adjoint bundle. We call such a vector A a
Higgs field. For a representable smooth map, the cotangent
pullback is injective and its image is the kernel of passage to the
relative cotangent; we use this in the form supplied by the cotangent
triangle.
[Transverse modification]
-Let $Pinmathcal B_s(k)$calB_s(k)$ and let $Ain T^*_Pmathcal B_s$calB_s$ be
-non-nilpotent. There exist a point $xin X(k)$, a cocharacter $lambda$,
-a modification $hin H^lambda_s(k)$ from $P$ to a bundle $P'$, and a
+Let Pinmathcal B_s(k)calB_s(k) and let Ain T^*_Pmathcal B_scalB_s be
+non-nilpotent. There exist a point xin X(k), a cocharacter lambda,
+a modification hin H^lambda_s(k) from P to a bundle P', and a
covector
[
- theta=(A',xi)in T^*_{(P',x)}(mathcal B_scalB_stimes X),qquad xine0,
+ theta=(A',xi)in T^*_{(P',x)}(mathcal B_scalB_stimes X), xine0,
]
-with $A'=A$ away from $x$, satisfying
+with A'=A away from x, satisfying
dp_h^*A=dq_h^*theta.
Moreover, on the fixed-side fibres
[
- H^{in}_{P,x}=widetilde p^{-1}(P,x),qquad
+ H^{in}_{P,x}=widetilde p^{-1}(P,x),
H^{out}_{P',x}=q^{-1}(P',x),
]
with fixed-bundle identifications retained, the following hold. The
-restriction of $p^*A$ to the $q$-relative tangent bundle defines a
-section $S_A$ of $Ω_{H_s/mathcal Z_s,q}calZ_s,q$; the restriction of
-$o^*A'$ to the $widetilde p$-relative tangent bundle defines a section
-$S_{A'}$. Both sections vanish at $h$, and both derivatives
+restriction of p^*A to the q-relative tangent bundle defines a
+section S_A of Ω_{H_s/mathcal Z_s,q}calZ_s,q; the restriction of
+o^*A' to the widetilde p-relative tangent bundle defines a section
+S_{A'}. Both sections vanish at h, and both derivatives
(dS_A)_h&colon T_hH^{in}_{P,x}
→ (T_hH^{out}_{P',x})^*,
@@ -1229,47 +1229,47 @@ $S_{A'}$. Both sections vanish at $h$, and both derivatives
→ (T_hH^{in}_{P,x})^*
are isomorphisms. In particular each section has an isolated reduced
-zero at $h$.
+zero at h.
Proof.
-Choice of $lambda$. Since $A$ is non-nilpotent, choose
-$xin X(k)$ where its value is non-nilpotent. Choose a formal coordinate
-$t$, a bundle frame at $x$, and write $A=a(t),dt$. Chevalley restriction
-identifies the nilpotent fibre with $chi^{-1}(chi(0))$, so
-$chi(a(0))nechi(0)$. Proposition~lie:input then supplies a
-formal change of frame, a Levi $M$, and a cocharacter
-$lambdacolonmathbb G_mbbG_mto Z(M)$ with
-
- a(t)inmathfrak m[[t]],frakm[[t]],quad
- ad_{a(0)} invertible on mathfrak g/frakg/mathfrak m,frakm,quad
+Choice of lambda. Since A is non-nilpotent, choose
+xin X(k) where its value is non-nilpotent. Choose a formal coordinate
+t, a bundle frame at x, and write A=a(t),dt. Chevalley restriction
+identifies the nilpotent fibre with chi^{-1}(chi(0)), so
+chi(a(0))nechi(0). Proposition~lie:input then supplies a
+formal change of frame, a Levi M, and a cocharacter
+lambdacolonmathbb G_mbbG_mto Z(M) with
+
+ a(t)inmathfrak m[[t]],frakm[[t]],
+ ad_{a(0)} invertible on mathfrak g/frakg/mathfrak m,frakm,
kappa(a(0),dlambda)ne0.
-After constant conjugation we regard $lambda$ as an element of the
+After constant conjugation we regard lambda as an element of the
fixed cocharacter lattice, so it labels an integral model of
-Lemma~hecke:models. Modify by $lambda(t)$. Since $lambda$
-centralises $M$, the transported field in the output frame is again
-$a(t),dt$; combined with $A$ away from $x$, this defines the global
-Higgs field $A'$ on $P'$. This central-in-a-Levi modification and the
+Lemma~hecke:models. Modify by lambda(t). Since lambda
+centralises M, the transported field in the output frame is again
+a(t),dt; combined with A away from x, this defines the global
+Higgs field A' on P'. This central-in-a-Levi modification and the
resulting leg covector originate in
cite[SS20.4--20.6]{AGKRRV}. What follows makes both derivative
assertions explicit.
Tangent spaces. In the input frame, put
-$L=mathfrak g[[t]]$,frakg[[t]]$, $L'=Ad(lambda(t))L$, $I=Lcap L'$. Let
-$mathfrak g_j$frakg_j$ be the weight-$j$ space of $lambda$; the integers $j$
-are genuine characters of $mathbb G_m$,bbG_m$, not reduced mod $p$. Then
+L=mathfrak g[[t]],frakg[[t]], L'=Ad(lambda(t))L, I=Lcap L'. Let
+mathfrak g_jfrakg_j be the weight-j space of lambda; the integers j
+are genuine characters of mathbb G_m,bbG_m, not reduced mod p. Then
- L'=⊕_j t^jmathfrak g_j[[t]],frakg_j[[t]],quad
- P_1:=L/I=⊕_{j>0}mathfrak g_jfrakg_j ⊗ k[[t]]/(t^j),quad
+ L'=⊕_j t^jmathfrak g_j[[t]],frakg_j[[t]],
+ P_1:=L/I=⊕_{j>0}mathfrak g_jfrakg_j ⊗ k[[t]]/(t^j),
P_2:=L'/I=⊕_{j<0}mathfrak g_jfrakg_j ⊗ (t^jk[[t]]/k[[t]]).
-An infinitesimal disc automorphism $1+ε D$, $Din L$, acts on
-the output lattice while fixing $(P,x)$, and its stabiliser tangent is
-$I$. Its orbit is smooth of dimension $dim_k(L/I)=m$ by the root
+An infinitesimal disc automorphism 1+ε D, Din L, acts on
+the output lattice while fixing (P,x), and its stabiliser tangent is
+I. Its orbit is smooth of dimension dim_k(L/I)=m by the root
formula, so this computes the full tangent space:
- T_hH^{in}_{P,x}=P_1,qquad
+ T_hH^{in}_{P,x}=P_1,
T_hH^{out}_{P',x}=P_2.
The same argument with output fixed gives the second identity. The
@@ -1278,98 +1278,98 @@ canonical isomorphism, no exponential map needed.
There is a perfect pairing
- betacolon P_1times P_2to k,qquad
+ betacolon P_1times P_2to k,
beta(D,C)=Res_{t=0}kappa(D,C),dt.
-Indeed, $L$ is its own residue-annihilator on $mathfrak g((t))$,frakg((t))$, and
-invariance of $kappa$ gives the same statement for $L'$. So $beta$
-descends to the quotients. Concretely, $kappa$ pairs $mathfrak g_j$frakg_j$
-perfectly with $mathfrak g_frakg_{-j}$ and the coefficient of $t^a$ in the
-first space, for $0le a<j$, pairs with the coefficient of $t^{-a-1}$
-in the second. This is perfect, including when $j$ is divisible by $p$.
+Indeed, L is its own residue-annihilator on mathfrak g((t)),frakg((t)), and
+invariance of kappa gives the same statement for L'. So beta
+descends to the quotients. Concretely, kappa pairs mathfrak g_jfrakg_j
+perfectly with mathfrak g_frakg_{-j} and the coefficient of t^a in the
+first space, for 0le a<j, pairs with the coefficient of t^{-a-1}
+in the second. This is perfect, including when j is divisible by p.
Cotangent identity and moving leg. Take the gluing sign
-convention where an infinitesimal transition $1+ε C$ gives the
-boundary class of the principal part $C$. Serre duality pairs it with
-$A$ by $Reskappa(a,C),dt$. For output-fixed tangent
-$Cin L'$ the residue is zero because $ain L'$ and $L'$ is self-annihilating.
-Therefore $dp_h^*A$ vanishes in the $q$-relative cotangent, so it is
-$dq_h^*theta$ for a unique covector $theta$.
-
-The bundle component of $theta$ is $A'$. Indeed, choose $x'ne x$ and
-vary both bundle transitions only at $x'$, leaving the modification at
-$x$ unchanged. Principal parts at $x'$ surject to
-$H^1(X,mathfrak g_frakg_{P'})$ (Serre vanishing for $Ngg0$), and the
-pairings agree on these directions since $A=A'$ near $x'$. Write
-$theta=(A',xi)$.
-
-Now keep the input fixed and move the leg from $t=0$ to $t=ε$.
-The output transition is $g_ε=lambda(t-ε)$ and to first
-order in $ε$,
-$(delta g)g^{-1}=-dlambda/t$. Evaluating the cotangent identity on
+convention where an infinitesimal transition 1+ε C gives the
+boundary class of the principal part C. Serre duality pairs it with
+A by Reskappa(a,C),dt. For output-fixed tangent
+Cin L' the residue is zero because ain L' and L' is self-annihilating.
+Therefore dp_h^*A vanishes in the q-relative cotangent, so it is
+dq_h^*theta for a unique covector theta.
+
+The bundle component of theta is A'. Indeed, choose x'ne x and
+vary both bundle transitions only at x', leaving the modification at
+x unchanged. Principal parts at x' surject to
+H^1(X,mathfrak g_frakg_{P'}) (Serre vanishing for Ngg0), and the
+pairings agree on these directions since A=A' near x'. Write
+theta=(A',xi).
+
+Now keep the input fixed and move the leg from t=0 to t=ε.
+The output transition is g_ε=lambda(t-ε) and to first
+order in ε,
+(delta g)g^{-1}=-dlambda/t. Evaluating the cotangent identity on
this motion gives
[
0=-kappa(a(0),dlambda)+xi(∂_t),
- qquad xi=kappa(a(0),dlambda),dtbig|_xne0,
+ xi=kappa(a(0),dlambda),dtbig|_xne0,
]
-using $kappa(a(0),dlambda)ne0$ from (hecke:lie-data). This is
+using kappa(a(0),dlambda)ne0 from (hecke:lie-data). This is
also the residue formula of cite[Eq.~(20.20)]{AGKRRV}; reversing the
gluing boundary convention reverses the common sign and leaves
everything below unchanged.
-The first derivative. The section $S_A$ is defined along the
-whole fixed-input fibre, using the specified identification with $P$.
-Its value at $h$ is zero from the previous paragraph. Deform the output
-lattice to $L'_ε=Ad(1+ε D)L'$ for $Din L$; a test
-vector $Cin L'$ extends as $C_ε=Ad(1+ε D)C$. The input
-field remains $a(t),dt$. Differentiating the residue pairing gives
+The first derivative. The section S_A is defined along the
+whole fixed-input fibre, using the specified identification with P.
+Its value at h is zero from the previous paragraph. Deform the output
+lattice to L'_ε=Ad(1+ε D)L' for Din L; a test
+vector Cin L' extends as C_ε=Ad(1+ε D)C. The input
+field remains a(t),dt. Differentiating the residue pairing gives
bigl<(dS_A)_h(Dbmod I),Cbmod Ibigr>
=Reskappa(a,[D,C]),dt
=Reskappa([a,D],C),dt
=beta(ad(a)D,C).
-Since $S_A(h)=0$ the derivative is well-defined, and since $ad(a)$
-preserves $L,L',I$, the formula is independent of representatives.
+Since S_A(h)=0 the derivative is well-defined, and since ad(a)
+preserves L,L',I, the formula is independent of representatives.
-For invertibility: $ad(a(t))$ preserves every weight summand in
-(hecke:weight-quotients). Because $lambda$ is central in $M$,
+For invertibility: ad(a(t)) preserves every weight summand in
+(hecke:weight-quotients). Because lambda is central in M,
each nonzero weight space injects as an invariant summand into
-$mathfrak g/frakg/mathfrak m$;frakm$; its endomorphism from $ad(a(0))$ is therefore
-invertible by (hecke:lie-data). The corresponding $k[[t]]$-linear
-block has unit determinant, so $ad(a(t))$ is invertible on every
-truncated summand of $P_1$ and $P_2$. Combined with
+mathfrak g/frakg/mathfrak m;frakm; its endomorphism from ad(a(0)) is therefore
+invertible by (hecke:lie-data). The corresponding k[[t]]-linear
+block has unit determinant, so ad(a(t)) is invertible on every
+truncated summand of P_1 and P_2. Combined with
(hecke:derivative) and (hecke:residue-pairing), this shows
-$(dS_A)_hcolon P_1to P_2^*$ is an isomorphism. Zero-weight directions
-do not appear in these quotients, so no extra regularity of $lambda$
-outside $M$ is needed.
-
-The reverse derivative. The value of $S_{A'}$ at $h$ is zero
-because $ain L$ and $L$ is self-annihilating. Keeping the output fixed,
-vary the input lattice by $Cin L'$ and extend $Din L$ to
-$Ad(1+ε C)D$. The same calculation, with the same gluing
+(dS_A)_hcolon P_1to P_2^* is an isomorphism. Zero-weight directions
+do not appear in these quotients, so no extra regularity of lambda
+outside M is needed.
+
+The reverse derivative. The value of S_{A'} at h is zero
+because ain L and L is self-annihilating. Keeping the output fixed,
+vary the input lattice by Cin L' and extend Din L to
+Ad(1+ε C)D. The same calculation, with the same gluing
convention, gives
[
bigl<(dS_{A'})_h(Cbmod I),Dbmod Ibigr>
=Reskappa(a,[C,D]),dt
=Reskappa([a,C],D),dt,
]
-and invertibility on $P_2$ with residue duality proves the second
-assertion. Both sections are sections of rank-$m$ vector bundles on
-smooth $m$-dimensional fibres; invertible derivatives make their zeros
+and invertibility on P_2 with residue duality proves the second
+assertion. Both sections are sections of rank-m vector bundles on
+smooth m-dimensional fibres; invertible derivatives make their zeros
reduced and isolated.
-When $m=0$, both tangent spaces vanish and both derivatives are
+When m=0, both tangent spaces vanish and both derivatives are
isomorphisms between zero spaces. The smooth zero-dimensional fibres
-already isolate $h$, and the moving-leg computation still applies.
+already isolate h, and the moving-leg computation still applies.
Modifications of this form, and tori, are thus included.
The proposition gives both properties we need. The nonvanishing
-$xine0$ allows a suitable hypersurface cut of the Hecke correspondence
-to be smooth over $mathcal B$,calB$, while the two relative derivatives
-isolate $h$ and set up the quadratic coordinates around it. In
+xine0 allows a suitable hypersurface cut of the Hecke correspondence
+to be smooth over mathcal B,calB, while the two relative derivatives
+isolate h and set up the quadratic coordinates around it. In
Section~sec:twohecke, these two facts are used in reverse order:
the isolating one first, and the nonvanishing one later.
@@ -1388,73 +1388,73 @@ Proposition~hecke:transverse identify the local neighbourhood of
that point with a split quadratic equation. The next section removes
the extra variables in that equation.
-Throughout this section, $FinShv(mathcal B_K)$calB_K)$ is the specialising
+Throughout this section, FinShv(mathcal B_K)calB_K) is the specialising
object of Proposition~spec:missing, and we use the vanishings of
Lemma~spec:vanishing freely.
[Vanishing at the quadratic vertex]
-Let $bcolon Utomathcal B$calB$ be a smooth chart with $U$ finite type over
-$S$. Let $finmathcal O(U)$calO(U)$ and $uin U_s(k)$ satisfy $f(u)=0$, and
-suppose $df_u=b^*A$ for some non-nilpotent $Ain T^*_{b(u)}mathcal B_s$.calB_s$.
-Then there exist $xinmathscr X_s(k)$scrX_s(k)$ and an integer $mge0$ with the
-following property. Put $T=Utimes_Smathscr X$scrX$ and let $F_T$ be the
-pullback of $F_U=b_K^*F$. Consider the affine quadric
+Let bcolon Utomathcal BcalB be a smooth chart with U finite type over
+S. Let finmathcal O(U)calO(U) and uin U_s(k) satisfy f(u)=0, and
+suppose df_u=b^*A for some non-nilpotent Ain T^*_{b(u)}mathcal B_s.calB_s.
+Then there exist xinmathscr X_s(k)scrX_s(k) and an integer mge0 with the
+following property. Put T=Utimes_Smathscr XscrX and let F_T be the
+pullback of F_U=b_K^*F. Consider the affine quadric
[
Q^a={sum_{i=1}^m u_iv_i+f=0}subset Ttimes_Smathbb A_S^bbA_S^{2m},
- qquad taucolon Q^ato T,
+ taucolon Q^ato T,
]
-where $u_i,v_i$ are affine coordinates (distinct symbols from the point
-$u$) and $f$ is pulled back from $U$. Then at the point
-$nu=(u,x,0,0)$,
+where u_i,v_i are affine coordinates (distinct symbols from the point
+u) and f is pulled back from U. Then at the point
+nu=(u,x,0,0),
(Psi_{Q^a}(tau_K^*F_T))_nu=0.
-When $m=0$, this is the vanishing on $Q^a=V(f)times_Smathscr X$.scrX$.
+When m=0, this is the vanishing on Q^a=V(f)times_Smathscr X.scrX.
Proof.
-Apply Proposition~hecke:transverse to $A$ to obtain a
-modification $h$, its leg $x$, its orbit dimension $m$, and a covector
-$theta=(A',xi)$ with $xine0$, satisfying
+Apply Proposition~hecke:transverse to A to obtain a
+modification h, its leg x, its orbit dimension m, and a covector
+theta=(A',xi) with xine0, satisfying
p^*A=q^*thetaquadat h.
-Use the exact-position correspondence $H=H^lambda$, its proper bound
-$overlineoverline H$,H$, and its generic Satake integrand $mathcal I_K$calI_K$ from
-Lemma~hecke:models. On $H_K$ this integrand is $p_K^*F$ tensored
+Use the exact-position correspondence H=H^lambda, its proper bound
+overlineoverline H,H, and its generic Satake integrand mathcal I_KcalI_K from
+Lemma~hecke:models. On H_K this integrand is p_K^*F tensored
with a fixed invertible shift and Tate line; we suppress that factor
after restricting to the open orbit.
Step 1: a proper fibre with vanishing cohomology.:
-Write $D=V(f)$. Take two copies $H_1,H_2$ of $H$ and form
+Write D=V(f). Take two copies H_1,H_2 of H and form
[
- W_0=H_2times_{p_2,mathcal B,b}calB,bU,qquad
+ W_0=H_2times_{p_2,mathcal B,b}calB,bU,
W=H_2times_{mathcal B}calBDsubset W_0.
]
-Both map to $mathcal Z=calZ=mathcal BcalBtimes_Smathscr X$scrX$ by
-$q_2=(o_2,ell_{H_2})$. The map $W_0tomathcal Z$calZ$ is smooth: its
-projection to $H_2$ is a base change of the smooth chart $b$, and $q_2$
-is smooth. At $w=(h,u)$, the differential of the equation cutting out
-$W$ from $W_0$ is the pullback of $theta$ by (eq:two-cotangent).
+Both map to mathcal Z=calZ=mathcal BcalBtimes_Smathscr XscrX by
+q_2=(o_2,ell_{H_2}). The map W_0tomathcal ZcalZ is smooth: its
+projection to H_2 is a base change of the smooth chart b, and q_2
+is smooth. At w=(h,u), the differential of the equation cutting out
+W from W_0 is the pullback of theta by (eq:two-cotangent).
Its relative differential over the output bundle stack is nonzero,
-because the leg component is $xine0$. The smooth-hypersurface
-criterion therefore makes $Wtomathcal B$calB$ (via the output bundle)
-smooth near $w$. The arbitrary-leg assertion of
+because the leg component is xine0. The smooth-hypersurface
+criterion therefore makes Wtomathcal BcalB (via the output bundle)
+smooth near w. The arbitrary-leg assertion of
Lemma~spec:vanishing gives
(Psi_W(q_{2,K}^*mathrm H_V(F)))_w=0.rmH_V(F)))_w=0.
-Note: smoothness is used only for the projection to $mathcal B$,calB$, matching
+Note: smoothness is used only for the projection to mathcal B,calB, matching
the hypothesis of Lemma~spec:vanishing.
Now form fibre products
[
- P=H_1times_{mathcal Z}calZW_0,qquad
+ P=H_1times_{mathcal Z}calZW_0,
P_D=H_1times_{mathcal Z}calZWsubsetoverlineoverline P_D=P_D=overlineoverline H_1H_1times_{mathcal Z}calZW.
]
The following squares are cartesian; in the second the vertical map
-$pi$ is proper. In the first, $Delta$ repeats the modification in the
+pi is proper. In the first, Delta repeats the modification in the
two factors.
[
[column sep=large,row sep=large]
@@ -1462,66 +1462,66 @@ two factors.
W_0 arrow[r,"q_2"'] arrow[u,bend left=38,"Delta"]
& mathcal ZcalZ
-qquad
+
[column sep=large,row sep=large]
overlineoverline P_DP_D arrow[r] arrow[d,"pi"']
& overlineoverline H_1H_1 arrow[d,"overlineoverline q_1"]q_1"]
W arrow[r,"q_2"'] & mathcal Z.calZ.
]
-Write $ccolon Pto U$ for the map through $W_0$, and
-$acolon Ptomathcal B$calB$ for the input map $p_1$. So $a$ records the
-input from which the sheaf is pulled back, and $c$ records the second
-input in $U$ on which we impose $f=0$. The distinguished point is
-$z=(h,h,u)in P_D(k)$.
+Write ccolon Pto U for the map through W_0, and
+acolon Ptomathcal BcalB for the input map p_1. So a records the
+input from which the sheaf is pulled back, and c records the second
+input in U on which we impose f=0. The distinguished point is
+z=(h,h,u)in P_D(k).
-Let $mathcal Ibar_K$calIbar_K$ be the pullback of the Satake integrand
-to $(overlineoverline P_D)_K$.P_D)_K$. Proper base change on the generic fibre gives
-$Rpi_{K,*}mathcal Ibar_KcalIbar_Ksimeq q_{2,K}^*mathrm H_V(F)$.rmH_V(F)$.
+Let mathcal Ibar_KcalIbar_K be the pullback of the Satake integrand
+to (overlineoverline P_D)_K.P_D)_K. Proper base change on the generic fibre gives
+Rpi_{K,*}mathcal Ibar_KcalIbar_Ksimeq q_{2,K}^*mathrm H_V(F).rmH_V(F).
Apply proper compatibility of specialisation, then proper base change
-at $w$, to (eq:two-hecke-zero). With
+at w, to (eq:two-hecke-zero). With
[
- Z_w=pi^{-1}(w),qquad
+ Z_w=pi^{-1}(w),
mathcal K=(calK=(Psi_{overlineoverline P_D}P_Dmathcal Ibar_K)|_calIbar_K)|_{Z_w},
]
we get
RGamma(Z_w,mathcal K)=0.calK)=0.
-Here $Z_w$ is the proper bounded Hecke fibre with output bundle and leg
+Here Z_w is the proper bounded Hecke fibre with output bundle and leg
fixed, identifications retained.
Step 2: isolating the distinguished stalk.:
-The map $acolon Ptomathcal B$calB$ is smooth: $Pto H_1$ is a base change
-of $W_0tomathcal Z$,calZ$, and $p_1$ is smooth. At a point $(h_1,w)$ in the
-exact-position part of $Z_w$, the differential of $f(c)$ is $q_1^*theta$
-pulled back. If the relative differential over $mathcal B$calB$ of $f(c)$
-were zero, cotangent injectivity for the smooth map $Pto H_1$ would
-imply that $q_1^*theta$ comes from the input cotangent under $p_1$.
-Then passing to the relative cotangent of $widetilde p_1=(p_1,ell_{H_1})$,
+The map acolon Ptomathcal BcalB is smooth: Pto H_1 is a base change
+of W_0tomathcal Z,calZ, and p_1 is smooth. At a point (h_1,w) in the
+exact-position part of Z_w, the differential of f(c) is q_1^*theta
+pulled back. If the relative differential over mathcal BcalB of f(c)
+were zero, cotangent injectivity for the smooth map Pto H_1 would
+imply that q_1^*theta comes from the input cotangent under p_1.
+Then passing to the relative cotangent of widetilde p_1=(p_1,ell_{H_1}),
the leg term drops out, and the necessary condition becomes
-$S_{A'}(h_1)=0$. The second half of
+S_{A'}(h_1)=0. The second half of
Proposition~hecke:transverse says this section has an isolated
-zero at $h$ on the fixed-output-and-leg fibre. Hence $P_Dtomathcal B$calB$
-via $a$ is smooth at every point of a punctured neighbourhood of $z$
-in $Z_w$. On the exact-position open, the generic integrand equals
-$a_K^*F$. Lemma~spec:vanishing therefore gives a neighbourhood
-$O$ of $z$ in $Z_w$ with
+zero at h on the fixed-output-and-leg fibre. Hence P_Dtomathcal BcalB
+via a is smooth at every point of a punctured neighbourhood of z
+in Z_w. On the exact-position open, the generic integrand equals
+a_K^*F. Lemma~spec:vanishing therefore gives a neighbourhood
+O of z in Z_w with
[
mathcal K|_calK|_{O∖{z}}=0.
]
This local vanishing prevents any cancellation with the boundary of the
-Hecke bound. Put $C_w=Z_w∖ O$ (reduced structure). The closed
-subsets ${z}$ and $C_w$ are disjoint, so open-closed localization
+Hecke bound. Put C_w=Z_w∖ O (reduced structure). The closed
+subsets {z} and C_w are disjoint, so open-closed localization
identifies
[
mathcal KcalKsimeq i_{z,*}mathcal K_zcalK_z ⊕ i_{C_w,*}(i_{C_w}^*mathcal K),calK),
- qquad
+
RGamma(Z_w,mathcal K)calK)simeq
mathcal K_zcalK_z ⊕ RGamma(C_w,i_{C_w}^*mathcal K).calK).
]
-The residue field of $z$ is algebraically closed, so the first summand
+The residue field of z is algebraically closed, so the first summand
is its stalk complex. Equation~(eq:two-fiber-zero) forces it to
vanish. Removing the invertible Satake normalization gives
@@ -1531,62 +1531,62 @@ No boundary behaviour of the Satake kernel is used.
Step 3: chart lift along the whole diagonal.:
We must now translate (eq:two-isolated-zero) into coordinates.
-First, the distinguished diagonal $Deltasimeq W_0subset P$ is a
-section of the smooth separated map $Pto W_0$ of relative dimension
-$m$, hence a closed regular immersion near $z$. On $Delta$ there is a
-specified isomorphism $bcsimeq a$, coming from the two identical
+First, the distinguished diagonal Deltasimeq W_0subset P is a
+section of the smooth separated map Pto W_0 of relative dimension
+m, hence a closed regular immersion near z. On Delta there is a
+specified isomorphism bcsimeq a, coming from the two identical
modifications.
-We claim that after an 'etale replacement of $P$ near $z$, there
-is a map $rcolon Pto U$ with an isomorphism
+We claim that after an 'etale replacement of P near z, there
+is a map rcolon Pto U with an isomorphism
- brsimeq a,qquad r|_Delta=c|_Delta,
+ brsimeq a, r|_Delta=c|_Delta,
-whose restriction to $Delta$ is the specified isomorphism. For this,
-form the smooth space $E_P=Ptimes_{a,mathcal B,b}calB,bUto P$. The
-diagonal data give a section over $Delta$. First take an 'etale
-scheme neighbourhood in $P$. Since $mathcal B$calB$ has affine diagonal
-(Section~sec:specialization), $E_Pto Ptimes_S U$ is affine, so
-$E_P$ is a scheme as well. Near the section point, smooth coordinates
-give an 'etale map $E_Ptomathbb A_P^bbA_P^{e}$
-cite[Tag~039P]{Stacks}. After shrinking, the section over $Delta$
+whose restriction to Delta is the specified isomorphism. For this,
+form the smooth space E_P=Ptimes_{a,mathcal B,b}calB,bUto P. The
+diagonal data give a section over Delta. First take an 'etale
+scheme neighbourhood in P. Since mathcal BcalB has affine diagonal
+(Section~sec:specialization), E_Pto Ptimes_S U is affine, so
+E_P is a scheme as well. Near the section point, smooth coordinates
+give an 'etale map E_Ptomathbb A_P^bbA_P^{e}
+cite[Tag~039P]{Stacks}. After shrinking, the section over Delta
lands in this coordinate neighbourhood. Extend its coordinate functions
-$Deltato P$, then pull back the 'etale map along the extended tuple.
-This gives an 'etale neighbourhood of $P$ with a map to $E_P$. Its
-inverse image of $Delta$ contains the open branch specified by the
+Deltato P, then pull back the 'etale map along the extended tuple.
+This gives an 'etale neighbourhood of P with a map to E_P. Its
+inverse image of Delta contains the open branch specified by the
original section; remove the closed complement of this branch inside
that inverse image. The retained diagonal is the prescribed one, and
-the map to $E_P$ proves (eq:two-chart-lift) including its
-isomorphism of bundles. We keep the notation $P,Delta,z$.
+the map to E_P proves (eq:two-chart-lift) including its
+isomorphism of bundles. We keep the notation P,Delta,z.
Step 4: exact quadratic coordinates.:
-After shrinking $U$, suppose $U/S$ has constant relative dimension $N$.
+After shrinking U, suppose U/S has constant relative dimension N.
The map
[
- Deltato T=Utimes_Smathscr X,scrX,qquad(c,leg),
+ Deltato T=Utimes_Smathscr X,scrX,(c,leg),
]
-is smooth of relative dimension $m$. So $T,Delta,P$ are smooth over
-$S$ of relative dimensions
+is smooth of relative dimension m. So T,Delta,P are smooth over
+S of relative dimensions
[
- N+1,qquad N+1+m,qquad N+1+2m.
+ N+1, N+1+m, N+1+2m.
]
-Choose generators $v_1,...,v_m$ of the diagonal ideal whose classes
+Choose generators v_1,...,v_m of the diagonal ideal whose classes
form a conormal frame. Under the identification with
-$Ω_{P/W_0}|_Delta$, the classes $dv_i$ form a basis. Because
-$r=c$ on $Delta$, there exist regular functions $u_1,...,u_m$ with
+Ω_{P/W_0}|_Delta, the classes dv_i form a basis. Because
+r=c on Delta, there exist regular functions u_1,...,u_m with
the exact identity
f(c)-f(r)=sum_{i=1}^m u_iv_i.
-The restrictions $u_i|_Delta$ are determined by the conormal class of
+The restrictions u_i|_Delta are determined by the conormal class of
the left side.
-Consider the whole local diagonal fibre $Delta_{(u,x)}$ on the special
-fibre. This parametrises modifications with input bundle $b(u)$, its
-identification, and leg $x$ fixed. The relative differential of $f(c)$
-for $Pto W_0$ is zero. On this diagonal fibre, the relative
-differential of $f(r)$ is induced by $df_u=b^*A$: the restriction
-$r|_{Delta_{(u,x)}}$ is constant at value $u$, and the bundle
+Consider the whole local diagonal fibre Delta_{(u,x)} on the special
+fibre. This parametrises modifications with input bundle b(u), its
+identification, and leg x fixed. The relative differential of f(c)
+for Pto W_0 is zero. On this diagonal fibre, the relative
+differential of f(r) is induced by df_u=b^*A: the restriction
+r|_{Delta_{(u,x)}} is constant at value u, and the bundle
isomorphism in (eq:two-chart-lift) is the canonical one along the
whole diagonal. The conormal coefficient in
(eq:two-exact-quadratic) is therefore
@@ -1594,48 +1594,48 @@ whole diagonal. The conormal coefficient in
sum_i(u_i|_Delta)[dv_i]=-S_A
quadon Delta_{(u,x)}.
-In particular $u_i(z)=0$, and the derivative of $(u_i|_Delta)$ is an
-isomorphism on $T_zDelta_{(u,x)}$ by
+In particular u_i(z)=0, and the derivative of (u_i|_Delta) is an
+isomorphism on T_zDelta_{(u,x)} by
Proposition~hecke:transverse. This uses the equality on the whole
-fibre in (eq:two-coefficients), not just at $z$.
+fibre in (eq:two-coefficients), not just at z.
Now set
[
Phi=(r,leg,u_1,...,u_m,v_1,...,v_m)colon
Pto Ttimes_Smathbb A_S^bbA_S^{2m}.
]
-Its relative differential at $z$ is an isomorphism: a tangent vector in
-its kernel is first killed by every $dv_i$, so lies in $T_zDelta$; its
-$(dr,dleg)$ then places it in $T_zDelta_{(u,x)}$; then the
+Its relative differential at z is an isomorphism: a tangent vector in
+its kernel is first killed by every dv_i, so lies in T_zDelta; its
+(dr,dleg) then places it in T_zDelta_{(u,x)}; then the
derivative isomorphism in (eq:two-coefficients) forces the vector
-to vanish. Source and target are both smooth over $S$ with the same
-relative dimension $N+1+2m$, so this injective differential is an
-isomorphism, and the relative Jacobian criterion makes $Phi$ 'etale
-at $z$.
-
-By (eq:two-exact-quadratic), the cut $P_D=V(f(c))$ is exactly
-$Phi^{-1}(Q^a)$. So the induced map $Phi_Dcolon P_Dto Q^a$ is
-'etale near $z$, sending $z$ to $nu$. Moreover (eq:two-chart-lift)
+to vanish. Source and target are both smooth over S with the same
+relative dimension N+1+2m, so this injective differential is an
+isomorphism, and the relative Jacobian criterion makes Phi 'etale
+at z.
+
+By (eq:two-exact-quadratic), the cut P_D=V(f(c)) is exactly
+Phi^{-1}(Q^a). So the induced map Phi_Dcolon P_Dto Q^a is
+'etale near z, sending z to nu. Moreover (eq:two-chart-lift)
identifies
[
a_K^*Fsimeq r_K^*F_UsimeqPhi_{D,K}^*tau_K^*F_T
quadon (P_D)_K,
]
-so $Phi_D$ identifies the actual generic sheaf and the equation
+so Phi_D identifies the actual generic sheaf and the equation
simultaneously. 'Etale base change transforms (eq:two-isolated-zero)
into (eq:two-vertex).
-If $m=0$, the diagonal is open near $z$, the sum in
+If m=0, the diagonal is open near z, the sum in
(eq:two-exact-quadratic) is empty, and the same argument gives
-$Q^a=Dtimes_Smathscr X$.scrX$. Then smooth base change along $Dtimes_S
-mathscr XscrXto D$ yields $(Psi_D(F_U|_{D_K}))_u=0$ directly.
+Q^a=Dtimes_Smathscr X.scrX. Then smooth base change along Dtimes_S
+mathscr XscrXto D yields (Psi_D(F_U|_{D_K}))_u=0 directly.
The argument used no division by 2 and no singular-support claim in
-mixed characteristic. Its output for $m>0$ is a vanishing on the quadric
-$Q^a$; the next section recovers the corresponding vanishing on the
-hypersurface $V(f)$ itself.
+mixed characteristic. Its output for m>0 is a vanishing on the quadric
+Q^a; the next section recovers the corresponding vanishing on the
+hypersurface V(f) itself.
% =========================================================================
@@ -1643,12 +1643,12 @@ hypersurface $V(f)$ itself.
% =========================================================================
-Section~sec:twohecke gave a vanishing on the affine quadric $Q^a$
-at its vertex. To get a test on the hypersurface $D=V(f)$ itself, we
+Section~sec:twohecke gave a vanishing on the affine quadric Q^a
+at its vertex. To get a test on the hypersurface D=V(f) itself, we
compactify the quadric to a projective family and use the relation
-between hyperplane powers and the closed support $D$. Concretely, the
+between hyperplane powers and the closed support D. Concretely, the
cone of the hyperplane-class map is a shifted constant Tate line
-supported on $f=0$; after tensoring with the ambient sheaf, the
+supported on f=0; after tensoring with the ambient sheaf, the
hyperplane terms have vanishing nearby cycles from the smooth-chart
result of Lemma~spec:vanishing, and the whole proper quadric
vanishing then forces the hypersurface vanishing. A second proper
@@ -1658,10 +1658,10 @@ of equations simultaneously.
## The projective quadric comparison
[Quadric comparison]
-Let $K_0$ be an algebraically closed field of characteristic $neell$,
-$E=mathbb Qbar_bbQbar_ell$, and $B$ a finite-type $K_0$-scheme. For
-$finGamma(B,mathcal O_B)$calO_B)$ put $icolon D=V(f)↪ B$. If
-$mge1$, form
+Let K_0 be an algebraically closed field of characteristic neell,
+E=mathbb Qbar_bbQbar_ell, and B a finite-type K_0-scheme. For
+finGamma(B,mathcal O_B)calO_B) put icolon D=V(f)↪ B. If
+mge1, form
[
gcolon Q_f=Bigl{sum_{a=1}^m u_av_a+fw^2=0Bigr}
subsetmathbb P_B^bbP_B^{2m}→ B.
@@ -1674,43 +1674,43 @@ Then the hyperplane-class map defines a distinguished triangle
→ (⊕_{j=0}^{2m-1}E_B(-j)[-2j])[1].
The identification of the third term with the Tate line requires only a
-fixed one-dimensional generator. Neither $B$ nor $D$ needs to be smooth
+fixed one-dimensional generator. Neither B nor D needs to be smooth
or reduced.
Proof.
-Write $h=c_1(mathcal O_calO_{Q_f}(1))in H^2(Q_f,E(1))$. Adjunction on each
-$h^j$ gives the morphism $E_B(-j)[-2j]to Rg_*E_{Q_f}$. Let
-$mathcal K$calK$ be the cone of their direct sum, and let
-$j_Bcolon B∖ D↪ B$.
+Write h=c_1(mathcal O_calO_{Q_f}(1))in H^2(Q_f,E(1)). Adjunction on each
+h^j gives the morphism E_B(-j)[-2j]to Rg_*E_{Q_f}. Let
+mathcal KcalK be the cone of their direct sum, and let
+j_Bcolon B∖ D↪ B.
We recall the cohomology of a split smooth projective quadric of
-dimension $n$: no odd cohomology, one $E(-r)$ in degree $2r$ for
-$0le rle n$, and one additional copy when $n$ is even and $r=n/2$
+dimension n: no odd cohomology, one E(-r) in degree 2r for
+0le rle n, and one additional copy when n is even and r=n/2
cite[Example~16.3,~Thm.~23.2]{MilneEtale}. This follows by induction on
-the equation $x_0x_1+q'(x_2,...)=0$; the chart $x_0ne0$ is affine
-$n$-space, whose complement is the projective cone over the quadric
-of dimension $n-2$ with the vertex removed (a line bundle over that
+the equation x_0x_1+q'(x_2,...)=0; the chart x_0ne0 is affine
+n-space, whose complement is the projective cone over the quadric
+of dimension n-2 with the vertex removed (a line bundle over that
smaller quadric). The resulting filtration by affine cells has one cell
-in every dimension $0,...,n$, and a second middle-dimensional cell
-when $n$ is even. Initial cases are a conic ($congmathbb P^1$)bbP^1$) and
+in every dimension 0,...,n, and a second middle-dimensional cell
+when n is even. Initial cases are a conic (congmathbb P^1)bbP^1) and
the split zero-dimensional quadric (two points). Localization plus
-$RGamma_c(mathbb A^r,E)=E(-r)[-2r]$bbA^r,E)=E(-r)[-2r]$ gives the cohomology. On a smooth
-quadric of positive dimension the top hyperplane power has degree $2$,
-hence is nonzero in $E$; all lower powers are nonzero as well.
+RGamma_c(mathbb A^r,E)=E(-r)[-2r]bbA^r,E)=E(-r)[-2r] gives the cohomology. On a smooth
+quadric of positive dimension the top hyperplane power has degree 2,
+hence is nonzero in E; all lower powers are nonzero as well.
-Over $B∖ D$, the geometric fibres of $g$ are smooth quadrics of
-odd dimension $2m-1$, so the calculation shows that the hyperplane-power
+Over B∖ D, the geometric fibres of g are smooth quadrics of
+odd dimension 2m-1, so the calculation shows that the hyperplane-power
map is an isomorphism on their cohomology. Proper base change therefore
-gives $j_B^*mathcal K=0$.calK=0$.
+gives j_B^*mathcal K=0.calK=0.
-Over $D$ the family, with its hyperplane line bundle, is the constant
+Over D the family, with its hyperplane line bundle, is the constant
family with fibre
[
V_m=Bigl{sum_{a=1}^m u_av_a=0Bigr}subsetmathbb P_bbP_{K_0}^{2m}.
]
-Let $e=[0:...:0:1]$ be its vertex. Projection away from $e$ identifies
-$V_m∖{e}$ with the total space of a line bundle over the split
+Let e=[0:...:0:1] be its vertex. Projection away from e identifies
+V_m∖{e} with the total space of a line bundle over the split
smooth quadric
[
C_m=Bigl{sum_{a=1}^m u_av_a=0Bigr}subsetmathbb P_bbP_{K_0}^{2m-1}.
@@ -1718,77 +1718,77 @@ smooth quadric
Compactly supported cohomology of this line bundle, plus localization at
the vertex, gives
- H^0(V_m,E)=E,qquad H^1(V_m,E)=0,qquad
- H^r(V_m,E)=H^{r-2}(C_m,E)(-1)quad(rge2).
+ H^0(V_m,E)=E, H^1(V_m,E)=0,
+ H^r(V_m,E)=H^{r-2}(C_m,E)(-1)(rge2).
-So $V_m$ has the cohomology of $mathbb P^bbP^{2m-1}$ plus one extra
-$E(-m)$ in degree $2m$.
+So V_m has the cohomology of mathbb P^bbP^{2m-1} plus one extra
+E(-m) in degree 2m.
-The hyperplane powers on $V_m$ remain nonzero. To check the top one
-without requiring smoothness of $V_m$, resolve the vertex by the
+The hyperplane powers on V_m remain nonzero. To check the top one
+without requiring smoothness of V_m, resolve the vertex by the
projective bundle of lines
[
picolonwidetilde V_m=mathbb P_bbP_{lines,C_m}
(mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m})
→ C_m.
]
-The inclusion $mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m}↪
-(trivial rank-(2m+1))$ defines $widetilde V_mto V_m$; its
-fibre over a point of $C_m$ is the projective line joining that point
-to $e$. If $zeta$ is the pullback of the hyperplane class and
-$h_C=c_1(mathcal O_calO_{C_m}(1))$, the projective-bundle relation is
-$zeta^2=pi^*h_C,zeta$ cite[S23]{MilneEtale}, hence
+The inclusion mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m}↪
+(trivial rank-(2m+1)) defines widetilde V_mto V_m; its
+fibre over a point of C_m is the projective line joining that point
+to e. If zeta is the pullback of the hyperplane class and
+h_C=c_1(mathcal O_calO_{C_m}(1)), the projective-bundle relation is
+zeta^2=pi^*h_C,zeta cite[S23]{MilneEtale}, hence
[
int_{widetilde V_m}zeta^{2m-1}=int_{C_m}h_C^{2m-2}=2.
]
-So the top hyperplane power on $V_m$ is nonzero; by multiplication, so
-are the lower powers. This includes $m=1$, where $C_1$ is two points
-and $V_1$ is two lines meeting at $e$.
+So the top hyperplane power on V_m is nonzero; by multiplication, so
+are the lower powers. This includes m=1, where C_1 is two points
+and V_1 is two lines meeting at e.
Therefore the cone of the vector-space morphism
-$⊕_{j=0}^{2m-1}E(-j)[-2j]to RGamma(V_m,E)$ has exactly one
-nonzero cohomology group, $E(-m)$ in degree $2m$: injectivity in every
+⊕_{j=0}^{2m-1}E(-j)[-2j]to RGamma(V_m,E) has exactly one
+nonzero cohomology group, E(-m) in degree 2m: injectivity in every
degree (from nonvanishing of the hyperplane powers) rules out any others.
-The cone is $E(-m)[-2m]$.
+The cone is E(-m)[-2m].
-Proper base change identifies $i^*mathcal K$calK$ with the constant
-pullback of this vector-space cone (both the family over $D$ and the
-hyperplane-power morphism are pulled back from $V_m$). Open-closed
+Proper base change identifies i^*mathcal KcalK with the constant
+pullback of this vector-space cone (both the family over D and the
+hyperplane-power morphism are pulled back from V_m). Open-closed
localization gives
[
(j_B)_!j_B^*mathcal KcalKtomathcal KcalKto i_*i^*mathcal KcalKtocdot,
]
-whose first term vanishes, so $mathcal KcalKsimeq i_*E_D(-m)[-2m]$. In
+whose first term vanishes, so mathcal KcalKsimeq i_*E_D(-m)[-2m]. In
particular the third term in (eq:slice-quadric-triangle) is
-constant along $D$; no middle-cohomology local system is left. Localization
+constant along D; no middle-cohomology local system is left. Localization
and proper base change hold with finite coefficients
cite[Tags 095L, 0DDE]{Stacks} and in their adic forms
-cite[Tags 09AH, 09C9]{Stacks}; we use them after inverting $ell$ and
-extending to $E$.
+cite[Tags 09AH, 09C9]{Stacks}; we use them after inverting ell and
+extending to E.
Lemma~slice:quadric uses rational coefficients, including
-$ell=2$. Its hyperplane-power comparison is not an isomorphism off $D$
-with $mathbb Z/2^a$bbZ/2^a$ coefficients: on a conic the hyperplane class has
-degree $2$. We use finite coefficients to build the standard
+ell=2. Its hyperplane-power comparison is not an isomorphism off D
+with mathbb Z/2^abbZ/2^a coefficients: on a conic the hyperplane class has
+degree 2. We use finite coefficients to build the standard
compatibilities, then use (eq:slice-quadric-triangle) after
rationalizing. This imposes no restriction on residue characteristic.
-The case $m=0$ is handled below directly; formally $Q_f=Dsubset
-mathbb P_B^0$bbP_B^0$ and the hyperplane sum is empty.
+The case m=0 is handled below directly; formally Q_f=Dsubset
+mathbb P_B^0bbP_B^0 and the hyperplane sum is empty.
## The hypersurface test
-Work over the trait $S$, with $mathcal B$calB$ and $Finmathcal D_C$calD_C$ as in
+Work over the trait S, with mathcal BcalB and Finmathcal D_CcalD_C as in
Proposition~spec:missing. For a smooth finite-type chart
-$bcolon Utomathcal B$,calB$, write $F_U=b_K^*F$, and let
-$mathcal N_UcalN_Usubset T^*U_s$ be the image of the global nilpotent cone
+bcolon Utomathcal B,calB, write F_U=b_K^*F, and let
+mathcal N_UcalN_Usubset T^*U_s be the image of the global nilpotent cone
under smooth cotangent pullback.
[Hypersurface test]
-Let $bcolon Utomathcal B$calB$ be smooth with $U$ finite type over $S$.
-Let $finmathcal O(U)$,calO(U)$, $D=V(f)$, and $uin D_s(k)$. If
+Let bcolon Utomathcal BcalB be smooth with U finite type over S.
+Let finmathcal O(U),calO(U), D=V(f), and uin D_s(k). If
[
d(f_s)_unotin(mathcal N_U)_u,calN_U)_u,
]
@@ -1799,58 +1799,58 @@ then
Proof.
-If $d(f_s)_u$ does not lie in the image of $T^*_{b(u)}mathcal B_s$,calB_s$,
-its image in the relative cotangent of $b$ is nonzero, so $Dtomathcal B$calB$
-is smooth near $u$; Lemma~spec:vanishing gives the claim.
+If d(f_s)_u does not lie in the image of T^*_{b(u)}mathcal B_s,calB_s,
+its image in the relative cotangent of b is nonzero, so Dtomathcal BcalB
+is smooth near u; Lemma~spec:vanishing gives the claim.
-Otherwise $d(f_s)_u=b^*A$ for some non-nilpotent
-$Ain T^*_{b(u)}mathcal B_s$.calB_s$. Proposition~two:vertex then provides
-$xinmathscr X_s(k)$scrX_s(k)$ and $mge0$ such that nearby cycles of the
-pullback of $F_U$ to
-$Q^a={sum_{a=1}^m u_av_a+f=0}$ vanish at $(t_0,0,0)$, where
-$T=Utimes_Smathscr X$scrX$ and $t_0=(u,x)$. If $m=0$ this affine model is
-$D_T$ and smooth compatibility for $D_Tto D$ gives the result.
+Otherwise d(f_s)_u=b^*A for some non-nilpotent
+Ain T^*_{b(u)}mathcal B_s.calB_s. Proposition~two:vertex then provides
+xinmathscr X_s(k)scrX_s(k) and mge0 such that nearby cycles of the
+pullback of F_U to
+Q^a={sum_{a=1}^m u_av_a+f=0} vanish at (t_0,0,0), where
+T=Utimes_Smathscr XscrX and t_0=(u,x). If m=0 this affine model is
+D_T and smooth compatibility for D_Tto D gives the result.
-Assume $mge1$. Compactify to
+Assume mge1. Compactify to
[
gcolon Q_f=Bigl{sum_{a=1}^m u_av_a+fw^2=0Bigr}
subsetmathbb P_T^bbP_T^{2m}to T.
]
-The fibre over $t_0$ is the split cone of Lemma~slice:quadric. At
-every point of this fibre except its vertex, some $u_a$ or $v_a$ is
-nonzero: in the chart $u_a=1$ its equation solves uniquely for $v_a$
-and $g$ is smooth there (works also in characteristic 2). Lemma~spec:vanishing
-gives $Psi_TF_T=0$, and smooth compatibility gives vanishing of nearby
-cycles of $g_K^*F_T$ at all those nonvertex points. The vertex is in
-the chart $w=1$, where Proposition~two:vertex gives the vanishing.
+The fibre over t_0 is the split cone of Lemma~slice:quadric. At
+every point of this fibre except its vertex, some u_a or v_a is
+nonzero: in the chart u_a=1 its equation solves uniquely for v_a
+and g is smooth there (works also in characteristic 2). Lemma~spec:vanishing
+gives Psi_TF_T=0, and smooth compatibility gives vanishing of nearby
+cycles of g_K^*F_T at all those nonvertex points. The vertex is in
+the chart w=1, where Proposition~two:vertex gives the vanishing.
So the nearby-cycle complex restricts to zero on the entire proper
fibre. Proper compatibility plus the projection formula give
(Psi_T(F_T ⊗ Rg_{K*}E))_{t_0}=0.
-Apply Lemma~slice:quadric on $T_K$ and tensor with $F_T$: the
+Apply Lemma~slice:quadric on T_K and tensor with F_T: the
projection formula identifies the third term with
-$i_{K,*}(F_T|_{(D_T)_K})(-m)[-2m]$ for
-$icolon D_T↪ T$. After $Psi_T$ and stalk at $t_0$,
-the first term vanishes ($Psi_TF_T=0$) and the second by
-(eq:slice-quadric-pushforward-zero); proper compatibility for $i$
+i_{K,*}(F_T|_{(D_T)_K})(-m)[-2m] for
+icolon D_T↪ T. After Psi_T and stalk at t_0,
+the first term vanishes (Psi_TF_T=0) and the second by
+(eq:slice-quadric-pushforward-zero); proper compatibility for i
identifies the third with
-$(Psi_{D_T}(F_T|_{(D_T)_K}))_{t_0}(-m)[-2m]$, which therefore also
-vanishes. Smooth compatibility for $D_Tto D$ finishes the proof.
+(Psi_{D_T}(F_T|_{(D_T)_K}))_{t_0}(-m)[-2m], which therefore also
+vanishes. Smooth compatibility for D_Tto D finishes the proof.
The passage through the projective quadric uses no purity theorem for
-the possibly singular hypersurface $D$; it uses the actual closed-support
+the possibly singular hypersurface D; it uses the actual closed-support
term of the comparison triangle. The same principle handles several
equations simultaneously.
## Projective incidence and higher codimension
[Incidence comparison]
-Let $B$ be a finite-type scheme over an algebraically closed field of
-characteristic $neell$, let $f_1,...,f_dinGamma(B,mathcal O_B)$calO_B)$
-with $dge1$, and set $icolon J=V(f_1,...,f_d)↪ B$.
+Let B be a finite-type scheme over an algebraically closed field of
+characteristic neell, let f_1,...,f_dinGamma(B,mathcal O_B)calO_B)
+with dge1, and set icolon J=V(f_1,...,f_d)↪ B.
For the incidence family
[
rcolon I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
@@ -1863,80 +1863,80 @@ the hyperplane powers give a distinguished triangle
→ i_*E_J(-(d-1))[-2(d-1)]
→ (⊕_{j=0}^{d-2}E_B(-j)[-2j])[1].
-For $d=1$ the first term is zero and $I=J$.
+For d=1 the first term is zero and I=J.
Proof.
-If $d=1$, the equation in $Btimesmathbb P^0$bbP^0$ is $f_1=0$ and the
-statement is immediate. Suppose $dge2$. Off $J$, the map
-$mathcal O_B^dcalO_B^dtomathcal O_B$calO_B$ with coefficients $(f_1,...,f_d)$ is
-surjective, so its kernel is a vector bundle of rank $d-1$ and $I$ is
+If d=1, the equation in Btimesmathbb P^0bbP^0 is f_1=0 and the
+statement is immediate. Suppose dge2. Off J, the map
+mathcal O_B^dcalO_B^dtomathcal O_BcalO_B with coefficients (f_1,...,f_d) is
+surjective, so its kernel is a vector bundle of rank d-1 and I is
its projective bundle of lines. The restricted hyperplane powers identify
its direct image with the first term of
(eq:slice-incidence-triangle) (check on geometric fibres, using
-proper base change and the cohomology of $mathbb P^bbP^{d-2}$).
+proper base change and the cohomology of mathbb P^bbP^{d-2}).
-On $J$, the family is exactly $Jtimesmathbb P^bbP^{d-1}$ and its hyperplane
+On J, the family is exactly Jtimesmathbb P^bbP^{d-1} and its hyperplane
class comes from the second factor. The restricted cone is therefore the
-constant top class $E_J(-(d-1))[-2(d-1)]$. The global cone vanishes off
-$J$, so open-closed localization identifies it with the pushforward of
+constant top class E_J(-(d-1))[-2(d-1)]. The global cone vanishes off
+J, so open-closed localization identifies it with the pushforward of
this restricted cone.
[Higher-codimension test]
-Let $bcolon Utomathcal B$calB$ be smooth with $U$ finite type over $S$,
-and let $F_U=b_K^*F$. Suppose $uin U_s(k)$ and $f_1,...,f_d$ are
-regular on $U$ vanishing at $u$. Suppose their special-fibre
-differentials at $u$ are linearly independent and
+Let bcolon Utomathcal BcalB be smooth with U finite type over S,
+and let F_U=b_K^*F. Suppose uin U_s(k) and f_1,...,f_d are
+regular on U vanishing at u. Suppose their special-fibre
+differentials at u are linearly independent and
[
span_k{d(f_{1,s})_u,...,d(f_{d,s})_u}
cap(mathcal N_U)_u=calN_U)_u={0}.
]
-Then for $J=V(f_1,...,f_d)$,
+Then for J=V(f_1,...,f_d),
[
(Psi_J(F_U|_{J_K}))_u=0.
]
-The case $d=0$ is allowed, with $J=U$.
+The case d=0 is allowed, with J=U.
Proof.
-For $d=0$ this is Lemma~spec:vanishing. For $dge1$ form the
+For d=0 this is Lemma~spec:vanishing. For dge1 form the
proper incidence morphism
[
rcolon I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
subset Utimes_Smathbb P_S^bbP_S^{d-1}→ U.
]
-Each affine projective-space chart in $Utimes_Smathbb P_S^bbP_S^{d-1}$ is
-still smooth over $mathcal B$.calB$. On the chart $c_a=1$, the hypersurface
-equation is $f_a+sum_{bne a}c_bf_b=0$, with special-fibre differential
-at $(u,[c])$ equal to $sum_{b=1}^dc_b,d(f_{b,s})_u$ (the projective
-parameter term is zero since every $f_b(u)=0$). This combination is
+Each affine projective-space chart in Utimes_Smathbb P_S^bbP_S^{d-1} is
+still smooth over mathcal B.calB. On the chart c_a=1, the hypersurface
+equation is f_a+sum_{bne a}c_bf_b=0, with special-fibre differential
+at (u,[c]) equal to sum_{b=1}^dc_b,d(f_{b,s})_u (the projective
+parameter term is zero since every f_b(u)=0). This combination is
nonzero by linear independence and, by hypothesis, lies outside
-$(mathcal N_U)_u$.calN_U)_u$. The nilpotent cone on the product chart is the
-pullback of $mathcal N_U$calN_U$ with zero parameter component, so
+(mathcal N_U)_u.calN_U)_u. The nilpotent cone on the product chart is the
+pullback of mathcal N_UcalN_U with zero parameter component, so
Proposition~slice:hypersurface gives vanishing of nearby cycles of
-$r_K^*F_U$ at every closed point over $u$.
+r_K^*F_U at every closed point over u.
The restriction of this nearby-cycle complex to the projective fibre is
constructible; a nonzero constructible complex on a finite-type scheme
-over $k$ has a nonzero stalk at some closed $k$-point, so it must
+over k has a nonzero stalk at some closed k-point, so it must
vanish on the entire fibre. Proper compatibility gives
[
(Psi_U Rr_{K*}r_K^*F_U)_u=0.
]
-Tensor the triangle of Lemma~slice:incidence on $U_K$ with $F_U$
+Tensor the triangle of Lemma~slice:incidence on U_K with F_U
and use the projection formula: the first term is a finite sum of shifts
-and twists of $F_U$, the second is $Rr_{K*}r_K^*F_U$, and the third is
-$i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)]$ for $icolon J↪ U$.
-After $Psi_U$, the first two terms vanish at $u$, and proper
-compatibility for $i$ shows that the third is a shift and twist of
-$(Psi_J(F_U|_{J_K}))_u$. Thus the third vanishes. For $d=1$ the first
+and twists of F_U, the second is Rr_{K*}r_K^*F_U, and the third is
+i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)] for icolon J↪ U.
+After Psi_U, the first two terms vanish at u, and proper
+compatibility for i shows that the third is a shift and twist of
+(Psi_J(F_U|_{J_K}))_u. Thus the third vanishes. For d=1 the first
sum is empty and the same argument applies.
We have thus obtained all finite transverse-slice tests, while applying
-the hypersurface argument only to schemes smooth over $mathcal B$.calB$. It
-remains to find one such slice that meets $Psi$ on the support of $F$
+the hypersurface argument only to schemes smooth over mathcal B.calB. It
+remains to find one such slice that meets Psi on the support of F
in a nonzero way.
% =========================================================================
@@ -1945,7 +1945,7 @@ in a nonzero way.
% =========================================================================
We finish the proof of Theorem~thm:full-support by applying the
-slicing theorem to the actual support of $F$. The structure of the
+slicing theorem to the actual support of F. The structure of the
argument: we first arrange that a nonzero generic stalk specializes to
the closed fibre; then we cut the support down to a finite scheme over
the trait; on a finite scheme nearby cycles are a finite direct sum of
@@ -1954,63 +1954,63 @@ stalks, so they cannot vanish if any single one is nonzero.
## Extending a generic bundle
-Let $mathscr XscrXto S$ be the smooth projective curve from
-Section~sec:specialization, and let $K$ be the fixed algebraic
-closure of $Frac(R)$. Every $G$-bundle on $mathscr X_K$scrX_K$
-extends to a $G$-bundle on $mathscr X_scrX_{R'}$ for some finite extension
-$Frac(R')/Frac(R)$ inside $K$, where
-$R'$ is the integral closure of $R$ in that extension.
+Let mathscr XscrXto S be the smooth projective curve from
+Section~sec:specialization, and let K be the fixed algebraic
+closure of Frac(R). Every G-bundle on mathscr X_KscrX_K
+extends to a G-bundle on mathscr X_scrX_{R'} for some finite extension
+Frac(R')/Frac(R) inside K, where
+R' is the integral closure of R in that extension.
Proof.
-Let $mathcal P_K$calP_K$ be the bundle. By
+Let mathcal P_KcalP_K be the bundle. By
cite[Thm.~2,~Rem.~2(b)]{DrinfeldSimpson} it is trivial on some
-nonempty open subset of $mathscr X_K$scrX_K$ (this theorem applies to connected
+nonempty open subset of mathscr X_KscrX_K (this theorem applies to connected
reductive groups over any field). Choose a trivialization away from
-distinct points $x_1,...,x_rinmathscr X(K)$.scrX(K)$. Gluing at those points
+distinct points x_1,...,x_rinmathscr X(K).scrX(K). Gluing at those points
gives a chain of one-point modifications
[
mathcal P_0calP_0dashrightarrowmathcal P_1calP_1dashrightarrow...
- dashrightarrowmathcal P_r=calP_r=mathcal P_K,calP_K,qquad
+ dashrightarrowmathcal P_r=calP_r=mathcal P_K,calP_K,
mathcal P_0calP_0 trivial,
]
-whose $i$th modification has leg $x_i$. Choose a Schubert bound for each
+whose ith modification has leg x_i. Choose a Schubert bound for each
of the finitely many relative positions.
-Start with the trivial bundle over $S=Spec R$ and form the space of
-bounded chains (legs varying on $mathscr X$).scrX$). At each stage the bounded
+Start with the trivial bundle over S=Spec R and form the space of
+bounded chains (legs varying on mathscr X).scrX). At each stage the bounded
Hecke space is representable and proper over the preceding bundle and
-leg. Since $mathscr X/S$scrX/S$ is proper, induction shows that the space of
-chains is a proper algebraic space of finite type over $S$. It carries
+leg. Since mathscr X/SscrX/S is proper, induction shows that the space of
+chains is a proper algebraic space of finite type over S. It carries
the universal final bundle and contains the chosen chain as a
-$K$-point. This point descends to a finite extension of
-$Frac(R)$ (the space is finite type and $K$ is algebraic).
-The valuative criterion extends the descended point over $R'$, and the
+K-point. This point descends to a finite extension of
+Frac(R) (the space is finite type and K is algebraic).
+The valuative criterion extends the descended point over R', and the
universal final bundle gives the extension. The construction allows
legs to coincide on the closed fibre, since it uses successive one-point
modifications.
-Now suppose a spectral component is missing and let $F$ be the nonzero
+Now suppose a spectral component is missing and let F be the nonzero
compact object from Proposition~spec:missing. It is constructible
-on finite-type smooth charts, so there is a $K$-valued bundle at which
+on finite-type smooth charts, so there is a K-valued bundle at which
its stalk is nonzero (a nonzero constructible complex on a finite-type
-scheme has a nonzero stalk at some closed $K$-point). Apply
-Lemma~support:extension to that bundle and replace $R$ by the
-resulting $R'$; all specialization statements from
+scheme has a nonzero stalk at some closed K-point). Apply
+Lemma~support:extension to that bundle and replace R by the
+resulting R'; all specialization statements from
Section~sec:specialization remain valid with the same geometric
-generic field $K$.
+generic field K.
-Choose a finite-type smooth affine chart $bcolon Utomathcal B$calB$
+Choose a finite-type smooth affine chart bcolon Utomathcal BcalB
through the special value of the extended bundle. The fibre product of
-this chart with its trait section is smooth over $S$ and has a
-$k$-point. Formal smoothness lifts that $k$-point to compatible points
-modulo every power of the uniformizer, and completeness of $R$ then
-gives an honest $R$-point of an affine 'etale neighbourhood; thus the
-trait section lifts to $U$. After shrinking around its closed point, $U/S$
-has constant relative dimension $N$. The generic stalk of $F_U=b_K^*F$
-on the lifted section is nonzero, so the support closure of $F_U$ meets
-$U_s$.
+this chart with its trait section is smooth over S and has a
+k-point. Formal smoothness lifts that k-point to compatible points
+modulo every power of the uniformizer, and completeness of R then
+gives an honest R-point of an affine 'etale neighbourhood; thus the
+trait section lifts to U. After shrinking around its closed point, U/S
+has constant relative dimension N. The generic stalk of F_U=b_K^*F
+on the lifted section is nonzero, so the support closure of F_U meets
+U_s.
## Finding a slice with nonzero nearby cycles
@@ -2020,185 +2020,185 @@ that only the support strata, not the sheaf itself, are descended
to a finite extension.
-Let $R$ be a complete discrete valuation ring with algebraically closed
-residue field $k$ of characteristic $neell$, $S=Spec R$, and $K$ an
-algebraic closure of its fraction field. Let $U/S$ be a smooth affine
-scheme of constant relative dimension $N$, $mathcal F$calF$ a bounded
-constructible $E$-complex on $U_K$, and
-$Λsubset T^*U_s$ a closed conical subset with $dimΛle N$.
-Suppose the closure in $U$ of the nonzero-stalk locus of $mathcal F$calF$
-meets $U_s$. Then after a finite extension of the trait and shrinking
-$U$, there exist $uin U_s(k)$, $0le dle N$ and functions
-$f_1,...,f_d$ vanishing at $u$ whose special-fibre differentials are
+Let R be a complete discrete valuation ring with algebraically closed
+residue field k of characteristic neell, S=Spec R, and K an
+algebraic closure of its fraction field. Let U/S be a smooth affine
+scheme of constant relative dimension N, mathcal FcalF a bounded
+constructible E-complex on U_K, and
+Λsubset T^*U_s a closed conical subset with dimΛle N.
+Suppose the closure in U of the nonzero-stalk locus of mathcal FcalF
+meets U_s. Then after a finite extension of the trait and shrinking
+U, there exist uin U_s(k), 0le dle N and functions
+f_1,...,f_d vanishing at u whose special-fibre differentials are
independent, such that
[
span{d(f_{1,s})_u,...,d(f_{d,s})_u}
capΛ_u⊆{0},
]
-and for $J=V(f_1,...,f_d)$,
+and for J=V(f_1,...,f_d),
(Psi_J(mathcal F|_calF|_{J_K}))_une0.
-For $d=0$ the list is empty and $J=U$.
+For d=0 the list is empty and J=U.
Proof.
Step 1: descend the finite support data.
-Let $A_1,...,A_t$ be the irreducible components of the closure of the
-nonzero-stalk locus in $U_K$. Bounded constructibility gives a proper
-closed subset $B_i⊊ A_i$ such that every stalk on
-$A_i∖ B_i$ is nonzero. Give these closed sets reduced
+Let A_1,...,A_t be the irreducible components of the closure of the
+nonzero-stalk locus in U_K. Bounded constructibility gives a proper
+closed subset B_i⊊ A_i such that every stalk on
+A_i∖ B_i is nonzero. Give these closed sets reduced
structures. Their finitely generated ideals descend to a common finite
-extension of $Frac(R)$; enlarge that extension to descend
+extension of Frac(R); enlarge that extension to descend
all containments too. Replace the trait accordingly. The new support
closure still meets the special fibre: the finite map from the new
model to the old is closed and agrees with the geometric generic
support, so it maps the new closure onto the old. This step does not
-require $mathcal F$calF$ to descend.
+require mathcal FcalF to descend.
-Write $overlineoverline A_i,A_i,overlineoverline B_i$B_i$ for the horizontal closures in $U$,
-and $e_i=dim A_i$. Each $overlineoverline A_i$A_i$ is integral and dominates $S$.
-Every nonempty special fibre $(overlineoverline A_i)_s$A_i)_s$ is pure of dimension
-$e_i$ cite[Tag~0B2J]{Stacks}. Applying this to the irreducible
-components of $B_i$ shows that whenever
-$(overlineoverline B_i)_sB_i)_sne∅$,
+Write overlineoverline A_i,A_i,overlineoverline B_iB_i for the horizontal closures in U,
+and e_i=dim A_i. Each overlineoverline A_iA_i is integral and dominates S.
+Every nonempty special fibre (overlineoverline A_i)_sA_i)_s is pure of dimension
+e_i cite[Tag~0B2J]{Stacks}. Applying this to the irreducible
+components of B_i shows that whenever
+(overlineoverline B_i)_sB_i)_sne∅,
[
dim(overlineoverline B_i)_s<e_i.B_i)_s<e_i.
]
The trait is excellent, hence universally catenary (a complete Noetherian
local ring is excellent cite[Tag~07QW]{Stacks}), so at any closed
-$vin(overlineoverline A_i)_s$,A_i)_s$,
+vin(overlineoverline A_i)_s,A_i)_s,
dimmathcal O_calO_{overlineoverline A_i,v}A_i,v=e_i+1
qquadcite[Tag~02JT]{Stacks}.
Step 2: choose the special point.
-Let $d$ be the maximum of the $e_i$ such that $(overlineoverline A_i)_sA_i)_sne
-∅$; such exists by the support assumption. Choose a reduced
-irreducible component $Tsubset(overlineoverline A_i)_s$A_i)_s$ of dimension $d$ for
-some such $i$. At a general closed point of $T$ the reduced structure is
-smooth, no other distinct reduced component of $∪_i(overlineoverline A_i)_s$A_i)_s$
-passes through the point, and no $overlineoverline B_i$B_i$ passes through it: an
-$i$ with empty special fibre contributes nothing, while every remaining
-bad special locus has dimension $<d$. Different horizontal components
-may share the same reduced $T$; we keep them all.
+Let d be the maximum of the e_i such that (overlineoverline A_i)_sA_i)_sne
+∅; such exists by the support assumption. Choose a reduced
+irreducible component Tsubset(overlineoverline A_i)_sA_i)_s of dimension d for
+some such i. At a general closed point of T the reduced structure is
+smooth, no other distinct reduced component of ∪_i(overlineoverline A_i)_sA_i)_s
+passes through the point, and no overlineoverline B_iB_i passes through it: an
+i with empty special fibre contributes nothing, while every remaining
+bad special locus has dimension <d. Different horizontal components
+may share the same reduced T; we keep them all.
We may also arrange
dimΛ_ule N-d.
-Indeed, restrict $Λ$ to $T$. Its components that do not dominate
-$T$ can be excluded over a proper closed subset of $T$. The generic
-fibre of each remaining component has dimension $le N-d$ because
-$dimΛle N$; the same bound holds over a nonempty open subset of
-$T$. Choose $uin T(k)$ satisfying all requirements.
+Indeed, restrict Λ to T. Its components that do not dominate
+T can be excluded over a proper closed subset of T. The generic
+fibre of each remaining component has dimension le N-d because
+dimΛle N; the same bound holds over a nonempty open subset of
+T. Choose uin T(k) satisfying all requirements.
Step 3: choose transverse equations.
-Let $V=T_u^*U_s$, so $dim V=N$. If $d>0$, a general $d$-plane
-$Lsubset V$ satisfies $LcapΛ_u⊆{0}$: from
+Let V=T_u^*U_s, so dim V=N. If d>0, a general d-plane
+Lsubset V satisfies LcapΛ_u⊆{0}: from
(eq:support-cone-fiber),
-$dimmathbb P(Λ_u)bbP(Λ_u)le N-d-1$, and the incidence of $d$-planes
-meeting this projective set is a proper closed subset of $Gr(d,V)$. The
-condition that $Lto T_u^*T$ be an isomorphism is another nonempty open
+dimmathbb P(Λ_u)bbP(Λ_u)le N-d-1, and the incidence of d-planes
+meeting this projective set is a proper closed subset of Gr(d,V). The
+condition that Lto T_u^*T be an isomorphism is another nonempty open
condition; two nonempty opens meet because the Grassmannian is
-irreducible. Pick such an $L$ and a basis of it; for $d=0$ take $L=0$.
+irreducible. Pick such an L and a basis of it; for d=0 take L=0.
-Lift the basis to functions $f_1,...,f_d$ vanishing at $u$, on an
-affine neighbourhood in $U$. Their restrictions to the smooth
-$d$-dimensional $T$ form a regular system of parameters at $u$. Hence
-for $J=V(f_1,...,f_d)$ and $Z=Jcap∪_ioverlineoverline A_i$,A_i$, the point
-$u$ is isolated in $Z_s$ after shrinking; the reduced special fibre of
-the union of all support closures near $u$ is precisely $T$, even if
-several horizontal components specialize to $T$.
+Lift the basis to functions f_1,...,f_d vanishing at u, on an
+affine neighbourhood in U. Their restrictions to the smooth
+d-dimensional T form a regular system of parameters at u. Hence
+for J=V(f_1,...,f_d) and Z=Jcap∪_ioverlineoverline A_i,A_i, the point
+u is isolated in Z_s after shrinking; the reduced special fibre of
+the union of all support closures near u is precisely T, even if
+several horizontal components specialize to T.
Step 4: a generic generization in the slice.
-Choose $i$ with $e_i=d$ and $T$ a component of $(overlineoverline A_i)_s$.A_i)_s$. By
-(eq:support-local-dimension), $dimmathcal O_calO_{overlineoverline A_i,u}A_i,u
-=d+1$. Quotienting by $f_1,...,f_d$ leaves dimension $ge1$; then
-quotienting by a uniformizer $pi$ gives dimension zero (isolation). So
-some prime of the cut local ring does not contain $pi$ -- otherwise
-$pi$ would be nilpotent and the ring would be dimension zero. This gives
-a generic-fibre point $z$ specializing to $u$.
-
-The morphism $Zto S$ is quasi-finite at $u$ (closed point isolated in
+Choose i with e_i=d and T a component of (overlineoverline A_i)_s.A_i)_s. By
+(eq:support-local-dimension), dimmathcal O_calO_{overlineoverline A_i,u}A_i,u
+=d+1. Quotienting by f_1,...,f_d leaves dimension ge1; then
+quotienting by a uniformizer pi gives dimension zero (isolation). So
+some prime of the cut local ring does not contain pi -- otherwise
+pi would be nilpotent and the ring would be dimension zero. This gives
+a generic-fibre point z specializing to u.
+
+The morphism Zto S is quasi-finite at u (closed point isolated in
its fibre cite[Tag~01TH]{Stacks}); its quasi-finite locus is open, so
-it contains $z$. Thus $z$ has residue field finite over
-$Frac(R)$. Its geometric lifts to $K$ avoid $B_i$: a
-point in the closed set $overlineoverline B_i$B_i$ cannot specialize to
-$unotinoverlineoverline B_i$.B_i$. Each such lift therefore has a nonzero stalk of
-$mathcal F|_calF|_{J_K}$.
+it contains z. Thus z has residue field finite over
+Frac(R). Its geometric lifts to K avoid B_i: a
+point in the closed set overlineoverline B_iB_i cannot specialize to
+unotinoverlineoverline B_i.B_i. Each such lift therefore has a nonzero stalk of
+mathcal F|_calF|_{J_K}.
Step 5: nearby cycles on a finite neighbourhood.
-Take an affine neighbourhood of $u$ inside the quasi-finite locus of
-$Zto S$. The henselian decomposition of a finite-type algebra
-cite[Tag~04GJ]{Stacks} gives a finite local factor containing $u$: an
-open neighbourhood $Q$ of $u$ in $Z$, finite over $S$, whose special
-fibre has only the point $u$ (equivalently, Zariski's main theorem
-embeds a quasi-finite neighbourhood into a finite $S$-scheme; the
-henselian local factor at $u$ lies in the original open because an open
+Take an affine neighbourhood of u inside the quasi-finite locus of
+Zto S. The henselian decomposition of a finite-type algebra
+cite[Tag~04GJ]{Stacks} gives a finite local factor containing u: an
+open neighbourhood Q of u in Z, finite over S, whose special
+fibre has only the point u (equivalently, Zariski's main theorem
+embeds a quasi-finite neighbourhood into a finite S-scheme; the
+henselian local factor at u lies in the original open because an open
subset of a local spectrum containing the closed point is the whole
-spectrum). Every generization of $u$, including $z$, lies in $Q$.
+spectrum). Every generization of u, including z, lies in Q.
-The restriction $mathcal F|_calF|_{J_K}$ is supported on $Z_K$. To use the
-finite neighbourhood, remove $Z∖ Q$ from $J$: this complement is
-closed in $J$ hence in $U$, so remove it from $U$ and pass to an affine
-neighbourhood of $u$. That neighbourhood contains $Q$ because every
-point of the finite local $Q$ specializes to $u$. The new $J$ is still
+The restriction mathcal F|_calF|_{J_K} is supported on Z_K. To use the
+finite neighbourhood, remove Z∖ Q from J: this complement is
+closed in J hence in U, so remove it from U and pass to an affine
+neighbourhood of u. That neighbourhood contains Q because every
+point of the finite local Q specializes to u. The new J is still
cut out by the same functions in a smooth affine chart, and the change
-leaves its nearby-cycle stalk at $u$ unchanged while making $Q$ closed
-in $J$. If $iotacolon Q↪ J$ is this closed immersion
-and $mathcal H=calH=mathcal F|_calF|_{Q_K}$, localization gives
-$mathcal F|_calF|_{J_K}simeqiota_{K,*}mathcal H$.calH$. Proper compatibility for
-$iota$ reduces (eq:support-nonzero) to the stalk of $Psi_Q
-mathcal H$calH$ at $u$.
+leaves its nearby-cycle stalk at u unchanged while making Q closed
+in J. If iotacolon Q↪ J is this closed immersion
+and mathcal H=calH=mathcal F|_calF|_{Q_K}, localization gives
+mathcal F|_calF|_{J_K}simeqiota_{K,*}mathcal H.calH. Proper compatibility for
+iota reduces (eq:support-nonzero) to the stalk of Psi_Q
+mathcal HcalH at u.
-Let $acolon Qto S$ be the finite structural map. Its special fibre
-has only $u$, so proper compatibility gives
+Let acolon Qto S be the finite structural map. Its special fibre
+has only u, so proper compatibility gives
[
(Psi_Qmathcal H)_ucalH)_usimeqPsi_S(RGamma(Q_K,mathcal H))calH))
simeqRGamma(Q_K,mathcal H).calH).
]
The second identification uses geometric specialization on the trait,
-which is the identity on coefficient complexes. The finite $K$-scheme
-$Q_K$ has the 'etale topos of a finite discrete set, so
-$RGamma(Q_K,mathcal H)calH)simeq⊕_{xin|Q_K|}mathcal H_x$.calH_x$. At
+which is the identity on coefficient complexes. The finite K-scheme
+Q_K has the 'etale topos of a finite discrete set, so
+RGamma(Q_K,mathcal H)calH)simeq⊕_{xin|Q_K|}mathcal H_x.calH_x. At
least one summand is nonzero (Step 4), so this complex is nonzero,
proving (eq:support-nonzero).
These are geometric nearby cycles with monodromy forgotten; no inertia
invariants. The finite direct-sum calculation is compatible with finite
-coefficients, their $ell$-adic limit, inversion of $ell$, and
-extension to $E$, so it applies to the specialization functor of
+coefficients, their ell-adic limit, inversion of ell, and
+extension to E, so it applies to the specialization functor of
Section~sec:specialization without requiring a field of descent
-for $mathcal F$.calF$.
+for mathcal F.calF.
## Completion of the full-support proof
Proof. [Proof of Theorem~thm:full-support]
-If $Y'ne Y$, Proposition~spec:missing supplies the nonzero
-compact $F$ used above. Lemma~support:extension and the
-subsequent chart construction give a smooth affine $Utomathcal B$calB$ of
-constant relative dimension $N$ for which the closure of the nonzero-stalk
-locus of $F_U$ meets $U_s$.
+If Y'ne Y, Proposition~spec:missing supplies the nonzero
+compact F used above. Lemma~support:extension and the
+subsequent chart construction give a smooth affine Utomathcal BcalB of
+constant relative dimension N for which the closure of the nonzero-stalk
+locus of F_U meets U_s.
-Let $mathcal N_UcalN_Usubset T^*U_s$ be the pullback of the global nilpotent
+Let mathcal N_UcalN_Usubset T^*U_s be the pullback of the global nilpotent
cone. Under either hypothesis regime, the nilpotent-parabolic
assumption and invariant form give the half-dimensionality theorem of
cite[App.~D.1.1--D.1.8]{AGKRRV}. In its smooth-chart formulation
-cite[App.~D.1.2]{AGKRRV}, this says $dimmathcal N_UcalN_Uledim U_s=N$.
+cite[App.~D.1.2]{AGKRRV}, this says dimmathcal N_UcalN_Uledim U_s=N.
-Apply Lemma~support:finite-slice with $mathcal F=F_U$calF=F_U$ and
-$Λ=mathcal N_U$.calN_U$. It gives a slice $J=V(f_1,...,f_d)$ and
-$uin J_s(k)$ such that
+Apply Lemma~support:finite-slice with mathcal F=F_UcalF=F_U and
+Λ=mathcal N_U.calN_U. It gives a slice J=V(f_1,...,f_d) and
+uin J_s(k) such that
[
(Psi_J(F_U|_{J_K}))_une0
]
while the independent differentials of its equations span a plane
-meeting $(mathcal N_U)_u$calN_U)_u$ only at zero. The higher-codimension vanishing
+meeting (mathcal N_U)_ucalN_U)_u only at zero. The higher-codimension vanishing
of Proposition~slice:higher applies to the same chart and slice
-and says that this stalk is zero. Contradiction. This also handles $d=0$,
+and says that this stalk is zero. Contradiction. This also handles d=0,
where the test is the original smooth-chart vanishing.
Thus the complementary union is empty. The primed equivalence recalled
@@ -2215,42 +2215,42 @@ Throughout this section assume Hypothesis~hyp:A or
Hypothesis~hyp:B, so Theorem~thm:full-support holds.
Put
[
- mathcal C_s=calC_s=Shv_{Nilp}(Bun_G),qquad
- mathcal A=calA=QCoh(Y),qquad
+ mathcal C_s=calC_s=Shv_{Nilp}(Bun_G),
+ mathcal A=calA=QCoh(Y),
mathcal D=calD=IndCoh_{Nilp}(Y),
]
-and write $mathbb L_G^bbL_G^{restr}colonmathcal C_scalC_stomathcal D$calD$
-for the restricted Langlands equivalence. Its $mathcal A$-linearcalA$-linear
+and write mathbb L_G^bbL_G^{restr}colonmathcal C_scalC_stomathcal DcalD
+for the restricted Langlands equivalence. Its mathcal A-linearcalA-linear
structure is the one furnished by cite[Lemma~1.3.7]{GR}.
## The category attached to one component
-For an open-and-closed component $icolon Z↪ Y$, define
+For an open-and-closed component icolon Z↪ Y, define
[
mathcal C_calC_{s,Z}:=QCoh(Z)underset{QCoh(Y)} ⊗ mathcal C_s,calC_s,
]
the tensor product being that of presentable dg categories. The
-idempotent quasi-coherent $mathcal O_Z$calO_Z$ (extended by zero) acts as the
+idempotent quasi-coherent mathcal O_ZcalO_Z (extended by zero) acts as the
projection to this summand.
-Under either hypothesis regime, every connected component $Zsubset Y$
+Under either hypothesis regime, every connected component Zsubset Y
has nonzero localized automorphic category, and the restriction of
-$mathbb L_G^bbL_G^{restr}$ induces an equivalence
-$mathcal C_calC_{s,Z}xrightarrow{sim}IndCoh_{Nilp}(Z)$.
+mathbb L_G^bbL_G^{restr} induces an equivalence
+mathcal C_calC_{s,Z}xrightarrow{sim}IndCoh_{Nilp}(Z).
Proof.
-The decomposition $Y=Zsqcup(Y∖ Z)$ decomposes quasi-coherent
+The decomposition Y=Zsqcup(Y∖ Z) decomposes quasi-coherent
and nilpotent ind-coherent sheaves into two summands, so
-$QCoh(Z) ⊗ _{mathcal A}calAmathcal DcalDsimeqIndCoh_{Nilp}(Z)$. Tensor
-the $mathcal A$-linearcalA$-linear equivalence of Theorem~thm:full-support
-with $QCoh(Z)$ over $mathcal A$,calA$, and the assertion follows.
+QCoh(Z) ⊗ _{mathcal A}calAmathcal DcalDsimeqIndCoh_{Nilp}(Z). Tensor
+the mathcal A-linearcalA-linear equivalence of Theorem~thm:full-support
+with QCoh(Z) over mathcal A,calA, and the assertion follows.
-Every component contains a semisimple $E$-point
+Every component contains a semisimple E-point
cite[Prop.~3.7.2,~Cor.~3.7.4]{AGKRRV}. Pullback to this point shows
-$mathcal O_ZcalO_Zne0$, and the fully faithful
-$QCoh(Z)↪IndCoh_{Nilp}(Z)$ then gives nonvanishing.
+mathcal O_ZcalO_Zne0, and the fully faithful
+QCoh(Z)↪IndCoh_{Nilp}(Z) then gives nonvanishing.
@@ -2263,15 +2263,15 @@ extension data, not just its semisimple base point.
## The arithmetic trace formula
-Assume Hypothesis~hyp:A. The finite-field models $X_0,G_0$
-determine geometric $q$-Frobenius, whose pullback action on $QLisse(X)$
-induces an automorphism $Frob$ of $Y$. Define the derived fixed-point
+Assume Hypothesis~hyp:A. The finite-field models X_0,G_0
+determine geometric q-Frobenius, whose pullback action on QLisse(X)
+induces an automorphism Frob of Y. Define the derived fixed-point
stack
[
Y^{arithm}:=Y^{Frob}=Yunderset{Ytimes_EY}{overset{mathrm R}rmRtimes}Y,
]
-where the two maps to $Ytimes_EY$ are the diagonal and the graph of
-$Frob$. An $E$-point of $Y^{arithm}$ is a geometric local
+where the two maps to Ytimes_EY are the diagonal and the graph of
+Frob. An E-point of Y^{arithm} is a geometric local
system together with a Weil structure. See cite[S24.1]{AGKRRV}
and cite[S1.5.1]{GR}.
@@ -2282,14 +2282,14 @@ canonical isomorphism
Funct_c(Bun_{G_0}(mathbb F_q),E)bbF_q),E)
simeqGamma^{IndCoh}(Y^{arithm},omega_{Y^{arithm}}).
]
-The left side is $H^0$ of compactly supported $E$-valued functions on
+The left side is H^0 of compactly supported E-valued functions on
isomorphism classes of rational bundles; the right side uses the
ind-coherent dualizing object and its ind-coherent global sections on the
derived fixed-point stack.
Proof.
-The primed union $Y'$ is Frobenius-stable. The spectral trace
+The primed union Y' is Frobenius-stable. The spectral trace
calculation of Beraldo--Lin--Reeves
cite[SS6.4.12--6.4.13]{BeraldoLinReeves} enters the known primed
formula cite[SS1.5.3--1.5.5,~Cor.~1.5.6]{GR}:
@@ -2298,18 +2298,18 @@ formula cite[SS1.5.3--1.5.5,~Cor.~1.5.6]{GR}:
simeqGamma^{IndCoh}((Y')^{Frob},omega_{(Y')^{Frob}}).
]
Its automorphic trace input is cite[Thm.~0.2.6]{AGKRRVTrace} with
-endofunctor $q$-Frobenius pushforward on $Bun_G$. By
-Theorem~thm:full-support, the inclusion $Y'↪ Y$ is
-equality. Taking derived fixed points identifies $(Y')^{Frob}$ with
-$Y^{arithm}$ and transports the dualizing object. Substitution
+endofunctor q-Frobenius pushforward on Bun_G. By
+Theorem~thm:full-support, the inclusion Y'↪ Y is
+equality. Taking derived fixed points identifies (Y')^{Frob} with
+Y^{arithm} and transports the dualizing object. Substitution
gives the stated formula.
-Keeping $Gamma^{IndCoh}$ explicit avoids a convention in
+Keeping Gamma^{IndCoh} explicit avoids a convention in
cite[S1.5.3]{GR}, whose "ordinary global sections" notation uses the
-image of $omega$ under the dual of $ΥcolonQCohtoIndCoh$. We
-neither truncate $Y^{arithm}$ classically nor replace its
+image of omega under the dual of ΥcolonQCohtoIndCoh. We
+neither truncate Y^{arithm} classically nor replace its
dualizing object by its structure sheaf.
@@ -2319,113 +2319,113 @@ We now prove that every parameter has a nonzero eigenobject. The
categorical input is simple.
-Let $mathcal A_0$calA_0$ be a compactly generated, presentable, stable,
-$E$-linear symmetric monoidal category whose compact objects are
+Let mathcal A_0calA_0 be a compactly generated, presentable, stable,
+E-linear symmetric monoidal category whose compact objects are
dualizable, and suppose its tensor product preserves colimits
-separately. Let $acolonmathcal A_0calA_0toVect_E$ be a continuous
-$E$-linear symmetric monoidal functor, and give $Vect_E$ the
-$mathcal A_0$-actioncalA_0$-action through $a$. Then the right adjoint $R_a$ of $a$
-is continuous, carries a coherent $mathcal A_0$-linearcalA_0$-linear structure, and
-$R_a(E)ne0$.
+separately. Let acolonmathcal A_0calA_0toVect_E be a continuous
+E-linear symmetric monoidal functor, and give Vect_E the
+mathcal A_0-actioncalA_0-action through a. Then the right adjoint R_a of a
+is continuous, carries a coherent mathcal A_0-linearcalA_0-linear structure, and
+R_a(E)ne0.
Proof.
-Presentability gives $R_a$. Every compact object of $mathcal A_0$calA_0$ is
-dualizable, so its image under $a$ is a perfect complex; thus $a$
+Presentability gives R_a. Every compact object of mathcal A_0calA_0 is
+dualizable, so its image under a is a perfect complex; thus a
preserves compact objects. Testing against compact generators and using
-adjunction shows $R_a$ preserves filtered colimits. Both categories are
-stable, so $R_a$ is exact, and exact + colimit-preserving implies
+adjunction shows R_a preserves filtered colimits. Both categories are
+stable, so R_a is exact, and exact + colimit-preserving implies
colimit-preserving.
-The canonical projection $b ⊗ R_a(V)to R_a(a(b) ⊗ V)$ is
-invertible for dualizable $b$: for any $c$,
+The canonical projection b ⊗ R_a(V)to R_a(a(b) ⊗ V) is
+invertible for dualizable b: for any c,
RHom_{mathcal A_0}calA_0(c,b ⊗ R_a(V))
&simeqRHom_{mathcal A_0}calA_0(b^vee ⊗ c,R_a(V))
&simeqRHom_E(a(b)^vee ⊗ a(c),V)
&simeqRHom_E(a(c),a(b) ⊗ V).
-Both sides are colimit-preserving in $b$, so compact generation proves
-invertibility for all $b$. These projections are adjunction mates of
-the module structure on $a$, hence satisfy the unit and associativity
+Both sides are colimit-preserving in b, so compact generation proves
+invertibility for all b. These projections are adjunction mates of
+the module structure on a, hence satisfy the unit and associativity
coherences before invertibility is imposed; invertibility then gives a
-coherent $mathcal A_0$-linearcalA_0$-linear structure on $R_a$.
+coherent mathcal A_0-linearcalA_0-linear structure on R_a.
-Finally $R_a(E)ne0$ because
-$RHom_{mathcal A_0}calA_0(mathbf 1,R_a(E))bf1,R_a(E))
-simeqRHom_E(E,E)simeq E$; this last argument does not require a
+Finally R_a(E)ne0 because
+RHom_{mathcal A_0}calA_0(mathbf 1,R_a(E))bf1,R_a(E))
+simeqRHom_E(E,E)simeq E; this last argument does not require a
compact unit.
-For our spectral stack, $mathcal A=calA=QCoh(Y)$ satisfies the
+For our spectral stack, mathcal A=calA=QCoh(Y) satisfies the
hypotheses of Lemma~lem:parameter-adjoint by
cite[S7.9,~Cor.~C.4.4]{AGKRRV}. On a disjoint union of components the
compact generators have finite component support. We use the continuous,
-fully faithful $mathcal A$-linearcalA$-linear functor
+fully faithful mathcal A-linearcalA-linear functor
[
jmath_YcolonQCoh(Y)xrightarrow[sim]{Υ_Y}
IndCoh_{{0}}(Y)↪IndCoh_{Nilp}(Y),
]
-where $Υ_Y(Q)=Q ⊗ omega_Y$
+where Υ_Y(Q)=Q ⊗ omega_Y
cite[S1.3.1,~Eq.~(1.2)]{GR}. Its essential image has zero singular
support, hence lies in the nilpotent category.
-For a finite set $I$, put $mathcal Q_I=calQ_I=QLisse(X)^{ ⊗ I}$. Given
-$VinRep(check G)^{ ⊗ I}$, let $H_I(V,-)$ denote the multi-leg
+For a finite set I, put mathcal Q_I=calQ_I=QLisse(X)^{ ⊗ I}. Given
+VinRep(check G)^{ ⊗ I}, let H_I(V,-) denote the multi-leg
Hecke functor and let
-$mathcal E_V^IcalE_V^Iinmathcal AcalA ⊗ mathcal Q_I$calQ_I$ be the universal
-evaluation object. For a parameter $sigmacolonSpec Eto Y$ define
-$sigma^I(V):=(sigma^* ⊗ id)(mathcal E_V^I)calE_V^I)inmathcal Q_I$.calQ_I$.
+mathcal E_V^IcalE_V^Iinmathcal AcalA ⊗ mathcal Q_IcalQ_I be the universal
+evaluation object. For a parameter sigmacolonSpec Eto Y define
+sigma^I(V):=(sigma^* ⊗ id)(mathcal E_V^I)calE_V^I)inmathcal Q_I.calQ_I.
-A Hecke eigenobject of eigenvalue $sigma$ with coherent structure is an
-$Minmathcal C_s$calC_s$ equipped, for all finite sets $I$, with isomorphisms
-$H_I(V,M)simeq Mboxtimessigma^I(V)$, natural in $Vin
-Rep(check G)^{ ⊗ I}$ and compatible with units, tensor products,
+A Hecke eigenobject of eigenvalue sigma with coherent structure is an
+Minmathcal C_scalC_s equipped, for all finite sets I, with isomorphisms
+H_I(V,M)simeq Mboxtimessigma^I(V), natural in Vin
+Rep(check G)^{ ⊗ I} and compatible with units, tensor products,
permutations of legs, collisions of legs, and all homotopy-coherent
compatibilities cite[S15.2]{AGKRRV}.
-Under either hypothesis regime, every parameter $sigmacolonSpec Eto
-Y$ admits a nonzero object $M_sigmainShv_{Nilp}(Bun_G)$ with a
-coherent Hecke eigenstructure of eigenvalue $sigma$. If $sigma$ lies
-in a component $Z$, then $M_sigmainmathcal C_calC_{s,Z}$. The assertion
+Under either hypothesis regime, every parameter sigmacolonSpec Eto
+Y admits a nonzero object M_sigmainShv_{Nilp}(Bun_G) with a
+coherent Hecke eigenstructure of eigenvalue sigma. If sigma lies
+in a component Z, then M_sigmainmathcal C_calC_{s,Z}. The assertion
is in the ind-constructible dg category; no boundedness, perversity
or Weil structure is included.
Proof.
-Apply Lemma~lem:parameter-adjoint to $a=sigma^*colonmathcal AcalA
-toVect_E$, and write $sigma_*$ for its continuous right adjoint.
+Apply Lemma~lem:parameter-adjoint to a=sigma^*colonmathcal AcalA
+toVect_E, and write sigma_* for its continuous right adjoint.
The composite
[
Phi_sigmacolonVect_sigmaxrightarrow{sigma_*}mathcal AcalA
xrightarrow{jmath_Y}mathcal DcalD
xrightarrow{(mathbb L_G^bbL_G^{restr})^{-1}}mathcal C_scalC_s
]
-is continuous $mathcal A$-linear,calA$-linear, where $Vect_sigma$ denotes
-$Vect_E$ with the action through $sigma^*$. Set
-$M_sigma=Phi_sigma(E)$. It is nonzero by the lemma, full faithfulness
-of $jmath_Y$, and the Langlands equivalence. If $sigma$ lies in $Z$,
-the idempotent $e_Z$ pulls back to $E$, while its complementary
-idempotent pulls back to zero; linearity puts $M_sigma$ in
-$mathcal C_calC_{s,Z}$.
+is continuous mathcal A-linear,calA-linear, where Vect_sigma denotes
+Vect_E with the action through sigma^*. Set
+M_sigma=Phi_sigma(E). It is nonzero by the lemma, full faithfulness
+of jmath_Y, and the Langlands equivalence. If sigma lies in Z,
+the idempotent e_Z pulls back to E, while its complementary
+idempotent pulls back to zero; linearity puts M_sigma in
+mathcal C_calC_{s,Z}.
-For the eigenstructure, fix a finite set $I$. The action of
-$mathcal E_V^I$calE_V^I$ on $mathcal C_s$calC_s$ is the multi-leg Hecke functor
-$H_I(V,-)$ cite[SS14.2--14.3]{AGKRRV}. Extend $Phi_sigma$ by the
-identity on $mathcal Q_I$.calQ_I$. Its module structure gives
+For the eigenstructure, fix a finite set I. The action of
+mathcal E_V^IcalE_V^I on mathcal C_scalC_s is the multi-leg Hecke functor
+H_I(V,-) cite[SS14.2--14.3]{AGKRRV}. Extend Phi_sigma by the
+identity on mathcal Q_I.calQ_I. Its module structure gives
[
H_I(V,M_sigma)simeq(Phi_sigma ⊗ id)(sigma^I(V))
simeq M_sigmaboxtimessigma^I(V),
]
-the second equivalence using that a continuous $E$-linear functor from
-$Vect_E$ is just tensoring with its value on $E$. Naturality, tensor
+the second equivalence using that a continuous E-linear functor from
+Vect_E is just tensoring with its value on E. Naturality, tensor
and unit compatibility, and collision coherence follow from the
universal symmetric monoidal evaluation system; equivalently, from
-restricting $Phi_sigma$ along the symmetric monoidal Ran-to-spectral
-action cite[S15.1]{AGKRRV}. The eigenvalue is the actual $sigma$,
+restricting Phi_sigma along the symmetric monoidal Ran-to-spectral
+action cite[S15.1]{AGKRRV}. The eigenvalue is the actual sigma,
semisimple or not.
@@ -2435,52 +2435,52 @@ semisimple or not.
% =========================================================================
We now prove the rational-coefficient form of full support. Put
-$E_0=mathbb Q_bbQ_ell$ and $E=mathbb Qbar_bbQbar_ell$, keep $k=mathbb Fbar_q$,bbFbar_q$,
+E_0=mathbb Q_bbQ_ell and E=mathbb Qbar_bbQbar_ell, keep k=mathbb Fbar_q,bbFbar_q,
and assume the curve and group satisfy the rational hypotheses in the
Introduction (Hypothesis~hyp:A plus a split dual group over
-$E_0$). Put $mathcal C_E=calC_E=Shv_{Nilp,E}(Bun_G)$, retain
-$mathcal C_0=calC_0=Shv_{Nilp,E_0}(Bun_G)$ and $Y_0$ from the Introduction,
-and let $Y$ denote the spectral prestack over $E$. All categorical tensor
+E_0). Put mathcal C_E=calC_E=Shv_{Nilp,E}(Bun_G), retain
+mathcal C_0=calC_0=Shv_{Nilp,E_0}(Bun_G) and Y_0 from the Introduction,
+and let Y denote the spectral prestack over E. All categorical tensor
products below are presentable dg categorical tensor products, so
-"extension of coefficients" means $Vect_E ⊗ _{E_0}(-)$, not
+"extension of coefficients" means Vect_E ⊗ _{E_0}(-), not
extension of the geometric field.
-Recall the standing datum: a continuous $QCoh(Y_0)$-action on
-$mathcal C_0$calC_0$ whose restriction along universal evaluation is the usual
+Recall the standing datum: a continuous QCoh(Y_0)-action on
+mathcal C_0calC_0 whose restriction along universal evaluation is the usual
multi-leg Hecke system with its tensor, permutation and collision
compatibilities. We call this the rational spectral-action datum.
[Rational spectral support]
For the rational datum above, every nonempty open-and-closed substack
-$Usubset Y_0$ satisfies
+Usubset Y_0 satisfies
[
QCoh(U)underset{QCoh(Y_0)} ⊗ mathcal C_0calC_0ne0.
]
-Consequently if $Y'_0subset Y_0$ is an open-and-closed support union
-whose complementary structure idempotent acts by zero on $mathcal C_0$,calC_0$,
-then $Y'_0=Y_0$ as spectral prestacks. In particular this applies to the
-support of any $QCoh(Y_0)$-linear primed equivalence.
+Consequently if Y'_0subset Y_0 is an open-and-closed support union
+whose complementary structure idempotent acts by zero on mathcal C_0,calC_0,
+then Y'_0=Y_0 as spectral prestacks. In particular this applies to the
+support of any QCoh(Y_0)-linear primed equivalence.
-The plan is: for $Xnemathbb P^1$bbP^1$ we compare the sheaf categories,
+The plan is: for Xnemathbb P^1bbP^1 we compare the sheaf categories,
prestacks and actions so that Theorem~thm:full-support in regime A
rules out the vanishing of any nonempty idempotent; faithful flatness of
-$E/E_0$ then descends the absence of a missing summand. For
-$X=mathbb P^1$bbP^1$ connectedness of the spectral prestack gives the result
+E/E_0 then descends the absence of a missing summand. For
+X=mathbb P^1bbP^1 connectedness of the spectral prestack gives the result
directly.
## Coefficient categories, including unbounded objects
-On a finite-type $T/k$, $Shv_L(T)$ is the ind-completion of bounded
-constructible $L$-adic complexes, $Lin{E_0,E}$. For a smooth $T$ and a
-closed conical $Nsubset T^*T$, let $mathcal P_calP_{N,L}(T)$ be the category
-of constructible perverse sheaves with singular support in $N$; the
-category $Shv_{N,L}(T)$ consists of objects whose perverse cohomology
-belongs in $Indmathcal P_calP_{N,L}(T)$ in every degree. This
+On a finite-type T/k, Shv_L(T) is the ind-completion of bounded
+constructible L-adic complexes, Lin{E_0,E}. For a smooth T and a
+closed conical Nsubset T^*T, let mathcal P_calP_{N,L}(T) be the category
+of constructible perverse sheaves with singular support in N; the
+category Shv_{N,L}(T) consists of objects whose perverse cohomology
+belongs in Indmathcal P_calP_{N,L}(T) in every degree. This
is the cohomological support definition
cite[SS E.5.1--E.5.8]{AGKRRV}; it is not necessarily the ind-completion
of support-bounded complexes. On a smooth scheme the zero-cone case is
-$QLisse_L$, with its equivalent ordinary-cohomology definition
+QLisse_L, with its equivalent ordinary-cohomology definition
cite[SS1.2.5--1.2.11]{AGKRRV}.
[Coefficient comparison]
@@ -2495,36 +2495,36 @@ Extension of coefficients induces equivalences
The first extends to locally finite-type algebraic stacks by smooth
descent. The second is symmetric monoidal and respects the t-structures
-used in the tensor-functor definition of $Y_0,Y$. On smooth charts the
+used in the tensor-functor definition of Y_0,Y. On smooth charts the
third respects cohomological nilpotent support. Free and forgetful
-functors here are $t$-exact for the ordinary and perverse t-structures
+functors here are t-exact for the ordinary and perverse t-structures
whenever these are used.
Proof.
-Ambient statement. For a finite extension $L/E_0$, coefficient
+Ambient statement. For a finite extension L/E_0, coefficient
extension and restriction preserve bounded constructibles, and on
-objects extended from $E_0$ the mapping complexes are obtained by
-tensoring the originals with $L$. Every $L$-constructible $M$ is a
-retract of $L ⊗ _{E_0}Res_{L/E_0}M$, the retraction
-being given by the separability idempotent of $L/E_0$. So the induced
+objects extended from E_0 the mapping complexes are obtained by
+tensoring the originals with L. Every L-constructible M is a
+retract of L ⊗ _{E_0}Res_{L/E_0}M, the retraction
+being given by the separability idempotent of L/E_0. So the induced
compact objects have the required mapping complexes and generate after
-retracts. Constructible $E$-complexes and their morphisms come from
+retracts. Constructible E-complexes and their morphisms come from
finite coefficient fields. These coefficient conventions and the
compact-object description are in
cite[Prop.~5.2,~Thm.~7.7,~Lemma~7.9,~Prop.~8.2,~Cor.~8.3]{HemoRicharzScholbach};
see also cite[S2]{HansenScholze}. Cohomological finiteness
cite[Lemma~8.6(1)]{HemoRicharzScholbach} holds on every finite-type
-chart over the separably closed $k$ (either algebraic
-$mathbb Q_bbQ_ell$-coefficient field), so its compactness statements apply.
+chart over the separably closed k (either algebraic
+mathbb Q_bbQ_ell-coefficient field), so its compactness statements apply.
Passing to ind-completions proves (eq:coeff-ambient).
Support condition. View (eq:coeff-ambient) as internal
-$E$-modules in $Shv_{E_0}(T)$. An internal $E$-module has the $E$-support
-condition iff its underlying $E_0$-object has the $E_0$-condition. Write
-$F$ for the free and $U$ for the forgetful functor; both are t-exact for
-ordinary and perverse cohomology. For $U$, using that a constructible
-$E$-object over a finite extension $L/E_0$ has underlying
+E-modules in Shv_{E_0}(T). An internal E-module has the E-support
+condition iff its underlying E_0-object has the E_0-condition. Write
+F for the free and U for the forgetful functor; both are t-exact for
+ordinary and perverse cohomology. For U, using that a constructible
+E-object over a finite extension L/E_0 has underlying
[
U(M_L ⊗ _L E)=
mathop{colim}_{substack{Lsubset L'subset E [L':E_0]<∞}}
@@ -2537,14 +2537,14 @@ ambient ind-categories.
For bounded constructibles, coefficient extension preserves and
reflects singular-support bounds. The defining test pairs express the
bound by universal local acyclicity; use the dualizability criterion in
-the algebraic $mathbb Q_bbQ_ell$-setting cite[Thm.~1.6]{HansenScholze}.
+the algebraic mathbb Q_bbQ_ell-setting cite[Thm.~1.6]{HansenScholze}.
The correspondence category has compositions formed from pullback,
-tensor, and $f_!$, all of which commute with coefficient extension by
+tensor, and f_!, all of which commute with coefficient extension by
the coefficient formalism and projection formula of S2 there; extending
duality data preserves universal local acyclicity. For reflection,
suppose a bounded constructible complex over a finite extension
-$L/E_0$ becomes universally locally acyclic over $E$. The relative dual
-of its $E$-extension is constructible by
+L/E_0 becomes universally locally acyclic over E. The relative dual
+of its E-extension is constructible by
cite[Prop.~3.4(ii)]{HansenScholze}; the dual, evaluation and
coevaluation maps plus both triangle identities are finite data on
fixed finite-type test schemes and their products, so by
@@ -2555,16 +2555,16 @@ acyclicity, giving reflection for each pair, hence for the bound.
Finite restriction preserves the same bound. The same arguments apply
to lissity.
-It follows that $F$ and $U$ preserve the relevant ind-perverse hearts;
-for $U$ use the colimit formula and filtered colimits, never treating a
-finite $E$-rank as a finite $E_0$-rank. For the converse, let $P$ be a
-perverse internal $E$-module with $U(P)$ in the $E_0$-ind-heart. The
-counit $F(U(P))to P$ is an epimorphism in the module heart because its
-underlying map splits as $umapsto1 ⊗ u$. Its source is in the
-$E_0$-ind-heart, which is closed under quotients by the Serre property
-of the support heart; hence $P$ is as well. So the cohomological support
-condition for an $E$-module coincides with the $E_0$-condition on its
-underlying $E_0$-object, degree by degree.
+It follows that F and U preserve the relevant ind-perverse hearts;
+for U use the colimit formula and filtered colimits, never treating a
+finite E-rank as a finite E_0-rank. For the converse, let P be a
+perverse internal E-module with U(P) in the E_0-ind-heart. The
+counit F(U(P))to P is an epimorphism in the module heart because its
+underlying map splits as umapsto1 ⊗ u. Its source is in the
+E_0-ind-heart, which is closed under quotients by the Serre property
+of the support heart; hence P is as well. So the cohomological support
+condition for an E-module coincides with the E_0-condition on its
+underlying E_0-object, degree by degree.
Scalar extension and smooth descent. We now realise these
subcategories as scalar extensions. The perverse Serre condition gives
@@ -2582,9 +2582,9 @@ Connective free modules generate the connective module category under
connective colimits, so the induced tensor t-structures agree.
Finally, take the smooth-chart limit defining sheaves on a stack.
-Internal modules commute with these limits. Equivalently, $Vect_E$ is
-the module category of the continuous monad $E ⊗ _{E_0}(-)$ on
-$Vect_{E_0}$ and is dualizable as a presentable $E_0$-linear category
+Internal modules commute with these limits. Equivalently, Vect_E is
+the module category of the continuous monad E ⊗ _{E_0}(-) on
+Vect_{E_0} and is dualizable as a presentable E_0-linear category
cite[Lemma~1.6.3]{GKRV}. Tensoring with a dualizable category preserves
limits, and the smooth pullbacks defining these limits are continuous;
chartwise support comparison therefore proves (eq:coeff-nilp).
@@ -2593,8 +2593,8 @@ chartwise support comparison therefore proves (eq:coeff-nilp).
The lemma exchanges scalar extension with the actual cohomological
categories, not with any unspecified left completion. It applies to
-$QLisse(mathbb P^1)$bbP^1)$ without identifying that category with
-$IndLisse(mathbb P^1)$.bbP^1)$.
+QLisse(mathbb P^1)bbP^1) without identifying that category with
+IndLisse(mathbb P^1).bbP^1).
## Affine families and coherent Hecke action
@@ -2605,7 +2605,7 @@ prestacks. We retain the right-t-exactness in their tensor functors.
[Spectral coefficient comparison]
There are natural identifications
- Y_0times_{Spec E_0}Spec Esimeq Y,qquad
+ Y_0times_{Spec E_0}Spec Esimeq Y,
Vect_E ⊗ _{E_0}QCoh(Y_0)simeqQCoh(Y),
with the second symmetric monoidal. These identify the universal
@@ -2613,8 +2613,8 @@ evaluation systems for every finite set of legs.
Proof.
-Let $A$ be a connective commutative $E$-algebra, $Mod_A
-=QCoh(Spec A)$. Lemma~lem:coeff-sheaves and associativity give
+Let A be a connective commutative E-algebra, Mod_A
+=QCoh(Spec A). Lemma~lem:coeff-sheaves and associativity give
[
Mod_A ⊗ _{E_0}QLisse_{E_0}(X)
simeqMod_A ⊗ _EQLisse_E(X).
@@ -2626,61 +2626,61 @@ identification by the connective assertion in
Lemma~lem:coeff-sheaves.
For the split dual group, scalar extension identifies
-$Vect_E ⊗ _{E_0}Rep_{E_0}(check G_0)$ with
-$Rep_E(check G)$ as a symmetric monoidal category with t-structure: in
+Vect_E ⊗ _{E_0}Rep_{E_0}(check G_0) with
+Rep_E(check G) as a symmetric monoidal category with t-structure: in
characteristic zero, finite-dimensional representations in the heart
generate connective objects, and irreducible highest-weight
-representations over a split reductive group are defined over $E_0$ and
+representations over a split reductive group are defined over E_0 and
remain irreducible after extension
cite[Thm.~22.2,~SS22.3--22.5,~Thm.~22.42]{Milne}. The universal
property of scalar extension therefore identifies tensor functors into
the displayed target, and right t-exactness is equivalent before and
after extension.
-Present $Y_0$ as a colimit of affine test schemes. Base change presents
+Present Y_0 as a colimit of affine test schemes. Base change presents
the left side of (eq:coeff-spectral) by base-changed affines, and
quasi-coherent categories form the corresponding limit. On an affine,
coefficient comparison is
-$Vect_E ⊗ _{E_0}Mod_Bsimeq
-Mod_{B ⊗ _{E_0}E}$. Affine pullbacks in this diagram
-are continuous, and $Vect_E$ is categorically dualizable, so tensoring
+Vect_E ⊗ _{E_0}Mod_Bsimeq
+Mod_{B ⊗ _{E_0}E}. Affine pullbacks in this diagram
+are continuous, and Vect_E is categorically dualizable, so tensoring
commutes with this limit and gives the second identification, monoidal
-included. No finite-dimensionality of $E$ over $E_0$ is required.
+included. No finite-dimensionality of E over E_0 is required.
-Evaluation at the tensor functor parametrized by $Spec A$ is unchanged
+Evaluation at the tensor functor parametrized by Spec A is unchanged
by these identifications; the assertion is natural in representations,
affine tests and maps of finite sets. It therefore identifies the whole
universal evaluation system, including tensor products and collisions
of legs.
-For $Xnemathbb P^1$bbP^1$ we can go further and compare the given rational
+For Xnemathbb P^1bbP^1 we can go further and compare the given rational
spectral action with the geometric one. This is the only case needed
-below; for $mathbb P^1$bbP^1$ we deduce support from connectedness of the
+below; for mathbb P^1bbP^1 we deduce support from connectedness of the
spectral prestack.
[Comparison of spectral actions]
-Suppose $Xnemathbb P^1$bbP^1$ and equip $mathcal C_0$calC_0$ with the rational
+Suppose Xnemathbb P^1bbP^1 and equip mathcal C_0calC_0 with the rational
spectral-action datum of Theorem~thm:rational-support. Under
Lemmas~lem:coeff-sheaves and~lem:coeff-spectral, its scalar
-extension is the established $QCoh(Y)$-action on $mathcal C_E$.calC_E$.
+extension is the established QCoh(Y)-action on mathcal C_E.calC_E.
Proof.
-First, $QLisse_E(X)=IndLisse_E(X)$ for this curve
-cite[S1.3.3,~S E.2.8]{AGKRRV}. The same equality holds over $E_0$:
-extend an object of $QLisse_{E_0}(X)$ and use the constructible lisse
-generators over $E$; forgetting coefficients takes those generators
-into the colimit closure of $E_0$-constructible lisse objects, and the
+First, QLisse_E(X)=IndLisse_E(X) for this curve
+cite[S1.3.3,~S E.2.8]{AGKRRV}. The same equality holds over E_0:
+extend an object of QLisse_{E_0}(X) and use the constructible lisse
+generators over E; forgetting coefficients takes those generators
+into the colimit closure of E_0-constructible lisse objects, and the
original object is a retract of its extension-then-forgetting, using an
-$E_0$-linear retraction $Eto E_0$ of the unit. Constructible lisse
+E_0-linear retraction Eto E_0 of the unit. Constructible lisse
objects are compact in the ambient category, so these
-$QLisse$-categories are compactly generated and dualizable; their
+QLisse-categories are compactly generated and dualizable; their
exterior-product and affine-limit evaluation constructions therefore
commute with coefficient comparison.
-Fix the geometric Satake identification over $E_0$
-cite[Thm.~14.1]{MirkovicVilonen}, and take scalar extension over $E$,
+Fix the geometric Satake identification over E_0
+cite[Thm.~14.1]{MirkovicVilonen}, and take scalar extension over E,
with the same dual-group identification, cohomological shifts and
geometric Tate-line conventions. We compare the whole finite-set
monoidal diagram giving the Hecke functors, not just objects indexed by
@@ -2690,7 +2690,7 @@ convolution pushforwards. It is perverse t-exact, so it also commutes
with intermediate extension (written as the image of the map from
perverse extension by zero to perverse direct image). So it takes the
Satake IC objects, their convolution maps, and the fusion diagrams with
-unit and symmetry constraints, to the corresponding diagrams over $E$.
+unit and symmetry constraints, to the corresponding diagrams over E.
The fiber and weight functors in geometric Satake identify the same
split dual group under this operation
cite[SS4--6,~Thm.~14.1]{MirkovicVilonen}.
@@ -2707,15 +2707,15 @@ correspondence action of cite[SS B.2.5--B.2.10]{GKRV} then compares
the continuous Hecke systems including units, composition maps and
base-change compatibilities.
-The factorisation into nilpotent sheaves with $QLisse$-valued legs goes
+The factorisation into nilpotent sheaves with QLisse-valued legs goes
through the fully faithful exterior-product functors
cite[S14.2.5]{AGKRRV}; Lemma~lem:coeff-sheaves compares these
subcategories, so the comparison holds after factorisation.
Lemma~lem:coeff-spectral identifies universal evaluation. By
assumption, restricting the extended spectral action along evaluation
-gives exactly the established coherent Hecke system over $E$. The
+gives exactly the established coherent Hecke system over E. The
equivalence of spaces of actions in cite[Thm.~8.1.4]{AGKRRV}, applied
-over the algebraically closed $E$, identifies it with the established
+over the algebraically closed E, identifies it with the established
spectral action of cite[Thm.~14.3.2]{AGKRRV}.
@@ -2726,57 +2726,57 @@ Theorem~thm:full-support.
## Descent of support and the arithmetic qualification
-For $Xnemathbb P^1$,bbP^1$, the sheaf, prestack and coherent-action
+For Xnemathbb P^1,bbP^1, the sheaf, prestack and coherent-action
comparisons are now in place.
[Detection of a clopen support]
Suppose the coefficient comparisons (eq:coeff-nilp)
and~(eq:coeff-spectral) identify the two spectral actions. If an
-open-and-closed substack $Usubset Y_0$ acts by zero on $mathcal C_0$,calC_0$,
-then $U=∅$.
+open-and-closed substack Usubset Y_0 acts by zero on mathcal C_0,calC_0,
+then U=∅.
Proof.
-Let $e_U=mathcal O_U$calO_U$ extended by zero in $QCoh(Y_0)$. Its extension
-is the analogous idempotent $e_{U_E}$, where $U_E=Utimes_{E_0}E$.
+Let e_U=mathcal O_UcalO_U extended by zero in QCoh(Y_0). Its extension
+is the analogous idempotent e_{U_E}, where U_E=Utimes_{E_0}E.
Compatibility of the actions and generation by scalar-extended objects
-show $e_{U_E}$ acts by zero on $mathcal C_E$.calC_E$.
+show e_{U_E} acts by zero on mathcal C_E.calC_E.
The curve and split group descend to a finite field, so
Theorem~thm:full-support in regime A identifies this action with
-the action on $IndCoh_{Nilp}(Y)$. Apply $e_{U_E}$ to
-$jmath_Y(mathcal O_Y)$;calO_Y)$; its value is $jmath_Y(e_{U_E})$. Full
-faithfulness of $jmath_Y$ implies $e_{U_E}=0$, hence $U_E=∅$.
-
-To descend, take any affine test $T=Spec Ato Y_0$. The inverse image
-of $U$ is an open-and-closed affine summand of $T$, cut out by an
-idempotent of $pi_0(A)$. Its base change to $A ⊗ _{E_0}E$ is
-empty, and faithful flatness of $E/E_0$ forces the idempotent to
-vanish. This holds on every affine test, so $U=∅$. No
+the action on IndCoh_{Nilp}(Y). Apply e_{U_E} to
+jmath_Y(mathcal O_Y);calO_Y); its value is jmath_Y(e_{U_E}). Full
+faithfulness of jmath_Y implies e_{U_E}=0, hence U_E=∅.
+
+To descend, take any affine test T=Spec Ato Y_0. The inverse image
+of U is an open-and-closed affine summand of T, cut out by an
+idempotent of pi_0(A). Its base change to A ⊗ _{E_0}E is
+empty, and faithful flatness of E/E_0 forces the idempotent to
+vanish. This holds on every affine test, so U=∅. No
rational-point description of components is used.
Proof. [Proof of Theorem~thm:rational-support]
-For $Xnemathbb P^1$,bbP^1$, Lemma~lem:coeff-action supplies the
+For Xnemathbb P^1,bbP^1, Lemma~lem:coeff-action supplies the
hypothesis of Lemma~lem:coeff-idempotent. An open-and-closed
localization is the image of its idempotent projection, so vanishing of
the localized category is exactly zero action of that idempotent. The
lemma gives the claim.
-If $X=mathbb P^1$,bbP^1$, the geometric spectral prestack $Y$ has one
+If X=mathbb P^1,bbP^1, the geometric spectral prestack Y has one
connected component: components are indexed by semisimple geometric
local systems cite[Prop.~3.7.2]{AGKRRV}, and the geometric 'etale
-fundamental group of $mathbb P^1$bbP^1$ is trivial (a connected finite
-'etale cover $Ctomathbb P^1$bbP^1$ of degree $n$ satisfies
-$2g(C)-2=-2n$ by Riemann--Hurwitz cite[Tag~0C1B]{Stacks}, so $n=1$).
+fundamental group of mathbb P^1bbP^1 is trivial (a connected finite
+'etale cover Ctomathbb P^1bbP^1 of degree n satisfies
+2g(C)-2=-2n by Riemann--Hurwitz cite[Tag~0C1B]{Stacks}, so n=1).
By Lemma~lem:coeff-spectral and the affine test of
-Lemma~lem:coeff-idempotent, $Y_0$ has no nonempty proper
-open-and-closed substack. The category $mathcal C_0$calC_0$ is nonzero (its
+Lemma~lem:coeff-idempotent, Y_0 has no nonempty proper
+open-and-closed substack. The category mathcal C_0calC_0 is nonzero (its
dualizing object has shifted constant restrictions on smooth charts and
zero singular support), so its only nonempty open-and-closed
localization is nonzero.
-Finally, the stipulated complement of $Y'_0$ acts by zero, so it is
+Finally, the stipulated complement of Y'_0 acts by zero, so it is
empty. In particular a given spectral-linear primed equivalence has
this property by linearity, and its specified inclusion is an equality
of prestacks.
@@ -2784,9 +2784,9 @@ of prestacks.
[Conditional rational unpriming]
-Let $Y'_0subset Y_0$ be a Frobenius-stable open-and-closed support union
+Let Y'_0subset Y_0 be a Frobenius-stable open-and-closed support union
as in Theorem~thm:rational-support. In addition to the rational
-hypotheses, assume a canonical primed trace isomorphism over $E_0$ is
+hypotheses, assume a canonical primed trace isomorphism over E_0 is
given,
[
Funct_c(Bun_{G_0}(mathbb F_q),E_0)bbF_q),E_0)
@@ -2795,12 +2795,12 @@ given,
]
intertwining the excursion and spherical Hecke actions in the same
Frobenius and Satake conventions. Then the same canonical formula and
-compatibilities hold with $Y_0^{Frob}$ in place of $(Y'_0)^{Frob}$.
+compatibilities hold with Y_0^{Frob} in place of (Y'_0)^{Frob}.
Proof.
Theorem~thm:rational-support identifies the actual inclusion
-$Y'_0subset Y_0$ with equality, so the derived fixed-point stacks,
+Y'_0subset Y_0 with equality, so the derived fixed-point stacks,
dualizing objects and ind-coherent global-sections functors agree.
Substitute.
@@ -2808,8 +2808,8 @@ Substitute.
The extra comparison map in Corollary~cor:rational-arithmetic
is not constructed here. The primed arithmetic theorem cited in
-Corollary~cor:arithmetic has coefficients $E$; its existence does
-not by itself produce a canonical map over $E_0$. An unconditional
+Corollary~cor:arithmetic has coefficients E; its existence does
+not by itself produce a canonical map over E_0. An unconditional
rational arithmetic formula still requires that map with its Frobenius,
dualizing-object, ind-coherent global-sections and excursion
compatibilities. Equality of derived fixed-point prestacks alone
@@ -2866,7 +2866,7 @@ C.~R. Acad. Sci. Paris S'er. I Math. 320 (1995), 335--340.
bibitem[DrinfeldSimpson]{DrinfeldSimpson}
V.~G.~Drinfeld and C.~Simpson,
-$B$-structures on {$G$-bundles and local triviality},
+B-structures on {G-bundles and local triviality},
Math. Res. Lett. 2 (1995), no.~6, 823--829.
bibitem[Springer]{Springer}
@@ -2917,7 +2917,7 @@ arXiv:1405.7680v3, 29 June 2016 (theorem locators use this version).
bibitem[BeraldoLinReeves]{BeraldoLinReeves}
D.~Beraldo, K.~Lin, W.~Reeves,
-Coherent sheaves, sheared {$D$-modules, and {Hochschild} cochains},
+Coherent sheaves, sheared {D-modules, and {Hochschild} cochains},
arXiv:2410.10692v2, 21 October 2024.
bibitem[Conrad]{Conrad}