diff b34550f..b1f9bc4 restricted_geometric_langlands_pos_v2.txt
diff --git a/restricted_geometric_langlands_pos_v2.txt b/restricted_geometric_langlands_pos_v2.txt
index 9659154..b94b315 100644
--- a/restricted_geometric_langlands_pos_v2.txt
+++ b/restricted_geometric_langlands_pos_v2.txt
@@ -56,26 +56,26 @@
-\allowdisplaybreaks[2]
+allowdisplaybreaks[2]
# The Restricted Geometric Langlands Equivalence in Positive Characteristic
-\maketitle
+maketitle
Abstract:
-We prove the $\mathbb Qbar_\ell$-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 $\ell\ne p$, in two regimes.
+group over a field of characteristic $p>0$, with $ellne p$, in two regimes.
-Over $\mathbb Fbar_q$, 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
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
+cite[Conj.~1.3.10]{GR}: the primed equivalence is already an equivalence
onto the entire spectral stack.
We then record three consequences: the local category on each spectral
@@ -86,7 +86,7 @@ of the spectral stack admits a nonzero Hecke eigenobject. A final section
handles rational coefficients by coefficient comparison.
-\tableofcontents
+tableofcontents
% =========================================================================
# Introduction
@@ -95,9 +95,9 @@ handles rational coefficients by coefficient comparison.
## 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
+$Bun_G(X)$ 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,
+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
@@ -115,89 +115,89 @@ 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 $\mathscr X/S$ 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,
+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,
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 $\ell\ne p$ be a prime, and set $E=\mathbb Qbar_\ell$. Let
+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
-$\mathfrak g=\Lie(G)$.
+$check G/E$ 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
-$\QLisse_E(X)\subset\Shv(X)$ is the full subcategory whose ordinary
+[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
+$E$-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) ⊗ _E\QLisse_E(X)),
+ 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
-$\sigma\in Y(E)$ 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 $V$.
+$sigmain Y(E)$ 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 $\mathcal N\subset T^*\Bun_G$; write this
-subcategory $\Shv_{\Nilp}(\Bun_G(X))$ (see \Ssec:intro-cat below
+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
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 $Y\times X$ 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 $G/k$.
-[label=(\roman*)]
-\item There is a nondegenerate $G$-invariant symmetric bilinear form
- $\kappa$ on $\mathfrak g$ whose restriction to $\Lie Z(M)$ is
+[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 $k[\Lie M]^M\to k[\Lie T_M]^{W_M}$ is an
- isomorphism for every Levi $M$ and every maximal torus $T_M\subset M$.
-\item The scheme-theoretic centraliser in $G$ of every semisimple
- element of $\mathfrak g$ is a Levi subgroup.
-\item Every nilpotent element of $\mathfrak g$ lies in $\Lie R_u(P)$ for
- some parabolic $P\subset G$, after any field extension.
+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.
-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
+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
-$\mathfrak g$ are exactly the ones geometrically conjugate into a
+$mathfrak g$frakg$ are exactly the ones geometrically conjugate into a
Cartan subalgebra, nilpotent ones the ones whose adjoint orbit closure
contains $0$.
[Regime A: finite-field descent]
There are $q=p^r$, a smooth projective geometrically connected curve
-$X_0/\mathbb F_q$, and a split connected reductive group
-$G_0/\mathbb F_q$ with $k=\mathbb Fbar_q$,
-$X=X_0 ⊗ _{\mathbb F_q}k$, $G=G_0 ⊗ _{\mathbb F_q}k$.
+$X_0/mathbb F_q$,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.
@@ -205,19 +205,19 @@ Hypothesis~hyp:common holds. No restriction on $|W_G|$ is imposed.
$k$ is any algebraically closed field of characteristic $p>0$.
Hypothesis~hyp:common holds, $p$ is very good for $G$, and
-$p\nmid|W_G|$. No finite-field descent is assumed.
+$pnmid|W_G|$. No finite-field descent is assumed.
``Very good'' is the usual root-system condition. For type $A_n$ it
-means $p\nmid n+1$; for $B_n,C_n,D_n$ it means $p\ne2$; for
-$G_2,F_4,E_6,E_7$ that $p\ne2,3$; for $E_8$ that $p\ne2,3,5$; applied on
+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
substitute for the other; both produce the identical statement of
@@ -232,11 +232,11 @@ it is worth pinning them down once, because several comparisons later
depend on the precise definition.
On the automorphic side, the global nilpotent cone
-$\mathcal N\subset T^*\Bun_G(X)$ consists of Higgs pairs $(P,A)$ with
-$A\in H^0(X,\Ad P ⊗ \omega_X)$ pointwise nilpotent. A bounded
-constructible $E$-sheaf on $\Bun_G(X)$ has nilpotent singular support
-if its wavefront set lies in $\mathcal N$; the full subcategory
-$\Shv_{\Nilp}(\Bun_G(X))\subset\Shv(\Bun_G(X))$ 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];
@@ -244,7 +244,7 @@ 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[\S\S21.2.1--21.2.5]{AGKRRV}.
+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
@@ -256,17 +256,17 @@ 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
+By cite[Main Thm.~1.3.9]{GR}, in either regime there is a
+$QCoh(Y)$-linear equivalence of the form
- \mathbb L_G^{restr}\colon
- \Shv_{\Nilp}(\Bun_G(X))
- \xrightarrow{\ \sim\ }\IndCoh_{\Nilp}(Y')
- ↪\IndCoh_{\Nilp}(Y),
+ 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.
@@ -276,42 +276,42 @@ 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
-\[
- \mathbb L_G^{restr}\colon
- \Shv_{\Nilp}(\Bun_G(X))\xrightarrow{\ \sim\ }
- \IndCoh_{\Nilp}(Y)
-\]
-is a $\QCoh(Y)$-linear equivalence.
+[
+ mathbb L_G^bbL_G^{restr}colon
+ Shv_{Nilp}(Bun_G(X))xrightarrow{ sim }
+ IndCoh_{Nilp}(Y)
+]
+is a $QCoh(Y)$-linear equivalence.
-The primed equivalence used as input comes from \cite[Main Thm.~1.3.9(i)]{GR};
+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 $GL_n$
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.
+in so doing proves cite[Conj.~1.3.10]{GR} in the two regimes.
## Consequences and rational coefficients
Three consequences are developed in Section~sec:consequences:
-[label=(\alph*)]
-\item localising at any one component $C\subset Y$ gives a nonzero
- automorphic category $\mathcal C_{s,C}$ 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 $\sigma\in Y(E)$ admits a nonzero Hecke
+[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
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 $\mathbb Q_\ell$. 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
-$\mathbb Q_\ell$ 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
@@ -321,61 +321,61 @@ spectral prestack.
The rest of the paper is organised so that each section proves one point.
-\smallskip\noindent
+smallskipnoindent
Section~sec:specialization (specialisation). We lift
-$(X,G)$ to $(\mathscr X/S,G/S)$ over the Witt trait, write
-$\mathcal B=\Bun_G(\mathscr X/S)$ and set up geometric specialisation
-$\Psi$ from the geometric generic fibre to the closed fibre. We show
+$(X,G)$ 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
-$\mathcal B_K$ with $\Psi(F)=0$, and that this vanishing persists
-after pulling back along smooth maps $a\colon T\to\mathcal B$ and after
-applying any Hecke operator $\mathrm H_V$. Crucially, the leg direction
-of the Hecke map is arbitrary; only the projection to $\mathcal B$ must
+$mathcal B_K$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
be smooth. This is the freedom later needed to cut Hecke correspondences
by a hypersurface.
-\smallskip\noindent
+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 $M\subset G$, a cocharacter $\lambda$ of its centre, and a
-change of formal frame, such that $A$ lies in $\mathfrak m[[t]]$,
-$\kappa(A,d\lambda)\ne0$, and $\ad_{A}$ is invertible on
-$\mathfrak g/\mathfrak m$. This is the input for constructing a Hecke
+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.
-\smallskip\noindent
+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.
-\smallskip\noindent
+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
-$\sum_{i=1}^m u_iv_i+f=0$.
+$sum_{i=1}^m u_iv_i+f=0$.
-\smallskip\noindent
+smallskipnoindent
Section~sec:slices (from quadric to slices). A proper
projective compactification of that quadric shows that the derived
restriction to the hypersurface is a constant Tate line; vanishing on
the quadric therefore forces vanishing on the hypersurface. A projective
incidence construction then handles higher-codimension transverse slices.
-\smallskip\noindent
+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 $\dim\mathcal B_s$, 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.
-\smallskip\noindent
-Sections~\ref{sec:consequences--sec:coefficients} apply the
+smallskipnoindent
+Sections~ref{sec:consequences--sec:coefficients} apply the
theorem to the previously listed consequences and prove the rational
coefficient statement by coefficient comparison.
@@ -386,7 +386,7 @@ 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
+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
@@ -397,181 +397,181 @@ transforms.
## The Witt trait and geometric specialisation
Let
-\[
- R=W(k),\qquad S=\Spec(R),\qquad K=Frac(R)bar,
-\]
+[
+ R=W(k),qquad S=Spec(R),qquad 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
excellent because it is a complete Noetherian local ring
-\cite[Tag~07QS]{Stacks}.
+cite[Tag~07QS]{Stacks}.
-The curve $X/k$ admits a smooth projective lift $\mathscr X/S$.
+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 ($H^2(X,\mathcal O_X)=0$); Grothendieck's existence theorem
+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
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[Tags~0DZQ,~0E7R]{Stacks}; the same lift is used in
+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
-$\mathbb Z$, 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
-\[
- \mathcal B=\Bun_G(\mathscr X/S),\qquad
- s=\Spec(k),\qquad \eta=\Spec(K).
-\]
-Then $\mathcal B=\Hom_S(\mathscr X,B_SG)$ is an algebraic stack locally
+$mathbb Z$,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).
+]
+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}: $\mathscr X/S$ is proper, flat and
+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
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 $\mathcal B$ is smooth over $S$.
+obstruction to deforming a bundle on a curve lies in $H^2(X,Ad P)=0$,
+so $mathcal B$calB$ 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
-excellent and strictly henselian \cite[Tags~0EXQ,~0323]{Stacks}. All
+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 $R,S,\mathscr X,\mathcal B$. 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 $K$.
For a finite-type $S$-scheme or algebraic space $T$, geometric
specialisation with monodromy forgotten is a functor
-\[
- \Psi_T\colon \Shv(T_K)→\Shv(T_s).
-\]
-Here $\Shv$ always means the ind-completion of bounded constructible
+[
+ 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
ordinary stalks. It extends to locally finite-type algebraic stacks by
-smooth descent, and $\Psi_T$ is defined on the whole sheaf category.
-The construction is as in \cite[\S\S4.1.1--4.1.5]{GR}: at finite
+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
+representable proper pushforward cite[S4.1.6]{GR}. These properties
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 $a\colon T\to\mathcal B$
-is smooth and $\mathcal L$ is a finite-rank lisse sheaf on $T$, then
-for every $D\in\Shv(\mathcal B_K)$
+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)$
- \Psi_T(a_K^*D ⊗ \mathcal L_K)
- \simeq a_s^*\Psi_{\mathcal B}(D) ⊗ \mathcal L_s.
+ Psi_T(a_K^*D ⊗ mathcal L_K)calL_K)
+ simeq a_s^*Psi_{mathcal B}calB(D) ⊗ mathcal L_s.calL_s.
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
+cite[Lemma~4.1.9]{GR}. Continuity makes it applicable to arbitrary
$D$ in the indicated ind-category.
## The spectral summand attached to a missing component
Set
-\[
- \mathcal C_s=\Shv_{\Nilp}(\mathcal B_s),\qquad
- \mathcal C_K=\Shv_{\Nilp}(\mathcal B_K),\qquad
- Y_K=LS^{restr}_{\check G}(\mathscr X_K).
-\]
+[
+ 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
+ Y_K=LS^{restr}_{check G}(mathscr X_K).scrX_K).
+]
Restriction to the closed fibre gives an identification
-$\QLisse(\mathscr X)\simeq\QLisse(X)$ \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
-\[
- \rho\colon\QLisse(X)\to\QLisse(\mathscr X_K).
-\]
+[
+ rhocolonQLisse(X)toQLisse(mathscr X_K).scrX_K).
+]
The corresponding map
-\[
- i\colon Y↪ Y_K
-\]
+[
+ icolon Y↪ 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
+cite[S3.1.3]{GR}. In particular the tensor product
- \mathcal D_Y:=\QCoh(Y)\underset{\QCoh(Y_K)}{ ⊗ }\mathcal C_K
+ mathcal D_Y:=calD_Y:=QCoh(Y)underset{QCoh(Y_K)}{ ⊗ }mathcal C_KcalC_K
-is a direct summand of $\mathcal C_K$.
+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 $\mathbb L_G$ across the
-specialisation. Corollary~4.3.6 and \S4.3.9 of [cite] show that the
+not merely a functor obtained by transporting $mathbb L_G$bbL_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
-\[
- \Psi_{\mathcal B}\colon \mathcal D_Y→ \mathcal C_s,
-\]
-and it is $\QCoh(Y)$-linear by \cite[Thm.~3.1.6(B) and \S4.3.10]{GR}.
+[
+ 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}.
-The characteristic-zero equivalence of \cite[Main Thm.~1.3.9(ii)]{GR},
+The characteristic-zero equivalence of cite[Main Thm.~1.3.9(ii)]{GR},
together with its spectral linearity, identifies
- \mathcal D_Y\simeq\IndCoh_{\Nilp}(Y).
+ mathcal D_YcalD_YsimeqIndCoh_{Nilp}(Y).
In positive characteristic we use only the primed version:
-$\mathcal C_s\simeq\IndCoh_{\Nilp}(Y')$ 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 $C\subset Y$ is disjoint from $Y'$, then the
+If a connected component $Csubset Y$ is disjoint from $Y'$, then the
summand
-\[
- \mathcal D_C:=\QCoh(C)\underset{\QCoh(Y)}{ ⊗ }\mathcal D_Y
-\]
-of $\mathcal D_Y$ is nonzero; $\Psi_{\mathcal B}(D)=0$ for every
-$D\in\mathcal D_C$; and there exists a nonzero $F\in\mathcal D_C$
-whose image under $\mathcal D_Y\subset\mathcal C_K\subset
-\Shv(\mathcal B_K)$ is compact, and which restricts to a bounded
-constructible complex on every finite-type smooth chart $U\to\mathcal B$.
+[
+ 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$.
Proof.
-Let $e_C\in\QCoh(Y)$ be the structure sheaf of $C$ extended by zero.
-Its action is the projection onto $\mathcal D_C$. On $\mathcal C_s$
-the same object acts by zero, because $C\cap Y'=∅$ 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 $mathcal C_s$calC_s$
+the same object acts by zero, because $Ccap Y'=∅$ and the
primed equivalence is spectral-linear. Therefore for every
-$D\in\mathcal D_C$,
-\[
- \Psi_{\mathcal B}(D)
- \simeq\Psi_{\mathcal B}(e_C\cdot D)
- \simeq e_C\cdot\Psi_{\mathcal B}(D)=0.
-\]
-Under (eq:spec-generic-equivalence), $\mathcal D_C$ is
-$\IndCoh_{\Nilp}(C)$, which is nonzero: the fully faithful
-$\QCoh(C)↪\IndCoh_{\Nilp}(C)$ sends $\mathcal O_C\ne0$ to
-a nonzero object \cite[\S\S1.3.1--1.3.4]{GR}.
-
-The category $\mathcal D_Y$ is compactly generated
-\cite[\S3.1.7]{GR}. In the binary decomposition
-$\mathcal D_Y\simeq\mathcal D_C\times\mathcal D_{Y∖ C}$ both
+$Dinmathcal D_C$,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
+a nonzero object cite[SS1.3.1--1.3.4]{GR}.
+
+The category $mathcal D_Y$calD_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
projections preserve compactness, because both their right adjoints
(the inclusions of the summands) are continuous. Project a set of
-compact generators of $\mathcal D_Y$ under $e_C$; at least one
+compact generators of $mathcal D_Y$calD_Y$ under $e_C$; at least one
projection is nonzero, since otherwise the colimit closure would give
-$\mathcal D_C=0$. Choose such an $F$.
+$mathcal D_C=0$.calD_C=0$. Choose such an $F$.
-Compactness in $\mathcal C_K$ is the same argument: $F$ is compact in
-$\mathcal D_Y$, which is a summand of $\mathcal C_K$, and the inclusion
+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$, and the inclusion
of a summand preserves compactness. The inclusion
-$\mathcal C_K↪\Shv(\mathcal B_K)$ preserves compactness
-by \cite[Thm.~1.1.7]{GR}. Finally, a compact sheaf on an algebraic
+$mathcal C_K↪calC_K↪Shv(mathcal B_K)$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}.
+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
+$e_C$. If $Y'ne Y$ then such a $C$ exists because $Y'$ is a union of
open-and-closed components of $Y$.
@@ -582,104 +582,104 @@ $Y'=Y$ there is nothing to prove.
## Hecke transforms of a specialising object
-Let $V$ be a finite-dimensional representation of $\check G$ and let
-\[
- \mathrm H_V\colon\mathcal C_K→
- \Shv(\mathcal B_K\times_K\mathscr X_K)
-\]
+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)
+]
be the one-leg Hecke functor, normalised so that it realises the
spectral action on the spectral side. The correspondence and kernel
are pinned down in Section~sec:hecke; here we only need two
facts about the universal family.
The universal objects are
-\[
- \mathcal E_{V,s}\in\QCoh(Y) ⊗ \QLisse(X),\qquad
- \mathcal E_{V,K}\in\QCoh(Y_K) ⊗ \QLisse(\mathscr X_K).
-\]
-On an affine test $T\to Y$ the restriction of $\mathcal E_V$ is the
+[
+ mathcal E_calE_{V,s}inQCoh(Y) ⊗ QLisse(X),qquad
+ 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
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 $\mathrm H_V$ on the spectral side
-\cite[\S14.3.3]{AGKRRV}.
+the exterior product gives $mathrm H_V$rmH_V$ on the spectral side
+cite[S14.3.3]{AGKRRV}.
By definition of $i$,
- (i^* ⊗ \id)(\mathcal E_{V,K})
- \simeq(\id ⊗ \rho)(\mathcal E_{V,s}).
+ (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,~\S\S E.2.6--E.2.7]{AGKRRV}, hence dualizable as a
+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}:
+The second, distinct input is cite[Rem.~14.3.9]{AGKRRV}:
- \mathcal E_{V,s}\in\QCoh(Y) ⊗ \IndLisse(X),
+ mathcal E_calE_{V,s}inQCoh(Y) ⊗ IndLisse(X),
-where $\IndLisse(X)=Ind(Lisse(X))$ and
+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
-$X=\mathbb P^1$. 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 $\mathcal D_C ⊗ \IndLisse(X)$. After applying
-$\id ⊗ \rho$ and the exterior-product functor, it is precisely
-$\mathrm H_V(F)$ by (eq:spec-universal-family). Elementary tensors
-generate $\mathcal D_C ⊗ \IndLisse(X)$ under colimits, and bounded
+$IndLisse(X)$ 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 $\mathrm H_V(F)$ lies in the stable
+finite-rank lisse sheaves. Therefore $mathrm H_V(F)$rmH_V(F)$ lies in the stable
colimit closure of objects
- D\boxtimes\mathcal L_K,\qquad D\in\mathcal D_C,
+ Dboxtimesmathcal L_K,calL_K,qquad Dinmathcal D_C,calD_C,
-where $\mathcal L_K$ is the restriction of a finite-rank lisse sheaf
-$\mathcal L_S$ on $\mathscr X$.
+where $mathcal L_K$calL_K$ is the restriction of a finite-rank lisse sheaf
+$mathcal L_S$calL_S$ on $mathscr X$.scrX$.
-Let $a\colon T\to\mathcal B$ be smooth, with $T$ a finite-type $S$-scheme
-or algebraic space, and let $\ell_T\colon T\to\mathscr X$ be any
+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 $V$,
-\[
- \Psi_T(a_K^*F)=0,
- \qquad
- \Psi_T((a,\ell_T)_K^*\mathrm H_V(F))=0.
-\]
+[
+ 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 finite trait replacements described above.
Proof.
-The first equality follows from $\Psi_{\mathcal B}(F)=0$ and smooth base
+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
-\[
- (a,\ell_T)_K^*(D\boxtimes\mathcal L_K)
- =a_K^*D ⊗ \ell_{T,K}^*\mathcal L_K.
-\]
-The sheaf $\ell_T^*\mathcal L_S$ is lisse on $T$, regardless of whether
-$\ell_T$ is smooth. Formula~(eq:spec-lisse-tensor) and
-$\Psi_{\mathcal B}(D)=0$ therefore give
-\[
- \Psi_T(a_K^*D ⊗ \ell_{T,K}^*\mathcal L_K)
- \simeq a_s^*\Psi_{\mathcal B}(D) ⊗ \ell_{T,s}^*\mathcal L_s=0.
-\]
+[
+ (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
+[
+ 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 $\mathrm H_V(F)$. 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 $a\colon T\to\mathcal B$; 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
freely over the legacy of the modification, and we do not control
@@ -695,106 +695,106 @@ nonzero incoming covector and both transverse derivatives.
% =========================================================================
Section~sec:specialization reduced the main theorem to the following
-local problem. Suppose $P\in\mathcal B_s(k)$ is a bundle and
-$A\in T^*_P\mathcal B_s$ is a non-nilpotent Higgs field: a section
-of $\Ad P ⊗ \omega_X$ whose values are not all nilpotent. We want to
+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,
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 $x\in X(k)$
+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 $a(t)\in\mathfrak g[[t]]$. Chevalley
-restriction identifies the nilpotent cone in $\mathfrak g$ with
-$\chi^{-1}(\chi(0))$, where $\chi\colon\mathfrak g\to\mathfrak g/\!/G$ is
+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 $\chi(a(0))\ne\chi(0)$.
+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
+Hypothesis~hyp:A or Hypothesis~hyp:B, and let $kappa$ be the
invariant symmetric form of Hypothesis~hyp:common(i). Let
-$a(t)\in\mathfrak g[[t]]$ satisfy $\chi(a(0))\ne\chi(0)$. Then after a
+$a(t)inmathfrak g[[t]]$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
-$M\subset G$ and a cocharacter $\lambda\colon\mathbb G_m\to Z(M)$ such
+$Msubset G$ and a cocharacter $lambdacolonmathbb G_mbbG_mto Z(M)$ such
that
- a(t)\in\mathfrak m[[t]],\qquad \kappa(a(0),d\lambda)\ne0,\qquad
- ad_{a(0)}\colon\mathfrak g/\mathfrak m
- \xrightarrow{\ \sim\ }\mathfrak g/\mathfrak m,
+ a(t)inmathfrak m[[t]],frakm[[t]],qquad kappa(a(0),dlambda)ne0,qquad
+ ad_{a(0)}colonmathfrak g/frakg/mathfrak mfrakm
+ xrightarrow{ sim }mathfrak g/frakg/mathfrak m,frakm,
-where $\mathfrak m=\Lie M$ and $d\lambda=d\lambda(1)\in\mathfrak g$ is the
-image of $1\in\Lie(\mathbb G_m)=k$. The subgroup $M$ may equal $G$; the
+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
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, $\mathfrak t$ is nondegenerate, and every element is conjugate
+(Ssec:lie-basic): $kappa$ pairs root vectors with their opposite
+partners, $mathfrak t$frakt$ 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
+(Ssec:lie-A and~Ssec:lie-B). Third, a formal smoothing
argument lifts the constant-term reduction to the full formal arc
-(\Ssec:lie-formal).
+(Ssec:lie-formal).
## Root pairings and constant terms
-Fix a maximal torus $T\subset G$. Set
-$Λ=X_*(T)$, $Λ^\vee=X^*(T)$, $\mathfrak t=\Lie T=Λ ⊗ k$.
-For a root $\alpha$ write $d\alpha\colon\mathfrak t\to k$ for the
-differential of the character $\alpha$ and $d\alpha^\vee\in\mathfrak t$
+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$
for the differential of the coroot.
-The form $\kappa$ is nondegenerate on $\mathfrak t$. For each root $\alpha$
-there is a scalar $c_\alpha\in k^\times$ 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(d\alpha^\vee,h)=c_\alpha\,d\alpha(h)\qquad(h\in\mathfrak t).
+ kappa(dalpha^vee,h)=c_alpha,dalpha(h)qquad(hinmathfrak t).frakt).
-In particular neither $d\alpha$ nor $d\alpha^\vee$ vanishes. Every element
-of $\mathfrak g$ is conjugate into the Lie algebra of a Borel subgroup
+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
-$\mathfrak t$.
+$mathfrak t$.frakt$.
Proof.
Under the $T$-weight decomposition
-$\mathfrak g=\mathfrak t ⊕ ⊕_\alpha\mathfrak g_\alpha$,
-invariance of $\kappa$ makes $\mathfrak t$ orthogonal to each root space
-and makes $\mathfrak g_\alpha$ pair only with $\mathfrak g_{-\alpha}$.
-Nondegeneracy of $\kappa$ on $\mathfrak g$ then gives nondegeneracy on
-$\mathfrak t$ and on each root pairing. Choose compatible root vectors
-$e_\alpha,e_{-\alpha}$ with $[e_\alpha,e_{-\alpha}]=d\alpha^\vee$; this
+$mathfrak g=frakg=mathfrak tfrakt ⊕ ⊕_alphamathfrak g_frakg_alpha$,
+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
-isogeny type \cite[\S\S8.1.1--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
-$d\alpha,d\alpha^\vee$ vanishes iff the other does. Simultaneous
-vanishing would put $\alpha\in pΛ^\vee$ and $\alpha^\vee\in pΛ$,
-so $<\alpha,\alpha^\vee>=2$ would be divisible by $p^2$,
+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 $p^2$,
impossible. This argument uses characters, not just their differentials,
and works in characteristic two.
For the Borel-conjugation statement: consider the incidence morphism
-\[
- G\times^B\mathfrak b→\mathfrak g,\qquad
- [g,x]\mapsto\Ad(g)x,
-\]
-where $B\supset T$ is a Borel subgroup and $\mathfrak b=\Lie B$. Its
-source is closed in $(G/B)\times\mathfrak g$, so the morphism is proper.
-Choose $c\in\mathfrak t$ outside the root differential hyperplanes.
+[
+ Gtimes^Bmathfrak b→frakb→mathfrak g,frakg,qquad
+ [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
negative-root tangent directions map isomorphically to the negative root
-spaces, and $\mathfrak b$ supplies the rest), so the morphism is dominant,
+spaces, and $mathfrak b$frakb$ supplies the rest), so the morphism is dominant,
hence surjective.
For the semisimple-conjugation statement: write a semisimple element of
-$\mathfrak b$ as $c+u$ with $c\in\mathfrak t$ and $u$ in the positive
+$mathfrak b$frakb$ as $c+u$ with $cinmathfrak t$frakt$ and $u$ in the positive
root Lie algebra. Conjugate by positive-root groups in increasing height.
-If $d\alpha(c)\ne0$, conjugation by the $\alpha$-root group changes the
-$\alpha$ coefficient by $-z\,d\alpha(c)$ 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
@@ -802,51 +802,51 @@ $(semisimple+nilpotent)$ uniqueness forces $u_0=0$.
Choose a closed faithful representation $G↪GL(V)$.
-Its differential realises $\mathfrak g$ as a restricted Lie subalgebra of
-$\End(V)$, with $x\mapsto x^{[p]}$ equal to matrix $p$th power. We use
+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
the ordinary matrix Jordan decomposition; the two lemmas below prove
-its two parts lie in $\mathfrak g$ in each regime; compare
-\cite[\S4.4, Thm.~4.4.20]{Springer}.
+its two parts lie in $mathfrak g$frakg$ in each regime; compare
+cite[S4.4, Thm.~4.4.20]{Springer}.
-Suppose $a_0=s+n\in\mathfrak g$ where $s,n\in\mathfrak g$, $s$ is
+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 $\chi(a_0)\ne\chi(0)$. After conjugation, $s$ is
-a nonzero element of $\mathfrak t$, $M=Z_G(s)$ is a Levi subgroup,
-$a_0\in\mathfrak m$, and $\ad_{a_0}$ is invertible on
-$\mathfrak g/\mathfrak m$. Moreover for every $\lambda\colon\mathbb G_m
-\to Z(M)$,
+$[s,n]=0$. 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,d\lambda)=\kappa(s,d\lambda).
+ kappa(a_0,dlambda)=kappa(s,dlambda).
Proof.
-The root-pairing lemma conjugates $s$ into $\mathfrak t$. If $s=0$ then
-$n=a_0$ is nilpotent. The projection $\mathfrak b\to\mathfrak t$ preserves
+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 $\mathfrak t$, its
+$p$-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
-$\chi(a_0)\ne\chi(0)$. Thus $s\ne0$.
+$chi(a_0)nechi(0)$. Thus $sne0$.
By Hypothesis~hyp:common(iii), $M=Z_G(s)$ is a Levi subgroup. Its
-Lie algebra is $\ker\ad_s$, which contains $a_0$. On
-$\mathfrak g/\mathfrak m$ 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.
+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 $\mathfrak m$ by
+nonzero differentials and $kappa$ is nondegenerate on $mathfrak m$frakm$ 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 $\mathfrak t$, so
-$\kappa(n,d\lambda)=0$ for $d\lambda\in\mathfrak t$ fixed by $M$; by
+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
invariance the same holds before conjugation.
@@ -857,58 +857,58 @@ diverge; everything else is common.
## Regime A: a lattice argument for actual central cocharacters
-Assume Hypothesis~hyp:A, so $k=\mathbb Fbar_q$. Two facts
+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 $\mathfrak g$; they start
-life only in an ambient $\End(V)$. Second, some $\lambda\in
-X_*(Z(M))$ has $\kappa(s,d\lambda)\ne0$. Under Hypothesis~hyp:A,
+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 $\mathbb F_{p^h}$; choose $e$ divisible by $h$ and large enough
+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 $s\in\mathfrak g$ as the $p^e$th restricted
-power of an element of $\mathfrak g$. Consequently $n\in\mathfrak g$ too.
-Lemma~lie:constant-parts then supplies a nonzero $s\in\mathfrak t$,
+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.
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
+Assume $kappa$ and Hypothesis~hyp:common(i),~(iii). Then every Levi
Cartan determinant is a $p$-adic unit. Moreover, for every Levi
-$M\supset T$,
+$Msupset T$,
- span_k\{d\lambda:\ \lambda\in X_*(Z(M))\}
- =\mathfrak z(\mathfrak m),
+ span_k{dlambda: lambdain X_*(Z(M))}
+ =mathfrak z(frakz(mathfrak m),frakm),
-where the $d\lambda\in\mathfrak t$ are differentials of actual
+where the $dlambdainmathfrak t$frakt$ are differentials of actual
central cocharacters of $Z(M)$.
Proof.
-Fix a positive system and let $I=\{\alpha_1,...,\alpha_r\}$ be the
+Fix a positive system and let $I={alpha_1,...,alpha_r}$ be the
simple roots of $M$. Set
-$J_I=span_k\{d\alpha_i^\vee\}\subset\mathfrak t$. A
-central element of $\mathfrak m$ has no root-space component because all
+$J_I=span_k{dalpha_i^vee}subsetmathfrak t$.frakt$. A
+central element of $mathfrak m$frakm$ has no root-space component because all
root differentials are nonzero (Lemma~lie:root-pairing), so
(eq:root-coroot-pairing) gives
-\[
- \mathfrak z(\mathfrak m)=∩_i\ker d\alpha_i=J_I^\perp.
-\]
+[
+ mathfrak z(frakz(mathfrak m)=∩_ifrakm)=∩_iker dalpha_i=J_I^perp.
+]
The form on this center is nondegenerate by hypothesis, and on
-$\mathfrak t$ by Lemma~lie:root-pairing; nondegeneracy on $J_I$
+$mathfrak t$frakt$ by Lemma~lie:root-pairing; nondegeneracy on $J_I$
follows.
-Define integral maps $U\colonΛ\to\mathbb Z^r$,
-$y\mapsto(<\alpha_i,y>)_i$, and $V\colon\mathbb Z^r\to
-Λ$, $e_j\mapsto\alpha_j^\vee$. Their composite is the Cartan matrix
+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
+(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).
@@ -916,51 +916,51 @@ $\ker V_k$, again by nondegeneracy on $J_I=\im V_k$. Hence
If some Levi Cartan determinant were divisible by $p$, the standard
determinant list
-\cite[Thm.~C.56,~\S24(c)]{Milne}
-\[
+cite[Thm.~C.56,~S24(c)]{Milne}
+[
{c|ccccccccc}
- type&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2\\ \hline
- \det&r+1&2&2&4&3&2&1&1&1
+ type&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2 hline
+ det&r+1&2&2&4&3&2&1&1&1
-\]
+]
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
(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 $r\times N_T$
-and $N_T\times r$. By (eq:cartan-ranks) both reductions have rank
+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
-\[
- \det C_I
- =\sum_{\substack{J\subset\{1,...,N_T\}\\|J|=r}}
- \det(U_{:,J})\,\det(V_{J,:}),
-\]
-each summand divisible by $p^2$, contradicting $\det C_I=p$. This uses
+[
+ 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
the full lattices, with no isogeny-type assumption.
-For an arbitrary Levi set $d_I=\det C_I$ and define
-$\Pi_I=d_I\id_Λ-Vadj(C_I)U\in\End(Λ)$. Then
-$U\Pi_I=0$; the lattice $\ker U$ is exactly the set of integral
-cocharacters into $Z(M)$. For $z\in\ker U_k$, $\Pi_{I,k}(z)=d_I z$; since
-$d_I$ is invertible in $k$, the differentials of $\ker U$ span
-$\ker U_k=\mathfrak z(\mathfrak m)$, 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 $s,n\in\mathfrak g$ have been constructed. Lemma~lie:constant-parts
-gives $s\in\mathfrak z(\mathfrak m)∖\{0\}$ and the quotient
-invertibility. Since $\kappa$ is nondegenerate on $\mathfrak z(\mathfrak
-m)$ by Hypothesis~hyp:common(i),
-Lemma~lie:lattice produces an actual $\lambda\in X_*(Z(M))$
-with $\kappa(s,d\lambda)\ne0$. Orthogonality
+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
(eq:nilpotent-orthogonality) gives the required
-$\kappa(a_0,d\lambda)\ne0$. This used no Weyl-order assumption.
+$kappa(a_0,dlambda)ne0$. This used no Weyl-order assumption.
## Regime B: Weyl-order averaging
@@ -968,94 +968,94 @@ $\kappa(a_0,d\lambda)\ne0$. This used no Weyl-order assumption.
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 $\mathbb F_p$-linear additive polynomial.
+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 $V_0\subset k$ be the
-finite-dimensional $\mathbb F_p$-span of the eigenvalues of $s$, and let
-$V_e=\{v^{p^e}:v\in V_0\}$. The inverse of Frobenius $V_e\to V_0$ is
-$\mathbb F_p$-linear. If $V_0=0$ then $s=0$ and $a_0=n\in\mathfrak g$,
-contradicting $\chi(a_0)\ne\chi(0)$
-(Lemma~lie:constant-parts). Hence $r=\dim_{\mathbb F_p}V_e\ge1$ and
-we may assume $V_0\ne0$.
-
-Take a basis $\beta_1,...,\beta_r$ of $V_e$ over $\mathbb F_p$. The
-Moore matrix $(\beta_i^{p^j})_{1\le i\le r,\ 0\le j<r}$ 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
+$n^{p^e}=0$, 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
+$Q(z)=sum_{j=0}^{r-1}c_j z^{p^j}$ agreeing with inverse-Frobenius on
$V_e$. Applying it to the diagonalisable matrix $s^{p^e}$ gives
-\[
- s=Q(a_0^{p^e})=\sum_{j=0}^{r-1}c_j a_0^{[p^{e+j}]}\in\mathfrak g.
-\]
-Hence $n\in\mathfrak g$ as well. Lemma~lie:constant-parts now
-supplies $s\in\mathfrak t∖\{0\}$, $M=Z_G(s)$ and the quotient
+[
+ 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
invertibility.
-To find $\lambda$: let $W_M=N_M(T)/T$ be the Weyl group of $M$ and set
-$h_M=|W_M|$. Since $W_M\subset W_G$, Hypothesis~hyp:B implies
+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
-\[
- e_M=\frac1{h_M}\sum_{w\in W_M}w\colon\mathfrak t\to\mathfrak t
-\]
-is a self-adjoint projection for $\kappa$, so $\kappa$ is nondegenerate
-on $\mathfrak t^{W_M}=\im e_M$. That subspace contains $s$ because $M$
-centralises $s$. In fact $\mathfrak t^{W_M}=\mathfrak z(\mathfrak m)$:
-the reflection formula is $s_\alpha(h)=h-d\alpha(h)d\alpha^\vee$
-and $d\alpha^\vee\ne0$ by Lemma~lie:root-pairing. Thus nondegeneracy
-on $\mathfrak z(\mathfrak m)$ again follows from $\kappa$ alone, without
+[
+ 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
needing Hypothesis~hyp:common(i)'s second clause.
-For an integral cocharacter, choose a basis $\mu_1,...,\mu_{N_T}$ of
-$Λ$ and set $\nu_i=\sum_{w\in W_M}w\mu_i\inΛ^{W_M}$. Their
-differentials span $\mathfrak t^{W_M}$ because $d\nu_i=h_M e_M(d\mu_i)$.
-The integral reflection formula $s_\alpha(\lambda)=\lambda-
-<\alpha,\lambda>\alpha^\vee$ and torsion-freeness of
+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
-$\mathfrak t^{W_M}$ now gives an integral $\lambda\in X_*(Z(M))$ with
-$\kappa(s,d\lambda)\ne0$, and (eq:nilpotent-orthogonality) gives
-$\kappa(a_0,d\lambda)\ne0$.
+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
-$p\nmid|W_G|$ is specific to Hypothesis~hyp:B.
+$pnmid|W_G|$ 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 $a_0=a(0)\in\mathfrak m$ with
-$\kappa(a_0,d\lambda)\ne0$ and $\ad_{a_0}$ invertible on
-$\mathfrak g/\mathfrak m$. The remaining step moves the entire formal
-arc $a(t)$ into $\mathfrak m[[t]]$.
+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]]$.
Let $G$ be a connected reductive group over an algebraically closed
-field $k$, $M\subset G$ a Levi subgroup, $a(t)\in\mathfrak g[[t]]$ with
-$a_0=a(0)\in\mathfrak m$. If $\ad_{a_0}$ is invertible on
-$\mathfrak g/\mathfrak m$, then there is $g(t)\in G(k[[t]])$ with
-$g(0)=1$ and $\Ad(g(t))a(t)\in\mathfrak m[[t]]$.
+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]]$ modulo $t^r$, $r\ge1$, and let $c_r\in
-\mathfrak g/\mathfrak m$ be the class of its $t^r$-coefficient. Choose
-$D\in\mathfrak g$ 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
-$g_r(t)\equiv1+t^rD\pmod{t^{r+1}}$; formally, this is a lift of a point
+$mathfrak m[[t]]$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
leaves all lower coefficients fixed, so it places the arc in
-$\mathfrak m[[t]]$ 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 $g_r\equiv1\pmod{t^r}$. The affine group $G$ then
-produces a single $g(t)\in G(k[[t]])$ with $g(0)=1$ and
-$\Ad(g(t))a(t)\in\mathfrak m[[t]]$. 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]
@@ -1064,17 +1064,17 @@ 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
-$\kappa(a_0,d\lambda)\ne0$. Apply Lemma~lie:formal after the
+$kappa(a_0,dlambda)ne0$. 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 $\mathfrak g$ injects into
-$\mathfrak g/\mathfrak m$, 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 $mathfrak g$frakg$ 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.
@@ -1086,14 +1086,14 @@ allowed characteristic.
% =========================================================================
Section~sec:lie produced Levi data. Now we turn it into a
-modification of a bundle. Let $P\in\mathcal B_s(k)$ and let
-$A\in T^*_P\mathcal B_s$ be non-nilpotent. We will choose a point
-$x\in X(k)$, a cocharacter $\lambda$, and a modification
-$(P,P',x)\in H^\lambda_s(k)$ such that:
-[label=(\roman*)]
-\item the moving-point displacement $(d\lambda/t)|_x$ pairs nonzero
- against $A$ to give a leg covector $\xi\ne0$;
-\item the fixed-input cotangent section $S_A$ and the fixed-output one
+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:
+[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
derivatives on their respective tangent spaces.
@@ -1101,41 +1101,41 @@ These are the "transversality" and "isolation" conditions.
## Correspondence and kernels
-We keep the trait $S$, the curve $\mathscr X/S$ and the bundle stack
-$\mathcal B$ from Section~sec:specialization. A one-point Hecke
-modification is a quadruple $(P,P',x,\alpha)$ where $x\in\mathscr X$
-and $\alpha$ identifies $P'$ and $P$ away from the graph of $x$. Write
-\[
- p(P,P',x,\alpha)=P,\qquad o(P,P',x,\alpha)=P',\qquad
- \ell_H(P,P',x,\alpha)=x.
-\]
+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
+[
+ p(P,P',x,alpha)=P,qquad o(P,P',x,alpha)=P',qquad
+ 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, $m=< 2\rho,\mu>$
-where $2\rho$ is the sum of positive roots, $\mu^\vee=-w_0\mu$, and
-$\mathcal Z=\mathcal B\times_S\mathscr X$. Denote the exact-position
-correspondence by $H=H^\lambda$, and set
-\[
- \widetilde p=(p,\ell_H)\colon H\to\mathcal Z,\qquad
- q=(o,\ell_H)\colon H\to\mathcal Z.
-\]
-Both $\widetilde p$ and $q$ forget some of the modification data.
+$lambda$ 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
+ q=(o,ell_H)colon Htomathcal Z.calZ.
+]
+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
+Both $widetilde p$ and $q$ are representable, separated and smooth of
relative dimension $m$. The correspondence $H$ is open in a bounded
-correspondence $\overline H$ for which both projections to $\mathcal Z$
+correspondence $overlineoverline H$H$ for which both projections to $mathcal Z$calZ$
are representable and proper. On the output side the bound on
-$\overline H$ has dominant label $\mu^\vee$.
+$overlineoverline H$H$ 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)=\overline q_{K,*}\mathcal I_K,
+ mathrm H_V(F)=rmH_V(F)=overlineoverline q_q_{K,*}mathcal I_K,calI_K,
-with $\mathcal I_K$ on $H_K$ equal to $p_K^*F$ tensored with a fixed
+with $mathcal I_K$calI_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).
@@ -1144,8 +1144,8 @@ Proof.
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[\S\S1.2,~2.1]{RicharzScholbach}. We use these
-models by base change from $\mathbb Z$.
+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
@@ -1159,11 +1159,11 @@ 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
+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
-$< 2\rho,\mu^\vee>=< 2\rho,\mu>=m$. Separatedness
+$< 2rho,mu^vee>=< 2rho,mu>=m$. 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.
@@ -1175,9 +1175,9 @@ identity; no quotient by automorphisms of the fixed bundle is taken.
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
+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[\S\S4.2.1--4.2.4]{GR}. Relabelling by $\mu^\vee$ if
+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
@@ -1187,45 +1187,45 @@ intersection complex.
## The cotangent identity and the transverse section
-On the special fibre a cotangent vector to $\mathcal B_s$ at $P$ is an
+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
+Serre duality cite[Tag~0BS4]{Stacks}, together with $kappa$, identify
this space as
-\[
- T_P^*\mathcal B_s=H^1(X,\mathfrak g_P)^*=H^0(X,\mathfrak g_P ⊗ \omega_X),
-\]
-where $\mathfrak g_P$ is the adjoint bundle. We call such a vector $A$ a
+[
+ 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
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 $P\in\mathcal B_s(k)$ and let $A\in T^*_P\mathcal B_s$ be
-non-nilpotent. There exist a point $x\in X(k)$, a cocharacter $\lambda$,
-a modification $h\in H^\lambda_s(k)$ from $P$ to a bundle $P'$, and a
+Let $Pinmathcal B_s(k)$calB_s(k)$ and let $Ain T^*_Pmathcal B_s$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
covector
-\[
- \theta=(A',\xi)\in T^*_{(P',x)}(\mathcal B_s\times X),\qquad \xi\ne0,
-\]
+[
+ theta=(A',xi)in T^*_{(P',x)}(mathcal B_scalB_stimes X),qquad xine0,
+]
with $A'=A$ away from $x$, satisfying
- dp_h^*A=dq_h^*\theta.
+ dp_h^*A=dq_h^*theta.
Moreover, on the fixed-side fibres
-\[
- H^{in}_{P,x}=\widetilde p^{-1}(P,x),\qquad
+[
+ H^{in}_{P,x}=widetilde p^{-1}(P,x),qquad
H^{out}_{P',x}=q^{-1}(P',x),
-\]
+]
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 $Ω_{H_s/\mathcal Z_s,q}$; the restriction of
-$o^*A'$ to the $\widetilde p$-relative tangent bundle defines a section
+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})^*,\\
- (dS_{A'})_h&\colon T_hH^{out}_{P',x}
+ (dS_A)_h&colon T_hH^{in}_{P,x}
+ → (T_hH^{out}_{P',x})^*,
+ (dS_{A'})_h&colon T_hH^{out}_{P',x}
→ (T_hH^{in}_{P,x})^*
are isomorphisms. In particular each section has an isolated reduced
@@ -1233,43 +1233,43 @@ zero at $h$.
Proof.
-Choice of $\lambda$. Since $A$ is non-nilpotent, choose
-$x\in X(k)$ 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
-$\chi(a(0))\ne\chi(0)$. Proposition~lie:input then supplies a
+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
-$\lambda\colon\mathbb G_m\to Z(M)$ with
+$lambdacolonmathbb G_mbbG_mto Z(M)$ with
- a(t)\in\mathfrak m[[t]],\quad
- \ad_{a(0)} invertible on \mathfrak g/\mathfrak m,\quad
- \kappa(a(0),d\lambda)\ne0.
+ a(t)inmathfrak m[[t]],frakm[[t]],quad
+ ad_{a(0)} invertible on mathfrak g/frakg/mathfrak m,frakm,quad
+ 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 $\lambda$
+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
+$a(t),dt$; 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[\S\S20.4--20.6]{AGKRRV}. What follows makes both derivative
+cite[SS20.4--20.6]{AGKRRV}. What follows makes both derivative
assertions explicit.
Tangent spaces. In the input frame, put
-$L=\mathfrak g[[t]]$, $L'=\Ad(\lambda(t))L$, $I=L\cap L'$. Let
-$\mathfrak g_j$ be the weight-$j$ space of $\lambda$; the integers $j$
-are genuine characters of $\mathbb G_m$, not reduced mod $p$. Then
+$L=mathfrak g[[t]]$,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'=⊕_j t^j\mathfrak g_j[[t]],\quad
- P_1:=L/I=⊕_{j>0}\mathfrak g_j ⊗ k[[t]]/(t^j),\quad
- P_2:=L'/I=⊕_{j<0}\mathfrak g_j ⊗ (t^jk[[t]]/k[[t]]).
+ 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
+ P_2:=L'/I=⊕_{j<0}mathfrak g_jfrakg_j ⊗ (t^jk[[t]]/k[[t]]).
-An infinitesimal disc automorphism $1+ε D$, $D\in L$, acts on
+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
+$I$. 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,qquad
T_hH^{out}_{P',x}=P_2.
The same argument with output fixed gives the second identity. The
@@ -1278,82 +1278,82 @@ canonical isomorphism, no exponential map needed.
There is a perfect pairing
- \beta\colon P_1\times P_2\to k,\qquad
- \beta(D,C)=Res_{t=0}\kappa(D,C)\,dt.
+ betacolon P_1times P_2to k,qquad
+ beta(D,C)=Res_{t=0}kappa(D,C),dt.
-Indeed, $L$ is its own residue-annihilator on $\mathfrak g((t))$, and
-invariance of $\kappa$ gives the same statement for $L'$. So $\beta$
-descends to the quotients. Concretely, $\kappa$ pairs $\mathfrak g_j$
-perfectly with $\mathfrak g_{-j}$ and the coefficient of $t^a$ in the
-first space, for $0\le a<j$, pairs with the coefficient of $t^{-a-1}$
+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 $p$.
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 $Res\kappa(a,C)\,dt$. For output-fixed tangent
-$C\in L'$ the residue is zero because $a\in L'$ and $L'$ is self-annihilating.
+$A$ 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 $\theta$.
+$dq_h^*theta$ for a unique covector $theta$.
-The bundle component of $\theta$ is $A'$. Indeed, choose $x'\ne x$ and
+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
-$H^1(X,\mathfrak g_{P'})$ (Serre vanishing for $N\gg0$), and the
+$H^1(X,mathfrak g_frakg_{P'})$ (Serre vanishing for $Ngg0$), and the
pairings agree on these directions since $A=A'$ near $x'$. Write
-$\theta=(A',\xi)$.
+$theta=(A',xi)$.
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
+The output transition is $g_ε=lambda(t-ε)$ and to first
order in $ε$,
-$(\delta g)g^{-1}=-d\lambda/t$. Evaluating the cotangent identity on
+$(delta g)g^{-1}=-dlambda/t$. Evaluating the cotangent identity on
this motion gives
-\[
- 0=-\kappa(a(0),d\lambda)+\xi(∂_t),
- \qquad \xi=\kappa(a(0),d\lambda)\,dt\big|_x\ne0,
-\]
-using $\kappa(a(0),d\lambda)\ne0$ from (hecke:lie-data). This is
-also the residue formula of \cite[Eq.~(20.20)]{AGKRRV}; reversing the
+[
+ 0=-kappa(a(0),dlambda)+xi(∂_t),
+ qquad xi=kappa(a(0),dlambda),dtbig|_xne0,
+]
+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 $D\in L$; a test
-vector $C\in L'$ extends as $C_ε=\Ad(1+ε D)C$. The input
-field remains $a(t)\,dt$. Differentiating the residue pairing gives
+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(D\bmod I),C\bmod I\bigr>
- =Res\kappa(a,[D,C])\,dt
- =Res\kappa([a,D],C)\,dt
- =\beta(\ad(a)D,C).
+ 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 $\ad(a)$
+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
-$\mathfrak g/\mathfrak m$; its endomorphism from $\ad(a(0))$ is therefore
+$mathfrak g/frakg/mathfrak m$;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
+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
-$(dS_A)_h\colon P_1\to P_2^*$ is an isomorphism. Zero-weight directions
-do not appear in these quotients, so no extra regularity of $\lambda$
+$(dS_A)_hcolon P_1to P_2^*$ 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 $a\in L$ and $L$ is self-annihilating. Keeping the output fixed,
-vary the input lattice by $C\in L'$ and extend $D\in L$ to
-$\Ad(1+ε C)D$. The same calculation, with the same gluing
+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
-\[
- \bigl<(dS_{A'})_h(C\bmod I),D\bmod I\bigr>
- =Res\kappa(a,[C,D])\,dt
- =Res\kappa([a,C],D)\,dt,
-\]
+[
+ 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
@@ -1367,8 +1367,8 @@ Modifications of this form, and tori, are thus included.
The proposition gives both properties we need. The nonvanishing
-$\xi\ne0$ allows a suitable hypersurface cut of the Hecke correspondence
-to be smooth over $\mathcal B$, while the two relative derivatives
+$xine0$ 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.
@@ -1388,247 +1388,247 @@ 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, $F\in\Shv(\mathcal B_K)$ 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 $b\colon U\to\mathcal B$ be a smooth chart with $U$ finite type over
-$S$. Let $f\in\mathcal O(U)$ and $u\in U_s(k)$ satisfy $f(u)=0$, and
-suppose $df_u=b^*A$ for some non-nilpotent $A\in T^*_{b(u)}\mathcal B_s$.
-Then there exist $x\in\mathscr X_s(k)$ and an integer $m\ge0$ with the
-following property. Put $T=U\times_S\mathscr X$ and let $F_T$ be the
+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
-\[
- Q^a={\sum_{i=1}^m u_iv_i+f=0}\subset T\times_S\mathbb A_S^{2m},
- \qquad \tau\colon Q^a\to T,
-\]
+[
+ Q^a={sum_{i=1}^m u_iv_i+f=0}subset Ttimes_Smathbb A_S^bbA_S^{2m},
+ qquad 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
-$\nu=(u,x,0,0)$,
+$nu=(u,x,0,0)$,
- (\Psi_{Q^a}(\tau_K^*F_T))_\nu=0.
+ (Psi_{Q^a}(tau_K^*F_T))_nu=0.
-When $m=0$, this is the vanishing on $Q^a=V(f)\times_S\mathscr X$.
+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 $\xi\ne0$, satisfying
+$theta=(A',xi)$ with $xine0$, satisfying
- p^*A=q^*\theta\quadat h.
+ p^*A=q^*thetaquadat h.
-Use the exact-position correspondence $H=H^\lambda$, its proper bound
-$\overline H$, and its generic Satake integrand $\mathcal I_K$ from
+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
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
-\[
- W_0=H_2\times_{p_2,\mathcal B,b}U,\qquad
- W=H_2\times_{\mathcal B}D\subset W_0.
-\]
-Both map to $\mathcal Z=\mathcal B\times_S\mathscr X$ by
-$q_2=(o_2,\ell_{H_2})$. The map $W_0\to\mathcal Z$ is smooth: its
+[
+ W_0=H_2times_{p_2,mathcal B,b}calB,bU,qquad
+ 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).
+$W$ 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 $\xi\ne0$. The smooth-hypersurface
-criterion therefore makes $W\to\mathcal B$ (via the output bundle)
+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
Lemma~spec:vanishing gives
- (\Psi_W(q_{2,K}^*\mathrm H_V(F)))_w=0.
+ (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 $\mathcal B$, matching
+Note: smoothness is used only for the projection to $mathcal B$,calB$, matching
the hypothesis of Lemma~spec:vanishing.
Now form fibre products
-\[
- P=H_1\times_{\mathcal Z}W_0,\qquad
- P_D=H_1\times_{\mathcal Z}W\subset\overline P_D=\overline H_1\times_{\mathcal Z}W.
-\]
+[
+ P=H_1times_{mathcal Z}calZW_0,qquad
+ 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.
-\[
+[
[column sep=large,row sep=large]
- P \arrow[r] \arrow[d] & H_1 \arrow[d,"q_1"] \\
- W_0 \arrow[r,"q_2"'] \arrow[u,bend left=38,"\Delta"]
- & \mathcal Z
+ P arrow[r] arrow[d] & H_1 arrow[d,"q_1"]
+ W_0 arrow[r,"q_2"'] arrow[u,bend left=38,"Delta"]
+ & mathcal ZcalZ
-\qquad
+qquad
[column sep=large,row sep=large]
- \overline P_D \arrow[r] \arrow[d,"\pi"']
- & \overline H_1 \arrow[d,"\overline q_1"] \\
- W \arrow[r,"q_2"'] & \mathcal Z.
+ overlineoverline P_DP_D arrow[r] arrow[d,"pi"']
+ & overlineoverline H_1H_1 arrow[d,"overlineoverline q_1"]q_1"]
+ W arrow[r,"q_2"'] & mathcal Z.calZ.
-\]
-Write $c\colon P\to U$ for the map through $W_0$, and
-$a\colon P\to\mathcal B$ for the input map $p_1$. So $a$ records the
+]
+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
-$z=(h,h,u)\in P_D(k)$.
+$z=(h,h,u)in P_D(k)$.
-Let $\mathcal Ibar_K$ be the pullback of the Satake integrand
-to $(\overline P_D)_K$. Proper base change on the generic fibre gives
-$R\pi_{K,*}\mathcal Ibar_K\simeq q_{2,K}^*\mathrm H_V(F)$.
+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)$.rmH_V(F)$.
Apply proper compatibility of specialisation, then proper base change
at $w$, to (eq:two-hecke-zero). With
-\[
- Z_w=\pi^{-1}(w),\qquad
- \mathcal K=(\Psi_{\overline P_D}\mathcal Ibar_K)|_{Z_w},
-\]
+[
+ Z_w=pi^{-1}(w),qquad
+ mathcal K=(calK=(Psi_{overlineoverline P_D}P_Dmathcal Ibar_K)|_calIbar_K)|_{Z_w},
+]
we get
- R\Gamma(Z_w,\mathcal K)=0.
+ RGamma(Z_w,mathcal K)=0.calK)=0.
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 $a\colon P\to\mathcal B$ is smooth: $P\to H_1$ is a base change
-of $W_0\to\mathcal Z$, 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 $\mathcal B$ of $f(c)$
-were zero, cotangent injectivity for the smooth map $P\to H_1$ 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 $acolon Ptomathcal B$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 leg term drops out, and the necessary condition becomes
$S_{A'}(h_1)=0$. The second half of
Proposition~hecke:transverse says this section has an isolated
-zero at $h$ on the fixed-output-and-leg fibre. Hence $P_D\to\mathcal B$
+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
-\[
- \mathcal K|_{O∖\{z\}}=0.
-\]
+[
+ 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
+subsets ${z}$ and $C_w$ are disjoint, so open-closed localization
identifies
-\[
- \mathcal K\simeq i_{z,*}\mathcal K_z ⊕ i_{C_w,*}(i_{C_w}^*\mathcal K),
- \qquad
- R\Gamma(Z_w,\mathcal K)\simeq
- \mathcal K_z ⊕ R\Gamma(C_w,i_{C_w}^*\mathcal K).
-\]
+[
+ 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
is its stalk complex. Equation~(eq:two-fiber-zero) forces it to
vanish. Removing the invertible Satake normalization gives
- (\Psi_{P_D}(a_K^*F))_z=0.
+ (Psi_{P_D}(a_K^*F))_z=0.
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 $\Delta\simeq W_0\subset P$ is a
-section of the smooth separated map $P\to W_0$ of relative dimension
-$m$, hence a closed regular immersion near $z$. On $\Delta$ there is a
-specified isomorphism $bc\simeq a$, 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 $r\colon P\to U$ with an isomorphism
+We claim that after an 'etale replacement of $P$ near $z$, there
+is a map $rcolon Pto U$ with an isomorphism
- br\simeq a,\qquad r|_\Delta=c|_\Delta,
+ brsimeq a,qquad r|_Delta=c|_Delta,
-whose restriction to $\Delta$ is the specified isomorphism. For this,
-form the smooth space $E_P=P\times_{a,\mathcal B,b}U\to P$. The
-diagonal data give a section over $\Delta$. First take an \'etale
-scheme neighbourhood in $P$. Since $\mathcal B$ has affine diagonal
-(Section~sec:specialization), $E_P\to P\times_S U$ is affine, so
+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 $E_P\to\mathbb A_P^{e}$
-\cite[Tag~039P]{Stacks}. After shrinking, the section over $\Delta$
+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
-$\Delta\to P$, 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 $P,\Delta,z$.
+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 $N$.
The map
-\[
- \Delta\to T=U\times_S\mathscr X,\qquad(c,leg),
-\]
-is smooth of relative dimension $m$. So $T,\Delta,P$ are smooth over
+[
+ Deltato T=Utimes_Smathscr X,scrX,qquad(c,leg),
+]
+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,qquad N+1+m,qquad N+1+2m.
+]
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.
+ 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
+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 $P\to W_0$ is zero. On this diagonal fibre, the relative
+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
+$r|_{Delta_{(u,x)}}$ 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
- \sum_i(u_i|_\Delta)[dv_i]=-S_A
- \quadon \Delta_{(u,x)}.
+ 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 $T_z\Delta_{(u,x)}$ 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 $z$.
Now set
-\[
- \Phi=(r,leg,u_1,...,u_m,v_1,...,v_m)\colon
- P\to T\times_S\mathbb A_S^{2m}.
-\]
+[
+ 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 $T_z\Delta$; its
-$(dr,dleg)$ then places it in $T_z\Delta_{(u,x)}$; then the
+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
+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 $\Phi_D\colon P_D\to Q^a$ is
-\'etale near $z$, sending $z$ to $\nu$. Moreover (eq:two-chart-lift)
+$Phi^{-1}(Q^a)$. So the induced map $Phi_Dcolon P_Dto Q^a$ is
+'etale near $z$, sending $z$ to $nu$. Moreover (eq:two-chart-lift)
identifies
-\[
- a_K^*F\simeq r_K^*F_U\simeq\Phi_{D,K}^*\tau_K^*F_T
- \quadon (P_D)_K,
-\]
-so $\Phi_D$ identifies the actual generic sheaf and the equation
-simultaneously. \'Etale base change transforms (eq:two-isolated-zero)
+[
+ 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
+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
(eq:two-exact-quadratic) is empty, and the same argument gives
-$Q^a=D\times_S\mathscr X$. Then smooth base change along $D\times_S
-\mathscr X\to D$ 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.
@@ -1658,14 +1658,14 @@ of equations simultaneously.
## The projective quadric comparison
[Quadric comparison]
-Let $K_0$ be an algebraically closed field of characteristic $\ne\ell$,
-$E=\mathbb Qbar_\ell$, and $B$ a finite-type $K_0$-scheme. For
-$f\in\Gamma(B,\mathcal O_B)$ put $i\colon D=V(f)↪ B$. If
-$m\ge1$, form
-\[
- g\colon Q_f=\Bigl\{\sum_{a=1}^m u_av_a+fw^2=0\Bigr\}
- \subset\mathbb P_B^{2m}→ B.
-\]
+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.
+]
Then the hyperplane-class map defines a distinguished triangle
⊕_{j=0}^{2m-1}E_B(-j)[-2j]
@@ -1679,165 +1679,165 @@ or reduced.
Proof.
-Write $h=c_1(\mathcal O_{Q_f}(1))\in H^2(Q_f,E(1))$. Adjunction on each
-$h^j$ gives the morphism $E_B(-j)[-2j]\to Rg_*E_{Q_f}$. Let
-$\mathcal K$ be the cone of their direct sum, and let
-$j_B\colon B∖ D↪ B$.
+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
+$mathcal K$calK$ 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
-$0\le r\le n$, 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 $x_0\ne0$ is affine
+$0le rle n$, 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
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 ($\cong\mathbb P^1$) and
+when $n$ is even. Initial cases are a conic ($congmathbb P^1$)bbP^1$) and
the split zero-dimensional quadric (two points). Localization plus
-$R\Gamma_c(\mathbb A^r,E)=E(-r)[-2r]$ gives the cohomology. On a smooth
+$RGamma_c(mathbb A^r,E)=E(-r)[-2r]$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
map is an isomorphism on their cohomology. Proper base change therefore
-gives $j_B^*\mathcal K=0$.
+gives $j_B^*mathcal K=0$.calK=0$.
Over $D$ the family, with its hyperplane line bundle, is the constant
family with fibre
-\[
- V_m=\Bigl\{\sum_{a=1}^m u_av_a=0\Bigr\}\subset\mathbb P_{K_0}^{2m}.
-\]
+[
+ 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
+$V_m∖{e}$ with the total space of a line bundle over the split
smooth quadric
-\[
- C_m=\Bigl\{\sum_{a=1}^m u_av_a=0\Bigr\}\subset\mathbb P_{K_0}^{2m-1}.
-\]
+[
+ C_m=Bigl{sum_{a=1}^m u_av_a=0Bigr}subsetmathbb P_bbP_{K_0}^{2m-1}.
+]
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(r\ge2).
+ 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).
-So $V_m$ has the cohomology of $\mathbb P^{2m-1}$ plus one extra
+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
projective bundle of lines
-\[
- \pi\colon\widetilde V_m=\mathbb P_{lines,C_m}
- (\mathcal O_{C_m}(-1) ⊕ \mathcal O_{C_m})
+[
+ picolonwidetilde V_m=mathbb P_bbP_{lines,C_m}
+ (mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m})
→ C_m.
-\]
-The inclusion $\mathcal O_{C_m}(-1) ⊕ \mathcal O_{C_m}↪
-(trivial rank-(2m+1))$ defines $\widetilde V_m\to V_m$; its
+]
+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
-$h_C=c_1(\mathcal O_{C_m}(1))$, the projective-bundle relation is
-$\zeta^2=\pi^*h_C\,\zeta$ \cite[\S23]{MilneEtale}, hence
-\[
- \int_{\widetilde V_m}\zeta^{2m-1}=\int_{C_m}h_C^{2m-2}=2.
-\]
+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 $e$.
Therefore the cone of the vector-space morphism
-$⊕_{j=0}^{2m-1}E(-j)[-2j]\to R\Gamma(V_m,E)$ has exactly one
+$⊕_{j=0}^{2m-1}E(-j)[-2j]to RGamma(V_m,E)$ 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 $E(-m)[-2m]$.
-Proper base change identifies $i^*\mathcal K$ with the constant
+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
localization gives
-\[
- (j_B)_!j_B^*\mathcal K\to\mathcal K\to i_*i^*\mathcal K\to\cdot,
-\]
-whose first term vanishes, so $\mathcal K\simeq i_*E_D(-m)[-2m]$. In
+[
+ (j_B)_!j_B^*mathcal KcalKtomathcal KcalKto i_*i^*mathcal KcalKtocdot,
+]
+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
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
+cite[Tags 095L, 0DDE]{Stacks} and in their adic forms
+cite[Tags 09AH, 09C9]{Stacks}; we use them after inverting $ell$ and
extending to $E$.
Lemma~slice:quadric uses rational coefficients, including
-$\ell=2$. Its hyperplane-power comparison is not an isomorphism off $D$
-with $\mathbb Z/2^a$ coefficients: on a conic the hyperplane class has
+$ell=2$. 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
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 $Q_f=D\subset
-\mathbb P_B^0$ and the hyperplane sum is empty.
+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 hypersurface test
-Work over the trait $S$, with $\mathcal B$ and $F\in\mathcal D_C$ as in
+Work over the trait $S$, with $mathcal B$calB$ and $Finmathcal D_C$calD_C$ as in
Proposition~spec:missing. For a smooth finite-type chart
-$b\colon U\to\mathcal B$, write $F_U=b_K^*F$, and let
-$\mathcal N_U\subset T^*U_s$ 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 $b\colon U\to\mathcal B$ be smooth with $U$ finite type over $S$.
-Let $f\in\mathcal O(U)$, $D=V(f)$, and $u\in D_s(k)$. If
-\[
- d(f_s)_u\notin(\mathcal N_U)_u,
-\]
+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
+[
+ d(f_s)_unotin(mathcal N_U)_u,calN_U)_u,
+]
then
-\[
- (\Psi_D(F_U|_{D_K}))_u=0.
-\]
+[
+ (Psi_D(F_U|_{D_K}))_u=0.
+]
Proof.
-If $d(f_s)_u$ does not lie in the image of $T^*_{b(u)}\mathcal B_s$,
-its image in the relative cotangent of $b$ is nonzero, so $D\to\mathcal B$
+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.
Otherwise $d(f_s)_u=b^*A$ for some non-nilpotent
-$A\in T^*_{b(u)}\mathcal B_s$. Proposition~two:vertex then provides
-$x\in\mathscr X_s(k)$ and $m\ge0$ such that nearby cycles of the
+$Ain T^*_{b(u)}mathcal B_s$.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
-$T=U\times_S\mathscr X$ and $t_0=(u,x)$. If $m=0$ this affine model is
-$D_T$ and smooth compatibility for $D_T\to D$ gives the result.
-
-Assume $m\ge1$. Compactify to
-\[
- g\colon Q_f=\Bigl\{\sum_{a=1}^m u_av_a+fw^2=0\Bigr\}
- \subset\mathbb P_T^{2m}\to T.
-\]
+$Q^a={sum_{a=1}^m u_av_a+f=0}$ 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.
+
+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
+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.
+ (Psi_T(F_T ⊗ Rg_{K*}E))_{t_0}=0.
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
-$i\colon D_T↪ T$. After $\Psi_T$ and stalk at $t_0$,
-the first term vanishes ($\Psi_TF_T=0$) and the second by
+$icolon D_T↪ T$. 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 $D_T\to D$ 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
@@ -1849,13 +1849,13 @@ equations simultaneously.
[Incidence comparison]
Let $B$ be a finite-type scheme over an algebraically closed field of
-characteristic $\ne\ell$, let $f_1,...,f_d\in\Gamma(B,\mathcal O_B)$
-with $d\ge1$, and set $i\colon J=V(f_1,...,f_d)↪ B$.
+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
-\[
- r\colon I=\Bigl\{\sum_{a=1}^d c_af_a=0\Bigr\}
- \subset B\times\mathbb P^{d-1}→ B,
-\]
+[
+ rcolon I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
+ subset Btimesmathbb P^bbP^{d-1}→ B,
+]
the hyperplane powers give a distinguished triangle
⊕_{j=0}^{d-2}E_B(-j)[-2j]
@@ -1867,16 +1867,16 @@ For $d=1$ the first term is zero and $I=J$.
Proof.
-If $d=1$, the equation in $B\times\mathbb P^0$ is $f_1=0$ and the
-statement is immediate. Suppose $d\ge2$. Off $J$, the map
-$\mathcal O_B^d\to\mathcal O_B$ with coefficients $(f_1,...,f_d)$ is
+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
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 $\mathbb P^{d-2}$).
+proper base change and the cohomology of $mathbb P^bbP^{d-2}$).
-On $J$, the family is exactly $J\times\mathbb P^{d-1}$ 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
@@ -1884,36 +1884,36 @@ this restricted cone.
[Higher-codimension test]
-Let $b\colon U\to\mathcal B$ be smooth with $U$ finite type over $S$,
-and let $F_U=b_K^*F$. Suppose $u\in U_s(k)$ and $f_1,...,f_d$ are
+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
-\[
- span_k\{d(f_{1,s})_u,...,d(f_{d,s})_u\}
- \cap(\mathcal N_U)_u=\{0\}.
-\]
+[
+ span_k{d(f_{1,s})_u,...,d(f_{d,s})_u}
+ cap(mathcal N_U)_u=calN_U)_u={0}.
+]
Then for $J=V(f_1,...,f_d)$,
-\[
- (\Psi_J(F_U|_{J_K}))_u=0.
-\]
+[
+ (Psi_J(F_U|_{J_K}))_u=0.
+]
The case $d=0$ is allowed, with $J=U$.
Proof.
-For $d=0$ this is Lemma~spec:vanishing. For $d\ge1$ form the
+For $d=0$ this is Lemma~spec:vanishing. For $dge1$ form the
proper incidence morphism
-\[
- r\colon I=\Bigl\{\sum_{a=1}^d c_af_a=0\Bigr\}
- \subset U\times_S\mathbb P_S^{d-1}→ U.
-\]
-Each affine projective-space chart in $U\times_S\mathbb P_S^{d-1}$ is
-still smooth over $\mathcal B$. On the chart $c_a=1$, the hypersurface
-equation is $f_a+\sum_{b\ne a}c_bf_b=0$, with special-fibre differential
-at $(u,[c])$ equal to $\sum_{b=1}^dc_b\,d(f_{b,s})_u$ (the projective
+[
+ 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
nonzero by linear independence and, by hypothesis, lies outside
-$(\mathcal N_U)_u$. The nilpotent cone on the product chart is the
-pullback of $\mathcal N_U$ with zero parameter component, so
+$(mathcal N_U)_u$.calN_U)_u$. The nilpotent cone on the product chart is the
+pullback of $mathcal N_U$calN_U$ with zero parameter component, so
Proposition~slice:hypersurface gives vanishing of nearby cycles of
$r_K^*F_U$ at every closed point over $u$.
@@ -1921,22 +1921,22 @@ 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
vanish on the entire fibre. Proper compatibility gives
-\[
- (\Psi_U Rr_{K*}r_K^*F_U)_u=0.
-\]
+[
+ (Psi_U Rr_{K*}r_K^*F_U)_u=0.
+]
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 $i\colon J↪ U$.
-After $\Psi_U$, the first two terms vanish at $u$, and proper
+$i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)]$ 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
+$(Psi_J(F_U|_{J_K}))_u$. 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 $\mathcal B$. 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.
% =========================================================================
@@ -1954,33 +1954,33 @@ stalks, so they cannot vanish if any single one is nonzero.
## Extending a generic bundle
-Let $\mathscr X\to S$ be the smooth projective curve from
+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 $\mathscr X_K$
-extends to a $G$-bundle on $\mathscr X_{R'}$ for some finite extension
+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.
Proof.
-Let $\mathcal P_K$ be the bundle. By
-\cite[Thm.~2,~Rem.~2(b)]{DrinfeldSimpson} it is trivial on some
-nonempty open subset of $\mathscr X_K$ (this theorem applies to connected
+Let $mathcal P_K$calP_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
reductive groups over any field). Choose a trivialization away from
-distinct points $x_1,...,x_r\in\mathscr X(K)$. 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
-\[
- \mathcal P_0\dashrightarrow\mathcal P_1\dashrightarrow...
- \dashrightarrow\mathcal P_r=\mathcal P_K,\qquad
- \mathcal P_0 trivial,
-\]
+[
+ mathcal P_0calP_0dashrightarrowmathcal P_1calP_1dashrightarrow...
+ dashrightarrowmathcal P_r=calP_r=mathcal P_K,calP_K,qquad
+ mathcal P_0calP_0 trivial,
+]
whose $i$th 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 $\mathscr X$). 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 $\mathscr X/S$ is proper, induction shows that the space of
+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
the universal final bundle and contains the chosen chain as a
$K$-point. This point descends to a finite extension of
@@ -2001,12 +2001,12 @@ resulting $R'$; all specialization statements from
Section~sec:specialization remain valid with the same geometric
generic field $K$.
-Choose a finite-type smooth affine chart $b\colon U\to\mathcal B$
+Choose a finite-type smooth affine chart $bcolon Utomathcal B$calB$
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
+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
@@ -2021,23 +2021,23 @@ to a finite extension.
Let $R$ be a complete discrete valuation ring with algebraically closed
-residue field $k$ of characteristic $\ne\ell$, $S=\Spec R$, and $K$ an
+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 $N$, $\mathcal F$ a bounded
+scheme of constant relative dimension $N$, $mathcal F$calF$ a bounded
constructible $E$-complex on $U_K$, and
-$Λ\subset T^*U_s$ a closed conical subset with $\dimΛ\le N$.
-Suppose the closure in $U$ of the nonzero-stalk locus of $\mathcal F$
+$Λsubset T^*U_s$ 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 $u\in U_s(k)$, $0\le d\le N$ and functions
+$U$, 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
-\[
- span\{d(f_{1,s})_u,...,d(f_{d,s})_u\}
- \capΛ_u⊆\{0\},
-\]
+[
+ span{d(f_{1,s})_u,...,d(f_{d,s})_u}
+ capΛ_u⊆{0},
+]
and for $J=V(f_1,...,f_d)$,
- (\Psi_J(\mathcal F|_{J_K}))_u\ne0.
+ (Psi_J(mathcal F|_calF|_{J_K}))_une0.
For $d=0$ the list is empty and $J=U$.
@@ -2054,85 +2054,85 @@ 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 $\mathcal F$ to descend.
+require $mathcal F$calF$ to descend.
-Write $\overline A_i,\overline B_i$ for the horizontal closures in $U$,
-and $e_i=\dim A_i$. Each $\overline A_i$ is integral and dominates $S$.
-Every nonempty special fibre $(\overline A_i)_s$ is pure of dimension
-$e_i$ \cite[Tag~0B2J]{Stacks}. Applying this to the irreducible
+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
-$(\overline B_i)_s\ne∅$,
-\[
- \dim(\overline B_i)_s<e_i.
-\]
+$(overlineoverline B_i)_sB_i)_sne∅$,
+[
+ 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
-$v\in(\overline A_i)_s$,
+local ring is excellent cite[Tag~07QW]{Stacks}), so at any closed
+$vin(overlineoverline A_i)_s$,A_i)_s$,
- \dim\mathcal O_{\overline A_i,v}=e_i+1
- \qquad\cite[Tag~02JT]{Stacks}.
+ 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 $(\overline A_i)_s\ne
+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 $T\subset(\overline A_i)_s$ of dimension $d$ for
+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 $∪_i(\overline A_i)_s$
-passes through the point, and no $\overline B_i$ passes through it: an
+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.
We may also arrange
- \dimΛ_u\le N-d.
+ 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
-$\dimΛ\le N$; the same bound holds over a nonempty open subset of
-$T$. Choose $u\in T(k)$ satisfying all requirements.
+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
-$L\subset V$ satisfies $L\capΛ_u⊆\{0\}$: 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),
-$\dim\mathbb P(Λ_u)\le N-d-1$, and the incidence of $d$-planes
-meeting this projective set is a proper closed subset of $\Gr(d,V)$. The
-condition that $L\to T_u^*T$ 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 $L=0$.
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 $Z=J\cap∪_i\overline A_i$, the point
+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 $(\overline A_i)_s$. By
-(eq:support-local-dimension), $\dim\mathcal O_{\overline A_i,u}
-=d+1$. 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
+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 $Z\to S$ is quasi-finite at $u$ (closed point isolated in
-its fibre \cite[Tag~01TH]{Stacks}); its quasi-finite locus is open, so
+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 $\overline B_i$ cannot specialize to
-$u\notin\overline B_i$. Each such lift therefore has a nonzero stalk of
-$\mathcal F|_{J_K}$.
+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}$.
Step 5: nearby cycles on a finite neighbourhood.
Take an affine neighbourhood of $u$ inside the quasi-finite locus of
-$Z\to S$. The henselian decomposition of a finite-type algebra
-\cite[Tag~04GJ]{Stacks} gives a finite local factor containing $u$: an
+$Zto S$. 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
@@ -2140,63 +2140,63 @@ 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 $Q$.
-The restriction $\mathcal F|_{J_K}$ is supported on $Z_K$. To use the
+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 $\iota\colon Q↪ J$ is this closed immersion
-and $\mathcal H=\mathcal F|_{Q_K}$, localization gives
-$\mathcal F|_{J_K}\simeq\iota_{K,*}\mathcal H$. Proper compatibility for
-$\iota$ reduces (eq:support-nonzero) to the stalk of $\Psi_Q
-\mathcal H$ at $u$.
+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 $u$.
-Let $a\colon Q\to S$ be the finite structural map. Its special fibre
+Let $acolon Qto S$ be the finite structural map. Its special fibre
has only $u$, so proper compatibility gives
-\[
- (\Psi_Q\mathcal H)_u\simeq\Psi_S(\RGamma(Q_K,\mathcal H))
- \simeq\RGamma(Q_K,\mathcal H).
-\]
+[
+ (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
-$\RGamma(Q_K,\mathcal H)\simeq⊕_{x\in|Q_K|}\mathcal H_x$. At
+$Q_K$ 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
+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 $\mathcal F$.
+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
+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 $U\to\mathcal B$ of
+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 $U_s$.
-Let $\mathcal N_U\subset T^*U_s$ 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 $\dim\mathcal N_U\le\dim U_s=N$.
-
-Apply Lemma~support:finite-slice with $\mathcal F=F_U$ and
-$Λ=\mathcal N_U$. It gives a slice $J=V(f_1,...,f_d)$ and
-$u\in J_s(k)$ such that
-\[
- (\Psi_J(F_U|_{J_K}))_u\ne0
-\]
+cite[App.~D.1.1--D.1.8]{AGKRRV}. In its smooth-chart formulation
+cite[App.~D.1.2]{AGKRRV}, this says $dimmathcal N_UcalN_Uledim U_s=N$.
+
+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
+[
+ (Psi_J(F_U|_{J_K}))_une0
+]
while the independent differentials of its equations span a plane
-meeting $(\mathcal N_U)_u$ only at zero. The higher-codimension vanishing
+meeting $(mathcal N_U)_u$calN_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 $d=0$,
where the test is the original smooth-chart vanishing.
@@ -2214,49 +2214,49 @@ equivalence asserted in Theorem~thm:full-support.
Throughout this section assume Hypothesis~hyp:A or
Hypothesis~hyp:B, so Theorem~thm:full-support holds.
Put
-\[
- \mathcal C_s=\Shv_{\Nilp}(\Bun_G),\qquad
- \mathcal A=\QCoh(Y),\qquad
- \mathcal D=\IndCoh_{\Nilp}(Y),
-\]
-and write $\mathbb L_G^{restr}\colon\mathcal C_s\to\mathcal D$
-for the restricted Langlands equivalence. Its $\mathcal A$-linear
-structure is the one furnished by \cite[Lemma~1.3.7]{GR}.
+[
+ mathcal C_s=calC_s=Shv_{Nilp}(Bun_G),qquad
+ mathcal A=calA=QCoh(Y),qquad
+ 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
+structure is the one furnished by cite[Lemma~1.3.7]{GR}.
## The category attached to one component
-For an open-and-closed component $i\colon Z↪ Y$, define
-\[
- \mathcal C_{s,Z}:=\QCoh(Z)\underset{\QCoh(Y)} ⊗ \mathcal C_s,
-\]
+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 $\mathcal O_Z$ (extended by zero) acts as the
+idempotent quasi-coherent $mathcal O_Z$calO_Z$ (extended by zero) acts as the
projection to this summand.
-Under either hypothesis regime, every connected component $Z\subset Y$
+Under either hypothesis regime, every connected component $Zsubset Y$
has nonzero localized automorphic category, and the restriction of
-$\mathbb L_G^{restr}$ induces an equivalence
-$\mathcal C_{s,Z}\xrightarrow{\sim}\IndCoh_{\Nilp}(Z)$.
+$mathbb L_G^bbL_G^{restr}$ induces an equivalence
+$mathcal C_calC_{s,Z}xrightarrow{sim}IndCoh_{Nilp}(Z)$.
Proof.
-The decomposition $Y=Z\sqcup(Y∖ Z)$ decomposes quasi-coherent
+The decomposition $Y=Zsqcup(Y∖ Z)$ decomposes quasi-coherent
and nilpotent ind-coherent sheaves into two summands, so
-$\QCoh(Z) ⊗ _{\mathcal A}\mathcal D\simeq\IndCoh_{\Nilp}(Z)$. Tensor
-the $\mathcal A$-linear equivalence of Theorem~thm:full-support
-with $\QCoh(Z)$ over $\mathcal A$, 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
-\cite[Prop.~3.7.2,~Cor.~3.7.4]{AGKRRV}. Pullback to this point shows
-$\mathcal O_Z\ne0$, and the fully faithful
-$\QCoh(Z)↪\IndCoh_{\Nilp}(Z)$ then gives nonvanishing.
+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.
The component index and the parameter are not the same object. Two
geometric parameters lie in the same component iff their semisimp-
-lifications are isomorphic \cite[Prop.~3.7.2]{AGKRRV}. The construction
+lifications are isomorphic cite[Prop.~3.7.2]{AGKRRV}. The construction
of eigenobjects below uses the actual parameter, including its
extension data, not just its semisimple base point.
@@ -2264,24 +2264,24 @@ 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
+determine geometric $q$-Frobenius, whose pullback action on $QLisse(X)$
+induces an automorphism $Frob$ of $Y$. Define the derived fixed-point
stack
-\[
- Y^{arithm}:=Y^{\Frob}=Y\underset{Y\times_EY}{\overset{\mathrm R}\times}Y,
-\]
-where the two maps to $Y\times_EY$ 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}.
+[
+ 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
+system together with a Weil structure. See cite[S24.1]{AGKRRV}
+and cite[S1.5.1]{GR}.
For the finite-field datum of Hypothesis~hyp:A there is a
canonical isomorphism
-\[
- \Funct_c(\Bun_{G_0}(\mathbb F_q),E)
- \simeq\Gamma^{\IndCoh}(Y^{arithm},\omega_{Y^{arithm}}).
-\]
+[
+ 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
isomorphism classes of rational bundles; the right side uses the
ind-coherent dualizing object and its ind-coherent global sections on the
@@ -2291,24 +2291,24 @@ derived fixed-point stack.
Proof.
The primed union $Y'$ is Frobenius-stable. The spectral trace
calculation of Beraldo--Lin--Reeves
-\cite[\S\S6.4.12--6.4.13]{BeraldoLinReeves} enters the known primed
-formula \cite[\S\S1.5.3--1.5.5,~Cor.~1.5.6]{GR}:
-\[
- \Funct_c(\Bun_{G_0}(\mathbb F_q),E)
- \simeq\Gamma^{\IndCoh}((Y')^{\Frob},\omega_{(Y')^{\Frob}}).
-\]
-Its automorphic trace input is \cite[Thm.~0.2.6]{AGKRRVTrace} with
-endofunctor $q$-Frobenius pushforward on $\Bun_G$. By
+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}:
+[
+ Funct_c(Bun_{G_0}(mathbb F_q),E)bbF_q),E)
+ 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
+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
-\cite[\S1.5.3]{GR}, whose "ordinary global sections" notation uses the
-image of $\omega$ under the dual of $Υ\colon\QCoh\to\IndCoh$. We
+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
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 $\mathcal A_0$ be a compactly generated, presentable, stable,
+Let $mathcal A_0$calA_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 $a\colon\mathcal A_0\to\Vect_E$ be a continuous
-$E$-linear symmetric monoidal functor, and give $\Vect_E$ the
-$\mathcal A_0$-action through $a$. Then the right adjoint $R_a$ of $a$
-is continuous, carries a coherent $\mathcal A_0$-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 $\mathcal A_0$ is
+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 $a$
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
colimit-preserving.
-The canonical projection $b ⊗ R_a(V)\to R_a(a(b) ⊗ V)$ is
+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}(c,b ⊗ R_a(V))
- &\simeq\RHom_{\mathcal A_0}(b^\vee ⊗ c,R_a(V))\\
- &\simeq\RHom_E(a(b)^\vee ⊗ a(c),V)\\
- &\simeq\RHom_E(a(c),a(b) ⊗ V).
+ 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
coherences before invertibility is imposed; invertibility then gives a
-coherent $\mathcal A_0$-linear structure on $R_a$.
+coherent $mathcal A_0$-linearcalA_0$-linear structure on $R_a$.
-Finally $R_a(E)\ne0$ because
-$\RHom_{\mathcal A_0}(\mathbf 1,R_a(E))
-\simeq\RHom_E(E,E)\simeq E$; 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, $\mathcal A=\QCoh(Y)$ 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
+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 $\mathcal A$-linear functor
-\[
- \jmath_Y\colon\QCoh(Y)\xrightarrow[\sim]{Υ_Y}
- \IndCoh_{\{0\}}(Y)↪\IndCoh_{\Nilp}(Y),
-\]
-where $Υ_Y(Q)=Q ⊗ \omega_Y$
-\cite[\S1.3.1,~Eq.~(1.2)]{GR}. Its essential image has zero singular
+fully faithful $mathcal A$-linearcalA$-linear functor
+[
+ jmath_YcolonQCoh(Y)xrightarrow[sim]{Υ_Y}
+ IndCoh_{{0}}(Y)↪IndCoh_{Nilp}(Y),
+]
+where $Υ_Y(Q)=Q ⊗ omega_Y$
+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 $\mathcal Q_I=\QLisse(X)^{ ⊗ I}$. Given
-$V\in\Rep(\check G)^{ ⊗ I}$, 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
-$\mathcal E_V^I\in\mathcal A ⊗ \mathcal Q_I$ be the universal
-evaluation object. For a parameter $\sigma\colon\Spec E\to Y$ define
-$\sigma^I(V):=(\sigma^* ⊗ \id)(\mathcal E_V^I)\in\mathcal Q_I$.
+$mathcal E_V^IcalE_V^Iinmathcal AcalA ⊗ mathcal Q_I$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$.
-A Hecke eigenobject of eigenvalue $\sigma$ with coherent structure is an
-$M\in\mathcal C_s$ equipped, for all finite sets $I$, with isomorphisms
-$H_I(V,M)\simeq M\boxtimes\sigma^I(V)$, natural in $V\in
-\Rep(\check G)^{ ⊗ I}$ and compatible with units, tensor products,
+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,
permutations of legs, collisions of legs, and all homotopy-coherent
-compatibilities \cite[\S15.2]{AGKRRV}.
+compatibilities cite[S15.2]{AGKRRV}.
-Under either hypothesis regime, every parameter $\sigma\colon\Spec E\to
-Y$ admits a nonzero object $M_\sigma\in\Shv_{\Nilp}(\Bun_G)$ with a
-coherent Hecke eigenstructure of eigenvalue $\sigma$. If $\sigma$ lies
-in a component $Z$, then $M_\sigma\in\mathcal C_{s,Z}$. 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 $a=\sigma^*\colon\mathcal A
-\to\Vect_E$, 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
-\[
- \Phi_\sigma\colon\Vect_\sigma\xrightarrow{\sigma_*}\mathcal A
- \xrightarrow{\jmath_Y}\mathcal D
- \xrightarrow{(\mathbb L_G^{restr})^{-1}}\mathcal C_s
-\]
-is continuous $\mathcal A$-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 $Z$,
+[
+ 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
-$\mathcal C_{s,Z}$.
+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
-$\mathcal E_V^I$ on $\mathcal C_s$ is the multi-leg Hecke functor
-$H_I(V,-)$ \cite[\S\S14.2--14.3]{AGKRRV}. Extend $\Phi_\sigma$ by the
-identity on $\mathcal Q_I$. Its module structure gives
-\[
- H_I(V,M_\sigma)\simeq(\Phi_\sigma ⊗ \id)(\sigma^I(V))
- \simeq M_\sigma\boxtimes\sigma^I(V),
-\]
+$mathcal E_V^I$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
+[
+ 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
+$Vect_E$ 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.
@@ -2435,62 +2435,62 @@ semisimple or not.
% =========================================================================
We now prove the rational-coefficient form of full support. Put
-$E_0=\mathbb Q_\ell$ and $E=\mathbb Qbar_\ell$, keep $k=\mathbb Fbar_q$,
+$E_0=mathbb Q_bbQ_ell$ 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 $\mathcal C_E=\Shv_{\Nilp,E}(\Bun_G)$, retain
-$\mathcal C_0=\Shv_{\Nilp,E_0}(\Bun_G)$ and $Y_0$ from the Introduction,
+$E_0$). 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
-$\mathcal C_0$ whose restriction along universal evaluation is the usual
+Recall the standing datum: a continuous $QCoh(Y_0)$-action on
+$mathcal C_0$calC_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
-$U\subset Y_0$ satisfies
-\[
- \QCoh(U)\underset{\QCoh(Y_0)} ⊗ \mathcal C_0\ne0.
-\]
-Consequently if $Y'_0\subset Y_0$ is an open-and-closed support union
-whose complementary structure idempotent acts by zero on $\mathcal C_0$,
+$Usubset Y_0$ 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.
+support of any $QCoh(Y_0)$-linear primed equivalence.
-The plan is: for $X\ne\mathbb P^1$ we compare the sheaf categories,
+The plan is: for $Xnemathbb P^1$bbP^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
-$X=\mathbb P^1$ connectedness of the spectral prestack gives the result
+$X=mathbb P^1$bbP^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, $L\in\{E_0,E\}$. For a smooth $T$ and a
-closed conical $N\subset T^*T$, let $\mathcal P_{N,L}(T)$ be the category
+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 $Ind\mathcal P_{N,L}(T)$ in every degree. This
+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[\S\S E.5.1--E.5.8]{AGKRRV}; it is not necessarily the ind-completion
+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
-\cite[\S\S1.2.5--1.2.11]{AGKRRV}.
+$QLisse_L$, with its equivalent ordinary-cohomology definition
+cite[SS1.2.5--1.2.11]{AGKRRV}.
[Coefficient comparison]
Extension of coefficients induces equivalences
- \Vect_E ⊗ _{E_0}\Shv_{E_0}(T)&\simeq\Shv_E(T),
- \\
- \Vect_E ⊗ _{E_0}\QLisse_{E_0}(X)&\simeq\QLisse_E(X),
- \\
- \Vect_E ⊗ _{E_0}\mathcal C_0&\simeq\mathcal C_E.
+ Vect_E ⊗ _{E_0}Shv_{E_0}(T)&simeqShv_E(T),
+
+ Vect_E ⊗ _{E_0}QLisse_{E_0}(X)&simeqQLisse_E(X),
+
+ Vect_E ⊗ _{E_0}mathcal C_0&calC_0&simeqmathcal C_E.calC_E.
The first extends to locally finite-type algebraic stacks by smooth
@@ -2512,24 +2512,24 @@ compact objects have the required mapping complexes and generate after
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
+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
-$\mathbb Q_\ell$-coefficient field), so its compactness statements apply.
+$mathbb Q_bbQ_ell$-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
+$E$-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
-\[
+[
U(M_L ⊗ _L E)=
- \mathop{colim}_{\substack{L\subset L'\subset E\\ [L':E_0]<∞}}
+ mathop{colim}_{substack{Lsubset L'subset E [L':E_0]<∞}}
Res_{L'/E_0}(M_L ⊗ _L L'),
-\]
+]
and that finite extension/restriction are t-exact and commute with the
ind-t-structure, one gets the claim first on constructibles, then on the
ambient ind-categories.
@@ -2537,18 +2537,18 @@ 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 $\mathbb Q_\ell$-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
-the coefficient formalism and projection formula of \S2 there; extending
+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
-\cite[Prop.~3.4(ii)]{HansenScholze}; the dual, evaluation and
+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
-\cite[Lemma~7.9]{HemoRicharzScholbach} they descend to a common finite
+cite[Lemma~7.9]{HemoRicharzScholbach} they descend to a common finite
coefficient extension. Thus the complex is universally locally acyclic
over that finite extension already. Finite extension reflects local
acyclicity, giving reflection for each pair, hence for the bound.
@@ -2559,8 +2559,8 @@ 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 $u\mapsto1 ⊗ u$. Its source is in 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
@@ -2582,10 +2582,10 @@ 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
+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
+$Vect_{E_0}$ 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
-$\QLisse(\mathbb P^1)$ without identifying that category with
-$\IndLisse(\mathbb P^1)$.
+$QLisse(mathbb P^1)$bbP^1)$ without identifying that category with
+$IndLisse(mathbb P^1)$.bbP^1)$.
## Affine families and coherent Hecke action
@@ -2605,8 +2605,8 @@ prestacks. We retain the right-t-exactness in their tensor functors.
[Spectral coefficient comparison]
There are natural identifications
- Y_0\times_{\Spec E_0}\Spec E\simeq Y,\qquad
- \Vect_E ⊗ _{E_0}\QCoh(Y_0)\simeq\QCoh(Y),
+ Y_0times_{Spec E_0}Spec Esimeq Y,qquad
+ Vect_E ⊗ _{E_0}QCoh(Y_0)simeqQCoh(Y),
with the second symmetric monoidal. These identify the universal
evaluation systems for every finite set of legs.
@@ -2614,25 +2614,25 @@ 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
-\[
- Mod_A ⊗ _{E_0}\QLisse_{E_0}(X)
- \simeqMod_A ⊗ _E\QLisse_E(X).
-\]
+=QCoh(Spec A)$. Lemma~lem:coeff-sheaves and associativity give
+[
+ Mod_A ⊗ _{E_0}QLisse_{E_0}(X)
+ simeqMod_A ⊗ _EQLisse_E(X).
+]
The tensor t-structures are generated by connective objects of the
factors; equivalently they are module t-structures with t-exact
-forgetful functor \cite[\S1.4.1]{AGKRRV}, and they agree under this
+forgetful functor cite[S1.4.1]{AGKRRV}, and they agree under this
identification by the connective assertion in
Lemma~lem:coeff-sheaves.
For the split dual group, scalar extension identifies
-$\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
remain irreducible after extension
-\cite[Thm.~22.2,~\S\S22.3--22.5,~Thm.~22.42]{Milne}. The universal
+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.
@@ -2641,46 +2641,46 @@ 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
-$\Vect_E ⊗ _{E_0}Mod_B\simeq
+$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
+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.
-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 $X\ne\mathbb P^1$ we can go further and compare the given rational
+For $Xnemathbb P^1$bbP^1$ we can go further and compare the given rational
spectral action with the geometric one. This is the only case needed
-below; for $\mathbb P^1$ we deduce support from connectedness of the
+below; for $mathbb P^1$bbP^1$ we deduce support from connectedness of the
spectral prestack.
[Comparison of spectral actions]
-Suppose $X\ne\mathbb P^1$ and equip $\mathcal C_0$ with the rational
+Suppose $Xnemathbb P^1$bbP^1$ and equip $mathcal C_0$calC_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 $\mathcal C_E$.
+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
+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 $E\to E_0$ 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 $E$,
+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
@@ -2693,30 +2693,30 @@ Satake IC objects, their convolution maps, and the fusion diagrams with
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[\S\S4--6,~Thm.~14.1]{MirkovicVilonen}.
+cite[SS4--6,~Thm.~14.1]{MirkovicVilonen}.
Derived functors here are derived extensions of naive geometric Satake.
-In \cite[\S\S B.3.5--B.3.8]{GKRV} the compatible monoidal diagram in
+In cite[SS B.3.5--B.3.8]{GKRV} the compatible monoidal diagram in
shifted perverse hearts is extended using the bounded-derived universal
property; the induced right-lax monoidal structure is strict, and one
restricts to compact representations and ind-extends. Applying this to
the whole coefficient-comparison diagram, naturality gives the
comparison of derived functors and all finite-set structural maps; the
monoidal equivalences persist under the exact coefficient functor. The
-correspondence action of \cite[\S\S B.2.5--B.2.10]{GKRV} then compares
+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
+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
-equivalence of spaces of actions in \cite[Thm.~8.1.4]{AGKRRV}, applied
+equivalence of spaces of actions in cite[Thm.~8.1.4]{AGKRRV}, applied
over the algebraically closed $E$, identifies it with the established
-spectral action of \cite[Thm.~14.3.2]{AGKRRV}.
+spectral action of cite[Thm.~14.3.2]{AGKRRV}.
No Frobenius structure or normalised rational Langlands functor was
@@ -2726,52 +2726,52 @@ Theorem~thm:full-support.
## Descent of support and the arithmetic qualification
-For $X\ne\mathbb P^1$, 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 $U\subset Y_0$ acts by zero on $\mathcal C_0$,
+open-and-closed substack $Usubset Y_0$ acts by zero on $mathcal C_0$,calC_0$,
then $U=∅$.
Proof.
-Let $e_U=\mathcal O_U$ extended by zero in $\QCoh(Y_0)$. Its extension
-is the analogous idempotent $e_{U_E}$, where $U_E=U\times_{E_0}E$.
+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 $U_E=Utimes_{E_0}E$.
Compatibility of the actions and generation by scalar-extended objects
-show $e_{U_E}$ acts by zero on $\mathcal C_E$.
+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
-$\jmath_Y(\mathcal O_Y)$; its value is $\jmath_Y(e_{U_E})$. Full
-faithfulness of $\jmath_Y$ implies $e_{U_E}=0$, hence $U_E=∅$.
+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 $T=\Spec A\to Y_0$. The inverse image
+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
+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 $X\ne\mathbb P^1$, 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 $X=\mathbb P^1$, 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 $\mathbb P^1$ is trivial (a connected finite
-\'etale cover $C\to\mathbb P^1$ of degree $n$ satisfies
-$2g(C)-2=-2n$ by Riemann--Hurwitz \cite[Tag~0C1B]{Stacks}, so $n=1$).
+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 $n=1$).
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 $\mathcal C_0$ is nonzero (its
+open-and-closed substack. The category $mathcal C_0$calC_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.
@@ -2784,23 +2784,23 @@ of prestacks.
[Conditional rational unpriming]
-Let $Y'_0\subset Y_0$ 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
given,
-\[
- \Funct_c(\Bun_{G_0}(\mathbb F_q),E_0)
- \xrightarrow{\ \sim\ }
- \Gamma^{\IndCoh}((Y'_0)^{\Frob},\omega_{(Y'_0)^{\Frob}}),
-\]
+[
+ Funct_c(Bun_{G_0}(mathbb F_q),E_0)bbF_q),E_0)
+ xrightarrow{ sim }
+ Gamma^{IndCoh}((Y'_0)^{Frob},omega_{(Y'_0)^{Frob}}),
+]
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
-$Y'_0\subset Y_0$ 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.
@@ -2823,128 +2823,128 @@ supplies none of these additional data.
{99}
-\bibitem[GR]{GR}
+bibitem[GR]{GR}
D.~Gaitsgory and S.~Raskin,
Geometric {Langlands in positive characteristic from characteristic zero},
arXiv:2508.02237v1, 4 August 2025.
-\bibitem[AGKRRV]{AGKRRV}
+bibitem[AGKRRV]{AGKRRV}
D.~Arinkin, D.~Gaitsgory, D.~Kazhdan, S.~Raskin, N.~Rozenblyum, Y.~Varshavsky,
The stack of local systems with restricted variation and geometric {Langlands theory with nilpotent singular support},
arXiv:2010.01906v2, 6 April 2022.
-\bibitem[AGKRRVTrace]{AGKRRVTrace}
+bibitem[AGKRRVTrace]{AGKRRVTrace}
D.~Arinkin, D.~Gaitsgory, D.~Kazhdan, S.~Raskin, N.~Rozenblyum, Y.~Varshavsky,
Automorphic functions as the trace of {Frobenius},
arXiv:2102.07906v2, 3 June 2022.
-\bibitem[GLCI]{GLCI}
+bibitem[GLCI]{GLCI}
D.~Gaitsgory and S.~Raskin,
Proof of the geometric {Langlands conjecture {I}: construction of the functor},
arXiv:2405.03599v3, 30 September 2025.
-\bibitem[GLCV]{GLCV}
+bibitem[GLCV]{GLCV}
D.~Gaitsgory and S.~Raskin,
Proof of the geometric {Langlands conjecture {V}: the multiplicity one theorem},
arXiv:2409.09856v3, 16 January 2026.
-\bibitem[RicharzScholbach]{RicharzScholbach}
+bibitem[RicharzScholbach]{RicharzScholbach}
T.~Richarz and J.~Scholbach,
The motivic {Satake equivalence},
-Math.\ Ann.\ 380 (2021), 1595--1653;
+Math. Ann. 380 (2021), 1595--1653;
arXiv:1909.08322v3, 24 June 2021 (section locators use this version).
-\bibitem[Richarz]{Richarz}
+bibitem[Richarz]{Richarz}
T.~Richarz,
A new approach to the geometric {Satake equivalence},
-Doc.\ Math.\ 19 (2014), 209--246.
+Doc. Math. 19 (2014), 209--246.
-\bibitem[BeauvilleLaszlo]{BeauvilleLaszlo}
+bibitem[BeauvilleLaszlo]{BeauvilleLaszlo}
A.~Beauville and Y.~Laszlo,
Un lemme de descente,
-C.~R.\ Acad.\ Sci.\ Paris S\'er.\ I Math.\ 320 (1995), 335--340.
+C.~R. Acad. Sci. Paris S'er. I Math. 320 (1995), 335--340.
-\bibitem[DrinfeldSimpson]{DrinfeldSimpson}
+bibitem[DrinfeldSimpson]{DrinfeldSimpson}
V.~G.~Drinfeld and C.~Simpson,
$B$-structures on {$G$-bundles and local triviality},
-Math.\ Res.\ Lett.\ 2 (1995), no.~6, 823--829.
+Math. Res. Lett. 2 (1995), no.~6, 823--829.
-\bibitem[Springer]{Springer}
+bibitem[Springer]{Springer}
T.~A.~Springer,
Linear Algebraic Groups (2nd ed.),
-Progress in Mathematics 9, Birkh\"auser, Boston, 1998.
+Progress in Mathematics 9, Birkh"auser, Boston, 1998.
-\bibitem[Stacks]{Stacks}
+bibitem[Stacks]{Stacks}
The Stacks Project Authors,
The {Stacks Project}, https://stacks.math.columbia.edu, accessed September 2026; cited by stable tag.
-\bibitem[HansenScholze]{HansenScholze}
+bibitem[HansenScholze]{HansenScholze}
D.~Hansen and P.~Scholze,
Relative perversity,
arXiv:2109.06766v3, 10 May 2023.
-\bibitem[HemoRicharzScholbach]{HemoRicharzScholbach}
+bibitem[HemoRicharzScholbach]{HemoRicharzScholbach}
T.~Hemo, T.~Richarz, J.~Scholbach,
Constructible sheaves on schemes,
arXiv:2305.18131v1, 29 May 2023.
-\bibitem[MirkovicVilonen]{MirkovicVilonen}
-I.~Mirkovi\'c and K.~Vilonen,
+bibitem[MirkovicVilonen]{MirkovicVilonen}
+I.~Mirkovi'c and K.~Vilonen,
Geometric {Langlands duality and representations of algebraic groups over commutative rings},
arXiv:math/0401222v5, 13 February 2018.
-\bibitem[GKRV]{GKRV}
+bibitem[GKRV]{GKRV}
D.~Gaitsgory, D.~Kazhdan, N.~Rozenblyum, Y.~Varshavsky,
A toy model for the {Drinfeld--Lafforgue shtuka construction},
arXiv:1908.05420v5, 7 February 2022.
-\bibitem[ArinkinGaitsgory]{ArinkinGaitsgory}
+bibitem[ArinkinGaitsgory]{ArinkinGaitsgory}
D.~Arinkin and D.~Gaitsgory,
Singular support of coherent sheaves, and the geometric {Langlands conjecture},
-Selecta Math.\ (N.S.) 21 (2015), no.~1, 1--199;
+Selecta Math. (N.S.) 21 (2015), no.~1, 1--199;
arXiv:1201.6343v4, 3 November 2014 (section locators use this version).
-\bibitem[BeilinsonSingularSupport]{BeilinsonSingularSupport}
+bibitem[BeilinsonSingularSupport]{BeilinsonSingularSupport}
A.~Beilinson,
Constructible sheaves are holonomic,
arXiv:1505.06768v8, 30 January 2017.
-\bibitem[HallRydh]{HallRydh}
+bibitem[HallRydh]{HallRydh}
J.~Hall and D.~Rydh,
Coherent {Tannaka duality and algebraicity of {Hom}-stacks},
Algebra Number Theory 13 (2019), 1633--1675;
arXiv:1405.7680v3, 29 June 2016 (theorem locators use this version).
-\bibitem[BeraldoLinReeves]{BeraldoLinReeves}
+bibitem[BeraldoLinReeves]{BeraldoLinReeves}
D.~Beraldo, K.~Lin, W.~Reeves,
Coherent sheaves, sheared {$D$-modules, and {Hochschild} cochains},
arXiv:2410.10692v2, 21 October 2024.
-\bibitem[Conrad]{Conrad}
+bibitem[Conrad]{Conrad}
B.~Conrad,
Reductive group schemes,
available at https://math.stanford.edu/~conrad/papers/luminysga3smf.pdf, 2014.
-\bibitem[Milne]{Milne}
+bibitem[Milne]{Milne}
J.~S.~Milne,
Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field,
Cambridge University Press, 2022; corrected text dated 5 October 2021.
-\bibitem[MilneEtale]{MilneEtale}
+bibitem[MilneEtale]{MilneEtale}
J.~S.~Milne,
-Lectures on \'Etale Cohomology,
+Lectures on 'Etale Cohomology,
version 2.21, 22 March 2013;
https://www.jmilne.org/math/CourseNotes/LEC.pdf.
-\bibitem[FrenkelGaitsgoryVilonen]{FrenkelGaitsgoryVilonen}
+bibitem[FrenkelGaitsgoryVilonen]{FrenkelGaitsgoryVilonen}
E.~Frenkel, D.~Gaitsgory, K.~Vilonen,
On the geometric {Langlands conjecture},
arXiv:math/0012255v3, 17 October 2001.
-\bibitem[GaitsgoryVanishing]{GaitsgoryVanishing}
+bibitem[GaitsgoryVanishing]{GaitsgoryVanishing}
D.~Gaitsgory,
On a vanishing conjecture appearing in the geometric {Langlands correspondence},
-Ann.\ of Math.\ 160 (2004), no.~2, 617--682;
+Ann. of Math. 160 (2004), no.~2, 617--682;
arXiv:math/0204081v2, 24 October 2003 (section locators use this version).