diff 56bc728..f6bf760 restricted_geometric_langlands_pos_v2.txt
diff --git a/restricted_geometric_langlands_pos_v2.txt b/restricted_geometric_langlands_pos_v2.txt
index 3ba507c..e8b6517 100644
--- a/restricted_geometric_langlands_pos_v2.txt
+++ b/restricted_geometric_langlands_pos_v2.txt
@@ -65,11 +65,11 @@ allowdisplaybreaks[2]
maketitle
Abstract:
-We prove the mathbb Qbar_bbQbar_ell-linear restricted geometric Langlands
+We prove the Qbar_-linear restricted geometric Langlands
equivalence for a smooth projective connected curve and a connected reductive
group over a field of characteristic p>0, with ellne p, in two regimes.
-Over mathbb Fbar_q,bbFbar_q, we assume only the four characteristic
+Over Fbar_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
@@ -116,7 +116,7 @@ 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/SscrX/S of X to the Witt trait, satisfying Psi(F)=0
+of a lift X/SscrX/S of X to the Witt trait, satisfying Psi(F)=0
for geometric nearby cycles Psi. We then construct a small, finite
slice transverse to the support of F on which Psi(F) cannot vanish,
provided the slice suitably avoids the nilpotent cone. That contradiction
@@ -125,7 +125,7 @@ finishes the proof.
## The spectral stack
Let k be an algebraically closed field of characteristic p>0,
-let ellne p be a prime, and set E=mathbb Qbar_bbQbar_ell. Let
+let ellne p be a prime, and set E=Qbar_ 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
@@ -152,7 +152,7 @@ as in characteristic zero. Ind-coherent sheaves alone with no support
condition do not capture the Arthur-type data that automorphic sheaves
must carry. On the automorphic side one restricts to sheaves with
nilpotent singular support, i.e. those whose wavefront set is contained
-in the global nilpotent cone mathcal NcalNsubset T^*Bun_G; write this
+in the global nilpotent cone N 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
@@ -195,9 +195,9 @@ contains 0.
[Regime A: finite-field descent]
There are q=p^r, a smooth projective geometrically connected curve
-X_0/mathbb F_q,bbF_q, and a split connected reductive group
-G_0/mathbb F_qbbF_q with k=mathbb Fbar_q,bbFbar_q,
-X=X_0 ⊗ _{mathbb F_q}bbF_qk, G=G_0 ⊗ _{mathbb F_q}bbF_qk.
+X_0/F_q, and a split connected reductive group
+G_0/F_q with k=Fbar_q,
+X=X_0 ⊗ _{F_q}, G=G_0 ⊗ _{F_q}
Hypothesis~hyp:common holds. No restriction on |W_G| is imposed.
@@ -232,10 +232,10 @@ it is worth pinning them down once, because several comparisons later
depend on the precise definition.
On the automorphic side, the global nilpotent cone
-mathcal NcalNsubset T^*Bun_G(X) consists of Higgs pairs (P,A) with
+N 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
+if its wavefront set lies in N; 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
@@ -259,7 +259,7 @@ correspondence we prove matches them.
By cite[Main Thm.~1.3.9]{GR}, in either regime there is a
QCoh(Y)-linear equivalence of the form
- mathbb L_G^bbL_G^{restr}colon
+ L_G^{restr}colon
Shv_{Nilp}(Bun_G(X))
xrightarrow{ sim }IndCoh_{Nilp}(Y')
↪IndCoh_{Nilp}(Y),
@@ -279,7 +279,7 @@ Under either Hypothesis~hyp:A or Hypothesis~hyp:B, the
inclusion Y'subset Y in (eq:primed) is equality of spectral
prestacks, and the natural functor
[
- mathbb L_G^bbL_G^{restr}colon
+ L_G^{restr}colon
Shv_{Nilp}(Bun_G(X))xrightarrow{ sim }
IndCoh_{Nilp}(Y)
]
@@ -297,7 +297,7 @@ in so doing proves cite[Conj.~1.3.10]{GR} in the two regimes.
Three consequences are developed in Section~sec:consequences:
[label=(alph*)]
localising at any one component Csubset Y gives a nonzero
- automorphic category mathcal C_calC_{s,C} equivalent to
+ automorphic category C_{s,C} equivalent to
IndCoh_{Nilp}(C);
in regime A, the arithmetic trace formula holds with the full
derived fixed-point stack Y^{Frob} rather than the primed
@@ -309,77 +309,14 @@ None of these requires any additional hypothesis beyond
Theorem~thm:full-support.
Section~sec:coefficients proves the same support statement for
-rational coefficients mathbb Q_bbQ_ell. The theorem there is conditional
+rational coefficients Q_ The theorem there is conditional
on a single input: the existence of the primed trace formula over
-mathbb Q_bbQ_ell itself, intertwining the excursion and spherical Hecke
+Q_ 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
spectral prestack.
-## Roadmap to the proof
-
-The rest of the paper is organised so that each section proves one point.
-
-smallskipnoindent
-Section~sec:specialization (specialisation). We lift
-(X,G) to (mathscr X/S,G/S)scrX/S,G/S) over the Witt trait, write
-mathcal B=calB=Bun_G(mathscr X/S)scrX/S) and set up geometric specialisation
-Psi from the geometric generic fibre to the closed fibre. We show
-that a missing component C produces a nonzero compact F on
-mathcal B_KcalB_K with Psi(F)=0, and that this vanishing persists
-after pulling back along smooth maps acolon Ttomathcal BcalB and after
-applying any Hecke operator mathrm H_V.rmH_V. Crucially, the leg direction
-of the Hecke map is arbitrary; only the projection to mathcal BcalB must
-be smooth. This is the freedom later needed to cut Hecke correspondences
-by a hypersurface.
-
-smallskipnoindent
-Section~sec:lie (Lie-theoretic reduction). Starting from a
-non-nilpotent Higgs field A at some point x on the curve, we find a
-Levi subgroup Msubset G, a cocharacter lambda of its centre, and a
-change of formal frame, such that A lies in mathfrak m[[t]],frakm[[t]],
-kappa(A,dlambda)ne0, and ad_{A} is invertible on
-mathfrak g/frakg/mathfrak m.frakm. This is the input for constructing a Hecke
-modification whose moving-point covector does not vanish.
-
-smallskipnoindent
-Section~sec:hecke (Hecke modification). Applying the above
-modification to a fixed bundle, we get a nonzero leg covector xi and
-two relative cotangent sections (fixed-input and fixed-output) whose
-derivatives are isomorphisms on the two Hecke fibres. This is the
-tranversality input.
-
-smallskipnoindent
-Section~sec:twohecke (two Hecke correspondences). Composing
-two Hecke correspondences with common output and common leg, and cutting
-by a hypersurface dual to xi, we isolate one stalk of Psi(F) and
-identify its local neighbourhood with a split quadratic equation
-sum_{i=1}^m u_iv_i+f=0.
-
-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.
-
-smallskipnoindent
-Section~sec:support (contradiction). The support of
-Psi(F) can be cut down to a finite scheme over the trait, on which
-nearby cycles are a finite direct sum of stalks. Since the summands
-cannot all vanish (the object is nonzero), we get a slice on which
-Psi does not vanish, contradicting the transverse test. Since the
-nilpotent cone has dimension at most dimmathcal B_s,calB_s, a slice of
-sufficiently high codimension always exists. This finishes
-Theorem~thm:full-support.
-
-smallskipnoindent
-Sections~ apply the
-theorem to the previously listed consequences and prove the rational
-coefficient statement by coefficient comparison.
-
-% =========================================================================
# Specialisation and a missing spectral component
% =========================================================================
@@ -406,11 +343,11 @@ p and residue field k. It is strictly henselian, and it is
excellent because it is a complete Noetherian local ring
cite[Tag~07QS]{Stacks}.
-The curve X/k admits a smooth projective lift mathscr X/S.scrX/S.
+The curve X/k admits a smooth projective lift X/S.scrX/S.
Concretely, smooth-lifting theory gives a smooth formal lift over
SpfR, since the obstruction to lifting a smooth curve
lies in H^2(X,T_X)=0; the ample line bundle O_X(1) lifts for the
-same reason (H^2(X,mathcal O_X)=0);calO_X)=0); Grothendieck's existence theorem
+same reason (H^2(X,O_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
@@ -420,33 +357,33 @@ cite[S3.1.1]{GR}. This construction does not require that X descends
to a finite field.
The root datum of the split G gives a split reductive group over
-mathbb Z,bbZ, and base change to S gives a lift, which we also denote
+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=calB=Bun_G(mathscr X/S),scrX/S),
+ B= X/S),scrX/S),
s=Spec(k), eta=Spec(K).
]
-Then mathcal B=calB=Hom_S(mathscr X,B_SG)scrX,B_SG) is an algebraic stack locally
+Then B= X,B_SG)scrX,B_SG) is an algebraic stack locally
of finite type over S with affine diagonal
-cite[Thm.~1.2(i)--(iii)]{HallRydh}: mathscr X/SscrX/S is proper, flat and
+cite[Thm.~1.2(i)--(iii)]{HallRydh}: X/SscrX/S is proper, flat and
of finite presentation, and B_SG is locally of finite presentation
and quasi-separated, with affine stabilisers and affine diagonal. The
obstruction to deforming a bundle on a curve lies in H^2(X,Ad P)=0,
-so mathcal BcalB is smooth over S.
+so B 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
subsequent models are understood to have been replaced in this way,
-and we continue to use R,S,mathscr X,scrX,mathcal B.calB. The point is that
+and we continue to use R,S,X,B. 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_Tcolon Shv(T_K)→Shv(T_s).
+ Psi_T: 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
@@ -463,12 +400,12 @@ are unchanged by the finite trait replacements above; the coefficient
functors and the models at successive stages agree on the identified
geometric-fibre coefficients.
-We will use one tensor identity repeatedly. If acolon Ttomathcal BcalB
-is smooth and mathcal LcalL is a finite-rank lisse sheaf on T, then
-for every DinShv(mathcal B_K)calB_K)
+We will use one tensor identity repeatedly. If a: TtoB
+is smooth and L is a finite-rank lisse sheaf on T, then
+for every DinShv(B_K)
- Psi_T(a_K^*D ⊗ mathcal L_K)calL_K)
- simeq a_s^*Psi_{mathcal B}calB(D) ⊗ mathcal L_s.calL_s.
+ Psi_T(a_K^*D ⊗ L_K)
+ ~= a_s^*Psi_{B} ⊗ L_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
@@ -481,88 +418,88 @@ D in the indicated ind-category.
Set
[
- mathcal C_s=calC_s=Shv_{Nilp}(mathcal B_s),calB_s),
- mathcal C_K=calC_K=Shv_{Nilp}(mathcal B_K),calB_K),
- Y_K=LS^{restr}_{check G}(mathscr X_K).scrX_K).
+ C_s={Nilp}(B_s),
+ C_K={Nilp}(B_K),
+ Y_K=LS^{restr}_{check G}(X_K).
]
Restriction to the closed fibre gives an identification
-QLisse(mathscr X)scrX)simeqQLisse(X) cite[S3.1.3]{GR}. Its inverse,
+QLisse(X)~=QLisse(X) cite[S3.1.3]{GR}. Its inverse,
composed with restriction to the geometric generic fibre, gives a
symmetric monoidal functor
[
- rhocolonQLisse(X)toQLisse(mathscr X_K).scrX_K).
+ rhocolonQLisse(X)toQLisse(X_K).
]
The corresponding map
[
- icolon Y↪ Y_K
+ i: 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
- mathcal D_Y:=calD_Y:=QCoh(Y)underset{QCoh(Y_K)}{ ⊗ }mathcal C_KcalC_K
+ D_Y:=calD_Y:=QCoh(Y)underset{QCoh(Y_K)}{ ⊗ }C_K
-is a direct summand of mathcal C_K.calC_K.
+is a direct summand of C_K.
We need the actual specialisation functor to commute with this summand,
-not merely a functor obtained by transporting mathbb L_GbbL_G across the
+not merely a functor obtained by transporting L_G across the
specialisation. Corollary~4.3.6 and S4.3.9 of [cite] show that the
chartwise geometric nearby-cycle functor we built above restricts to
[
- Psi_{mathcal B}calBcolon mathcal D_Y→calD_Y→ mathcal C_s,calC_s,
+ Psi_{B}: D_Y→calD_Y→ C_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},
together with its spectral linearity, identifies
- mathcal D_YcalD_YsimeqIndCoh_{Nilp}(Y).
+ D_Y~=IndCoh_{Nilp}(Y).
In positive characteristic we use only the primed version:
-mathcal C_scalC_ssimeqIndCoh_{Nilp}(Y') for some open-and-closed
+C_s~=IndCoh_{Nilp}(Y') for some open-and-closed
Y'subset Y.
If a connected component Csubset Y is disjoint from Y', then the
summand
[
- mathcal D_C:=calD_C:=QCoh(C)underset{QCoh(Y)}{ ⊗ }mathcal D_YcalD_Y
+ D_C:=calD_C:=QCoh(C)underset{QCoh(Y)}{ ⊗ }D_Y
]
-of mathcal D_YcalD_Y is nonzero; Psi_{mathcal B}calB(D)=0 for every
-Dinmathcal D_C;calD_C; and there exists a nonzero Finmathcal D_CcalD_C
-whose image under mathcal D_YcalD_Ysubsetmathcal C_KcalC_Ksubset
-Shv(mathcal B_K)calB_K) is compact, and which restricts to a bounded
-constructible complex on every finite-type smooth chart Utomathcal B.calB.
+of D_Y is nonzero; Psi_{B} for every
+DinD_C; and there exists a nonzero FinD_C
+whose image under D_Y C_KcalC_Ksubset
+Shv(B_K) is compact, and which restricts to a bounded
+constructible complex on every finite-type smooth chart UtoB.
Proof.
Let e_CinQCoh(Y) be the structure sheaf of C extended by zero.
-Its action is the projection onto mathcal D_C.calD_C. On mathcal C_scalC_s
+Its action is the projection onto D_C. On C_s
the same object acts by zero, because Ccap Y'=∅ and the
primed equivalence is spectral-linear. Therefore for every
-Dinmathcal D_C,calD_C,
+DinD_C,
[
- Psi_{mathcal B}calB(D)
- simeqPsi_{mathcal B}calB(e_Ccdot D)
- simeq e_CcdotPsi_{mathcal B}calB(D)=0.
+ Psi_{B}
+ ~=Psi_{B}· D)
+ ~= e_C·Psi_{B}
]
-Under (eq:spec-generic-equivalence), mathcal D_CcalD_C is
+Under (eq:spec-generic-equivalence), D_C is
IndCoh_{Nilp}(C), which is nonzero: the fully faithful
-QCoh(C)↪IndCoh_{Nilp}(C) sends mathcal O_CcalO_Cne0 to
+QCoh(C)↪IndCoh_{Nilp}(C) sends O_C to
a nonzero object cite[SS1.3.1--1.3.4]{GR}.
-The category mathcal D_YcalD_Y is compactly generated
+The category D_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
+D_Y~=D_C 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_YcalD_Y under e_C; at least one
+compact generators of D_Y under e_C; at least one
projection is nonzero, since otherwise the colimit closure would give
-mathcal D_C=0.calD_C=0. Choose such an F.
+D_C=0. Choose such an F.
-Compactness in mathcal C_KcalC_K is the same argument: F is compact in
-mathcal D_Y,calD_Y, which is a summand of mathcal C_K,calC_K, and the inclusion
+Compactness in C_K is the same argument: F is compact in
+D_Y, which is a summand of C_K, and the inclusion
of a summand preserves compactness. The inclusion
-mathcal C_K↪calC_K↪Shv(mathcal B_K)calB_K) preserves compactness
+C_K↪calC_K↪Shv(B_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}.
@@ -584,8 +521,8 @@ Y'=Y there is nothing to prove.
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)
+ mathrm H_VrmH_VcolonC_K→calC_K→
+ Shv(B_K 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
@@ -594,10 +531,10 @@ facts about the universal family.
The universal objects are
[
- mathcal E_calE_{V,s}inQCoh(Y) ⊗ QLisse(X),
- mathcal E_calE_{V,K}inQCoh(Y_K) ⊗ QLisse(mathscr X_K).scrX_K).
+ E_{V,s}inQCoh(Y) ⊗ QLisse(X),
+ E_{V,K}inQCoh(Y_K) ⊗ QLisse(X_K).
]
-On an affine test Tto Y the restriction of mathcal E_VcalE_V is the
+On an affine test Tto Y the restriction of E_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_VrmH_V on the spectral side
@@ -605,8 +542,8 @@ cite[S14.3.3]{AGKRRV}.
By definition of i,
- (i^* ⊗ id)(mathcal E_calE_{V,K})
- simeq(id ⊗ rho)(mathcal E_calE_{V,s}).
+ (i^* ⊗ id)(E_{V,K})
+ ~=(id ⊗ rho)(E_{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
@@ -618,32 +555,32 @@ commutes with the descent limit defining QCoh(Y).
The second, distinct input is cite[Rem.~14.3.9]{AGKRRV}:
- mathcal E_calE_{V,s}inQCoh(Y) ⊗ IndLisse(X),
+ E_{V,s}inQCoh(Y) ⊗ IndLisse(X),
where IndLisse(X)=Ind(Lisse(X)) and
Lisse(X) consists of bounded constructible complexes
with finite-rank lisse cohomology sheaves. We do not identify
IndLisse(X) with QLisse(X); that identification fails for
-X=mathbb P^1.bbP^1. The two properties used above (dualizability for
+X=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_CcalD_C ⊗ IndLisse(X). After applying
+result is an object of D_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
+generate D_C ⊗ IndLisse(X) under colimits, and bounded
truncations build every object of Lisse(X) from
finite-rank lisse sheaves. Therefore mathrm H_V(F)rmH_V(F) lies in the stable
colimit closure of objects
- Dboxtimesmathcal L_K,calL_K, Dinmathcal D_C,calD_C,
+ DboxtimesL_K, DinD_C,
-where mathcal L_KcalL_K is the restriction of a finite-rank lisse sheaf
-mathcal L_ScalL_S on mathscr X.scrX.
+where L_K is the restriction of a finite-rank lisse sheaf
+L_S on X.
-Let acolon Ttomathcal BcalB be smooth, with T a finite-type S-scheme
-or algebraic space, and let ell_Tcolon Ttomathscr XscrX be any
+Let a: TtoB be smooth, with T a finite-type S-scheme
+or algebraic space, and let ell_T: TtoX be any
S-morphism. Then for F as in Proposition~spec:missing and
every finite-dimensional V,
[
@@ -656,19 +593,19 @@ the finite trait replacements described above.
Proof.
-The first equality follows from Psi_{mathcal B}calB(F)=0 and smooth base
+The first equality follows from Psi_{B} and smooth base
change along a. For an exterior generator in
(eq:spec-exterior-generators), ordinary pullback gives
[
- (a,ell_T)_K^*(Dboxtimesmathcal L_K)calL_K)
- =a_K^*D ⊗ ell_{T,K}^*mathcal L_K.calL_K.
+ (a,ell_T)_K^*(DboxtimesL_K)
+ =a_K^*D ⊗ ell_{T,K}^*L_K.
]
-The sheaf ell_T^*mathcal L_ScalL_S is lisse on T, regardless of whether
+The sheaf ell_T^*L_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_{B} 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.
+ Psi_T(a_K^*D ⊗ ell_{T,K}^*L_K)
+ ~= a_s^*Psi_{B} ⊗ ell_{T,s}^*L_s=0.
]
Both pullback and specialisation are continuous, so the same vanishing
holds for the stable colimit closure of these exterior generators. That
@@ -677,7 +614,7 @@ geometric specialisation, so no new hypothesis is needed there.
[What is and is not used]
-The proof uses smoothness only of acolon Ttomathcal B;calB; the leg
+The proof uses smoothness only of a: TtoB; 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
@@ -695,8 +632,8 @@ nonzero incoming covector and both transverse derivatives.
% =========================================================================
Section~sec:specialization reduced the main theorem to the following
-local problem. Suppose Pinmathcal B_s(k)calB_s(k) is a bundle and
-Ain T^*_Pmathcal B_scalB_s is a non-nilpotent Higgs field: a section
+local problem. Suppose PinB_s(k) is a bundle and
+Ain T^*_PB_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
@@ -717,7 +654,7 @@ Hypothesis~hyp:A or Hypothesis~hyp:B, and let kappa be the
invariant symmetric form of Hypothesis~hyp:common(i). Let
a(t)inmathfrak g[[t]]frakg[[t]] satisfy chi(a(0))nechi(0). Then after a
change of frame by an element of G(k[[t]]), there are a Levi subgroup
-Msubset G and a cocharacter lambdacolonmathbb G_mbbG_mto Z(M) such
+Msubset G and a cocharacter lambdacolonG_m Z(M) such
that
a(t)inmathfrak m[[t]],frakm[[t]], kappa(a(0),dlambda)ne0,
@@ -725,7 +662,7 @@ that
xrightarrow{ sim }mathfrak g/frakg/mathfrak m,frakm,
where mathfrak m=frakm=Lie M and dlambda=dlambda(1)inmathfrak gfrakg is the
-image of 1inLie(mathbb G_m)=k.bbG_m)=k. The subgroup M may equal G; the
+image of 1inLie(G_m)=k. The subgroup M may equal G; the
invertibility includes the zero vector space.
@@ -780,7 +717,7 @@ and works in characteristic two.
For the Borel-conjugation statement: consider the incidence morphism
[
Gtimes^Bmathfrak b→frakb→mathfrak g,frakg,
- [g,x]mapstoAd(g)x,
+ [g,x]|->Ad(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.
@@ -803,7 +740,7 @@ semisimple, u_0 is nilpotent, and their sum is semisimple, so the
Choose a closed faithful representation G↪GL(V).
Its differential realises mathfrak gfrakg as a restricted Lie subalgebra of
-End(V), with xmapsto x^{[p]} equal to matrix pth power. We use
+End(V), with x|-> x^{[p]} equal to matrix pth power. We use
the ordinary matrix Jordan decomposition; the two lemmas below prove
its two parts lie in mathfrak gfrakg in each regime; compare
cite[S4.4, Thm.~4.4.20]{Springer}.
@@ -814,7 +751,7 @@ semisimple, n is nilpotent in a faithful representation, and
[s,n]=0. Suppose also chi(a_0)nechi(0). After conjugation, s is
a nonzero element of mathfrak t,frakt, M=Z_G(s) is a Levi subgroup,
a_0inmathfrak m,frakm, and ad_{a_0} is invertible on
-mathfrak g/frakg/mathfrak m.frakm. Moreover for every lambdacolonmathbb G_mbbG_m
+mathfrak g/frakg/mathfrak m.frakm. Moreover for every lambdacolonG_m
to Z(M),
kappa(a_0,dlambda)=kappa(s,dlambda).
@@ -857,7 +794,7 @@ diverge; everything else is common.
## Regime A: a lattice argument for actual central cocharacters
-Assume Hypothesis~hyp:A, so k=mathbb Fbar_q.bbFbar_q. Two facts
+Assume Hypothesis~hyp:A, so k=Fbar_q. Two facts
are needed. First, the Jordan parts s,n of a_0 constructed in
Lemma~lie:constant-parts actually lie in mathfrak g;frakg; they start
life only in an ambient End(V). Second, some lambdain
@@ -866,7 +803,7 @@ both are integral.
The Jordan parts first. In a faithful representation write a_0=s+n for
the matrix Jordan decomposition. The eigenvalues of s lie in a finite
-field mathbb F_bbF_{p^h}; choose e divisible by h and large enough
+field F_{p^h}; choose e divisible by h and large enough
to have n^{p^e}=0. Then a_0^{[p^e]}=a_0^{p^e}=s^{p^e}+n^{p^e}=s^{p^e}
so s=a_0^{[p^e]}, hence sinmathfrak gfrakg as the p^eth restricted
power of an element of mathfrak g.frakg. Consequently ninmathfrak gfrakg too.
@@ -902,9 +839,9 @@ The form on this center is nondegenerate by hypothesis, and on
mathfrak tfrakt by Lemma~lie:root-pairing; nondegeneracy on J_I
follows.
-Define integral maps UcolonΛtomathbb Z^r,bbZ^r,
-ymapsto(<alpha_i,y>)_i, and Vcolonmathbb Z^rbbZ^rto
-Λ, e_jmapstoalpha_j^vee. Their composite is the Cartan matrix
+Define integral maps UcolonΛtoZ^r,
+y|->(<alpha_i,y>)_i, and VcolonZ^r
+Λ, e_j|->alpha_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
@@ -968,18 +905,18 @@ kappa(a_0,dlambda)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-linearbbF_p-linear additive polynomial.
+Frobenius using an F_p-linearbbF_p-linear additive polynomial.
In the faithful representation write a_0=s+n and choose e with
n^{p^e}=0, so a_0^{p^e}=s^{p^e}. Let V_0subset k be the
-finite-dimensional mathbb F_p-spanbbF_p-span of the eigenvalues of s, and let
+finite-dimensional 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,
+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
+(Lemma~lie:constant-parts). Hence r=dim_{F_p} and
we may assume V_0ne0.
-Take a basis beta_1,...,beta_r of V_e over mathbb F_p.bbF_p. The
+Take a basis beta_1,...,beta_r of V_e over F_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
@@ -1086,8 +1023,8 @@ allowed characteristic.
% =========================================================================
Section~sec:lie produced Levi data. Now we turn it into a
-modification of a bundle. Let Pinmathcal B_s(k)calB_s(k) and let
-Ain T^*_Pmathcal B_scalB_s be non-nilpotent. We will choose a point
+modification of a bundle. Let PinB_s(k) and let
+Ain T^*_PB_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*)]
@@ -1101,9 +1038,9 @@ These are the "transversality" and "isolation" conditions.
## Correspondence and kernels
-We keep the trait S, the curve mathscr X/SscrX/S and the bundle stack
-mathcal BcalB from Section~sec:specialization. A one-point Hecke
-modification is a quadruple (P,P',x,alpha) where xinmathscr XscrX
+We keep the trait S, the curve X/SscrX/S and the bundle stack
+B from Section~sec:specialization. A one-point Hecke
+modification is a quadruple (P,P',x,alpha) where xinX
and alpha identifies P' and P away from the graph of x. Write
[
p(P,P',x,alpha)=P, o(P,P',x,alpha)=P',
@@ -1114,18 +1051,18 @@ 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
+Z= BcalBtimes_SX. Denote the exact-position
correspondence by H=H^lambda, and set
[
- widetilde p=(p,ell_H)colon Htomathcal Z,calZ,
- q=(o,ell_H)colon Htomathcal Z.calZ.
+ widetilde p=(p,ell_H)colon HtoZ,
+ q=(o,ell_H)colon HtoZ.
]
Both widetilde p and q forget some of the modification data.
[Hecke models and kernel]
Both widetilde p and q are representable, separated and smooth of
relative dimension m. The correspondence H is open in a bounded
-correspondence overlineoverline HH for which both projections to mathcal ZcalZ
+correspondence overlineoverline HH for which both projections to Z
are representable and proper. On the output side the bound on
overlineoverline HH has dominant label mu^vee.
@@ -1133,9 +1070,9 @@ Let V be the irreducible representation of check G corresponding
to this bound, with the indicated input/output convention. Then the
generic Hecke functor has the form
- mathrm H_V(F)=rmH_V(F)=overlineoverline q_q_{K,*}mathcal I_K,calI_K,
+ mathrm H_V(F)=rmH_V(F)=overlineoverline q_q_{K,*}I_K,
-with mathcal I_KcalI_K on H_K equal to p_K^*F tensored with a fixed
+with I_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).
@@ -1145,7 +1082,7 @@ Consider the split integral affine Grassmannian with loops G(A((t)))
and positive loops G(A[[t]]). Its integral Schubert bound is
projective and the positive-loop orbit is smooth, open, fibre-dense of
relative dimension m cite[SS1.2,~2.1]{RicharzScholbach}. We use these
-models by base change from mathbb Z.bbZ.
+models by base change from Z.
A graph coordinate and a formal-disc frame identify modifications of
P with the corresponding Grassmannian spaces; these descriptions glue
@@ -1187,12 +1124,12 @@ intersection complex.
## The cotangent identity and the transverse section
-On the special fibre a cotangent vector to mathcal B_scalB_s at P is an
+On the special fibre a cotangent vector to B_s at P is an
element of H^0 of the cotangent-complex fibre. Deformation theory and
Serre duality cite[Tag~0BS4]{Stacks}, together with kappa, identify
this space as
[
- T_P^*mathcal B_s=H^1(X,calB_s=H^1(X,mathfrak g_P)^*=H^0(X,frakg_P)^*=H^0(X,mathfrak g_Pfrakg_P ⊗ omega_X),
+ T_P^*B_s=H^1(X,mathfrak g_P)^*=H^0(X,frakg_P)^*=H^0(X,mathfrak g_Pfrakg_P ⊗ omega_X),
]
where mathfrak g_Pfrakg_P is the adjoint bundle. We call such a vector A a
Higgs field. For a representable smooth map, the cotangent
@@ -1201,12 +1138,12 @@ relative cotangent; we use this in the form supplied by the cotangent
triangle.
[Transverse modification]
-Let Pinmathcal B_s(k)calB_s(k) and let Ain T^*_Pmathcal B_scalB_s be
+Let PinB_s(k) and let Ain T^*_PB_s be
non-nilpotent. There exist a point xin X(k), a cocharacter lambda,
a modification hin H^lambda_s(k) from P to a bundle P', and a
covector
[
- theta=(A',xi)in T^*_{(P',x)}(mathcal B_scalB_stimes X), xine0,
+ theta=(A',xi)in T^*_{(P',x)}(B_s X), xine0,
]
with A'=A away from x, satisfying
@@ -1219,7 +1156,7 @@ Moreover, on the fixed-side fibres
]
with fixed-bundle identifications retained, the following hold. The
restriction of p^*A to the q-relative tangent bundle defines a
-section S_A of Ω_{H_s/mathcal Z_s,q}calZ_s,q; the restriction of
+section S_A of Ω_{H_s/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
@@ -1239,7 +1176,7 @@ t, a bundle frame at x, and write A=a(t),dt. Chevalley restriction
identifies the nilpotent fibre with chi^{-1}(chi(0)), so
chi(a(0))nechi(0). Proposition~lie:input then supplies a
formal change of frame, a Levi M, and a cocharacter
-lambdacolonmathbb G_mbbG_mto Z(M) with
+lambdacolonG_m Z(M) with
a(t)inmathfrak m[[t]],frakm[[t]],
ad_{a(0)} invertible on mathfrak g/frakg/mathfrak m,frakm,
@@ -1258,7 +1195,7 @@ assertions explicit.
Tangent spaces. In the input frame, put
L=mathfrak g[[t]],frakg[[t]], L'=Ad(lambda(t))L, I=Lcap L'. Let
mathfrak g_jfrakg_j be the weight-j space of lambda; the integers j
-are genuine characters of mathbb G_m,bbG_m, not reduced mod p. Then
+are genuine characters of G_m, not reduced mod p. Then
L'=⊕_j t^jmathfrak g_j[[t]],frakg_j[[t]],
P_1:=L/I=⊕_{j>0}mathfrak g_jfrakg_j ⊗ k[[t]]/(t^j),
@@ -1278,7 +1215,7 @@ canonical isomorphism, no exponential map needed.
There is a perfect pairing
- betacolon P_1times P_2to k,
+ beta: P_1times P_2to k,
beta(D,C)=Res_{t=0}kappa(D,C),dt.
Indeed, L is its own residue-annihilator on mathfrak g((t)),frakg((t)), and
@@ -1340,7 +1277,7 @@ invertible by (hecke:lie-data). The corresponding k[[t]]-linear
block has unit determinant, so ad(a(t)) is invertible on every
truncated summand of P_1 and P_2. Combined with
(hecke:derivative) and (hecke:residue-pairing), this shows
-(dS_A)_hcolon P_1to P_2^* is an isomorphism. Zero-weight directions
+(dS_A)_h: 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.
@@ -1368,7 +1305,7 @@ Modifications of this form, and tori, are thus included.
The proposition gives both properties we need. The nonvanishing
xine0 allows a suitable hypersurface cut of the Hecke correspondence
-to be smooth over mathcal B,calB, while the two relative derivatives
+to be smooth over B, 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.
@@ -1388,21 +1325,21 @@ Proposition~hecke:transverse identify the local neighbourhood of
that point with a split quadratic equation. The next section removes
the extra variables in that equation.
-Throughout this section, FinShv(mathcal B_K)calB_K) is the specialising
+Throughout this section, FinShv(B_K) is the specialising
object of Proposition~spec:missing, and we use the vanishings of
Lemma~spec:vanishing freely.
[Vanishing at the quadratic vertex]
-Let bcolon Utomathcal BcalB be a smooth chart with U finite type over
-S. Let finmathcal O(U)calO(U) and uin U_s(k) satisfy f(u)=0, and
-suppose df_u=b^*A for some non-nilpotent Ain T^*_{b(u)}mathcal B_s.calB_s.
-Then there exist xinmathscr X_s(k)scrX_s(k) and an integer mge0 with the
-following property. Put T=Utimes_Smathscr XscrX and let F_T be the
+Let b: UtoB be a smooth chart with U finite type over
+S. Let finO(U) and uin U_s(k) satisfy f(u)=0, and
+suppose df_u=b^*A for some non-nilpotent Ain T^*_{b(u)}B_s.
+Then there exist xinX_s(k) and an integer mge0 with the
+following property. Put T=Utimes_SX and let F_T be the
pullback of F_U=b_K^*F. Consider the affine quadric
[
- Q^a={sum_{i=1}^m u_iv_i+f=0}subset Ttimes_Smathbb A_S^bbA_S^{2m},
- taucolon Q^ato T,
+ Q^a={sum_{i=1}^m u_iv_i+f=0}subset Ttimes_SA_S^{2m},
+ tau: 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
@@ -1410,7 +1347,7 @@ nu=(u,x,0,0),
(Psi_{Q^a}(tau_K^*F_T))_nu=0.
-When m=0, this is the vanishing on Q^a=V(f)times_Smathscr X.scrX.
+When m=0, this is the vanishing on Q^a=V(f)times_SX.
Proof.
@@ -1421,7 +1358,7 @@ theta=(A',xi) with xine0, satisfying
p^*A=q^*thetaquadat h.
Use the exact-position correspondence H=H^lambda, its proper bound
-overlineoverline H,H, and its generic Satake integrand mathcal I_KcalI_K from
+overlineoverline H,H, and its generic Satake integrand I_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.
@@ -1429,29 +1366,29 @@ 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_2times_{p_2,mathcal B,b}calB,bU,
- W=H_2times_{mathcal B}calBDsubset W_0.
+ W_0=H_2times_{p_2,B,b}calB,bU,
+ W=H_2times_{B} W_0.
]
-Both map to mathcal Z=calZ=mathcal BcalBtimes_Smathscr XscrX by
-q_2=(o_2,ell_{H_2}). The map W_0tomathcal ZcalZ is smooth: its
+Both map to Z= BcalBtimes_SX by
+q_2=(o_2,ell_{H_2}). The map W_0toZ is smooth: its
projection to H_2 is a base change of the smooth chart b, and q_2
is smooth. At w=(h,u), the differential of the equation cutting out
W from W_0 is the pullback of theta by (eq:two-cotangent).
Its relative differential over the output bundle stack is nonzero,
because the leg component is xine0. The smooth-hypersurface
-criterion therefore makes Wtomathcal BcalB (via the output bundle)
+criterion therefore makes WtoB (via the output bundle)
smooth near w. The arbitrary-leg assertion of
Lemma~spec:vanishing gives
(Psi_W(q_{2,K}^*mathrm H_V(F)))_w=0.rmH_V(F)))_w=0.
-Note: smoothness is used only for the projection to mathcal B,calB, matching
+Note: smoothness is used only for the projection to B, matching
the hypothesis of Lemma~spec:vanishing.
Now form fibre products
[
- P=H_1times_{mathcal Z}calZW_0,
- P_D=H_1times_{mathcal Z}calZWsubsetoverlineoverline P_D=P_D=overlineoverline H_1H_1times_{mathcal Z}calZW.
+ P=H_1times_{Z},
+ P_D=H_1times_{Z} P_D=P_D=overlineoverline H_1H_1times_{Z}
]
The following squares are cartesian; in the second the vertical map
pi is proper. In the first, Delta repeats the modification in the
@@ -1460,55 +1397,55 @@ 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 ZcalZ
+ & Z
[column sep=large,row sep=large]
overlineoverline P_DP_D arrow[r] arrow[d,"pi"']
& overlineoverline H_1H_1 arrow[d,"overlineoverline q_1"]q_1"]
- W arrow[r,"q_2"'] & mathcal Z.calZ.
+ W arrow[r,"q_2"'] & Z.
]
-Write ccolon Pto U for the map through W_0, and
-acolon Ptomathcal BcalB for the input map p_1. So a records the
+Write c: Pto U for the map through W_0, and
+a: PtoB for the input map p_1. So a records the
input from which the sheaf is pulled back, and c records the second
input in U on which we impose f=0. The distinguished point is
z=(h,h,u)in P_D(k).
-Let mathcal Ibar_KcalIbar_K be the pullback of the Satake integrand
+Let Ibar_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).
+Rpi_{K,*}Ibar_K~= 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),
- mathcal K=(calK=(Psi_{overlineoverline P_D}P_Dmathcal Ibar_K)|_calIbar_K)|_{Z_w},
+ K=({overlineoverline P_D}P_DIbar_K)|_calIbar_K)|_{Z_w},
]
we get
- RGamma(Z_w,mathcal K)=0.calK)=0.
+ RGamma(Z_w,K)=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 acolon Ptomathcal BcalB is smooth: Pto H_1 is a base change
-of W_0tomathcal Z,calZ, and p_1 is smooth. At a point (h_1,w) in the
+The map a: PtoB is smooth: Pto H_1 is a base change
+of W_0toZ, and p_1 is smooth. At a point (h_1,w) in the
exact-position part of Z_w, the differential of f(c) is q_1^*theta
-pulled back. If the relative differential over mathcal BcalB of f(c)
+pulled back. If the relative differential over B 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_Dtomathcal BcalB
+zero at h on the fixed-output-and-leg fibre. Hence P_DtoB
via a is smooth at every point of a punctured neighbourhood of z
in Z_w. On the exact-position open, the generic integrand equals
a_K^*F. Lemma~spec:vanishing therefore gives a neighbourhood
O of z in Z_w with
[
- mathcal K|_calK|_{O∖{z}}=0.
+ K|_calK|_{O∖{z}}=0.
]
This local vanishing prevents any cancellation with the boundary of the
@@ -1516,10 +1453,10 @@ Hecke bound. Put C_w=Z_w∖ O (reduced structure). The closed
subsets {z} and C_w are disjoint, so open-closed localization
identifies
[
- mathcal KcalKsimeq i_{z,*}mathcal K_zcalK_z ⊕ i_{C_w,*}(i_{C_w}^*mathcal K),calK),
+ K~= i_{z,*}K_z ⊕ i_{C_w,*}(i_{C_w}^*K),
- RGamma(Z_w,mathcal K)calK)simeq
- mathcal K_zcalK_z ⊕ RGamma(C_w,i_{C_w}^*mathcal K).calK).
+ RGamma(Z_w,K)~=
+ K_z ⊕ RGamma(C_w,i_{C_w}^*K).
]
The residue field of z is algebraically closed, so the first summand
is its stalk complex. Equation~(eq:two-fiber-zero) forces it to
@@ -1531,24 +1468,24 @@ No boundary behaviour of the Satake kernel is used.
Step 3: chart lift along the whole diagonal.:
We must now translate (eq:two-isolated-zero) into coordinates.
-First, the distinguished diagonal Deltasimeq W_0subset P is a
+First, the distinguished diagonal Delta~= 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
+specified isomorphism bc~= a, coming from the two identical
modifications.
We claim that after an 'etale replacement of P near z, there
-is a map rcolon Pto U with an isomorphism
+is a map r: Pto U with an isomorphism
- brsimeq a, r|_Delta=c|_Delta,
+ br~= a, r|_Delta=c|_Delta,
whose restriction to Delta is the specified isomorphism. For this,
-form the smooth space E_P=Ptimes_{a,mathcal B,b}calB,bUto P. The
+form the smooth space E_P=Ptimes_{a,B,b}calB,bUto P. The
diagonal data give a section over Delta. First take an 'etale
-scheme neighbourhood in P. Since mathcal BcalB has affine diagonal
+scheme neighbourhood in P. Since B has affine diagonal
(Section~sec:specialization), E_Pto Ptimes_S U is affine, so
E_P is a scheme as well. Near the section point, smooth coordinates
-give an 'etale map E_Ptomathbb A_P^bbA_P^{e}
+give an 'etale map E_PtoA_P^{e}
cite[Tag~039P]{Stacks}. After shrinking, the section over Delta
lands in this coordinate neighbourhood. Extend its coordinate functions
Deltato P, then pull back the 'etale map along the extended tuple.
@@ -1563,7 +1500,7 @@ Step 4: exact quadratic coordinates.:
After shrinking U, suppose U/S has constant relative dimension N.
The map
[
- Deltato T=Utimes_Smathscr X,scrX,(c,leg),
+ Deltato T=Utimes_SX,(c,leg),
]
is smooth of relative dimension m. So T,Delta,P are smooth over
S of relative dimensions
@@ -1602,7 +1539,7 @@ fibre in (eq:two-coefficients), not just at z.
Now set
[
Phi=(r,leg,u_1,...,u_m,v_1,...,v_m)colon
- Pto Ttimes_Smathbb A_S^bbA_S^{2m}.
+ Pto Ttimes_SA_S^{2m}.
]
Its relative differential at z is an isomorphism: a tangent vector in
its kernel is first killed by every dv_i, so lies in T_zDelta; its
@@ -1614,11 +1551,11 @@ isomorphism, and the relative Jacobian criterion makes Phi 'etale
at z.
By (eq:two-exact-quadratic), the cut P_D=V(f(c)) is exactly
-Phi^{-1}(Q^a). So the induced map Phi_Dcolon P_Dto Q^a is
+Phi^{-1}(Q^a). So the induced map Phi_D: P_Dto Q^a is
'etale near z, sending z to nu. Moreover (eq:two-chart-lift)
identifies
[
- a_K^*Fsimeq r_K^*F_UsimeqPhi_{D,K}^*tau_K^*F_T
+ a_K^*F~= r_K^*F_U~=Phi_{D,K}^*tau_K^*F_T
quadon (P_D)_K,
]
so Phi_D identifies the actual generic sheaf and the equation
@@ -1627,8 +1564,8 @@ 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=Dtimes_Smathscr X.scrX. Then smooth base change along Dtimes_S
-mathscr XscrXto D yields (Psi_D(F_U|_{D_K}))_u=0 directly.
+Q^a=Dtimes_SX. Then smooth base change along Dtimes_S
+X D yields (Psi_D(F_U|_{D_K}))_u=0 directly.
@@ -1659,12 +1596,12 @@ of equations simultaneously.
[Quadric comparison]
Let K_0 be an algebraically closed field of characteristic neell,
-E=mathbb Qbar_bbQbar_ell, and B a finite-type K_0-scheme. For
-finGamma(B,mathcal O_B)calO_B) put icolon D=V(f)↪ B. If
+E=Qbar_, and B a finite-type K_0-scheme. For
+finGamma(B,O_B) put i: 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.
+ g: Q_f=Bigl{sum_{a=1}^m u_av_a+fw^2=0Bigr}
+ subsetP_B^{2m}→ B.
]
Then the hyperplane-class map defines a distinguished triangle
@@ -1679,10 +1616,10 @@ or reduced.
Proof.
-Write h=c_1(mathcal O_calO_{Q_f}(1))in H^2(Q_f,E(1)). Adjunction on each
+Write h=c_1(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 KcalK be the cone of their direct sum, and let
-j_Bcolon B∖ D↪ B.
+K be the cone of their direct sum, and let
+j_B: 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
@@ -1693,27 +1630,27 @@ n-space, whose complement is the projective cone over the quadric
of dimension n-2 with the vertex removed (a line bundle over that
smaller quadric). The resulting filtration by affine cells has one cell
in every dimension 0,...,n, and a second middle-dimensional cell
-when n is even. Initial cases are a conic (congmathbb P^1)bbP^1) and
+when n is even. Initial cases are a conic (congP^1) and
the split zero-dimensional quadric (two points). Localization plus
-RGamma_c(mathbb A^r,E)=E(-r)[-2r]bbA^r,E)=E(-r)[-2r] gives the cohomology. On a smooth
+RGamma_c(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.calK=0.
+gives j_B^*K=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=0Bigr}subsetmathbb P_bbP_{K_0}^{2m}.
+ V_m=Bigl{sum_{a=1}^m u_av_a=0Bigr}subsetP_{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
smooth quadric
[
- C_m=Bigl{sum_{a=1}^m u_av_a=0Bigr}subsetmathbb P_bbP_{K_0}^{2m-1}.
+ C_m=Bigl{sum_{a=1}^m u_av_a=0Bigr}subsetP_{K_0}^{2m-1}.
]
Compactly supported cohomology of this line bundle, plus localization at
the vertex, gives
@@ -1721,22 +1658,22 @@ the vertex, gives
H^0(V_m,E)=E, H^1(V_m,E)=0,
H^r(V_m,E)=H^{r-2}(C_m,E)(-1)(rge2).
-So V_m has the cohomology of mathbb P^bbP^{2m-1} plus one extra
+So V_m has the cohomology of P^{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
[
- picolonwidetilde V_m=mathbb P_bbP_{lines,C_m}
- (mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m})
+ picolonwidetilde V_m=P_{lines,C_m}
+ (O_{C_m}(-1) ⊕ O_{C_m})
→ C_m.
]
-The inclusion mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m}↪
+The inclusion O_{C_m}(-1) ⊕ O_{C_m}↪
(trivial rank-(2m+1)) defines widetilde V_mto V_m; its
fibre over a point of C_m is the projective line joining that point
to e. If zeta is the pullback of the hyperplane class and
-h_C=c_1(mathcal O_calO_{C_m}(1)), the projective-bundle relation is
+h_C=c_1(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.
@@ -1751,14 +1688,14 @@ 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 KcalK with the constant
+Proper base change identifies i^*K with the constant
pullback of this vector-space cone (both the family over D and the
hyperplane-power morphism are pulled back from V_m). Open-closed
localization gives
[
- (j_B)_!j_B^*mathcal KcalKtomathcal KcalKto i_*i^*mathcal KcalKtocdot,
+ (j_B)_!j_B^*K KcalKto i_*i^*K·,
]
-whose first term vanishes, so mathcal KcalKsimeq i_*E_D(-m)[-2m]. In
+whose first term vanishes, so K~= 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
@@ -1770,27 +1707,27 @@ 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^abbZ/2^a coefficients: on a conic the hyperplane class has
+with Z/2^abbZ/2^a coefficients: on a conic the hyperplane class has
degree 2. We use finite coefficients to build the standard
compatibilities, then use (eq:slice-quadric-triangle) after
rationalizing. This imposes no restriction on residue characteristic.
The case m=0 is handled below directly; formally Q_f=Dsubset
-mathbb P_B^0bbP_B^0 and the hyperplane sum is empty.
+P_B^0 and the hyperplane sum is empty.
## The hypersurface test
-Work over the trait S, with mathcal BcalB and Finmathcal D_CcalD_C as in
+Work over the trait S, with B and FinD_C as in
Proposition~spec:missing. For a smooth finite-type chart
-bcolon Utomathcal B,calB, write F_U=b_K^*F, and let
-mathcal N_UcalN_Usubset T^*U_s be the image of the global nilpotent cone
+b: UtoB, write F_U=b_K^*F, and let
+N_U T^*U_s be the image of the global nilpotent cone
under smooth cotangent pullback.
[Hypersurface test]
-Let bcolon Utomathcal BcalB be smooth with U finite type over S.
-Let finmathcal O(U),calO(U), D=V(f), and uin D_s(k). If
+Let b: UtoB be smooth with U finite type over S.
+Let finO(U), D=V(f), and uin D_s(k). If
[
- d(f_s)_unotin(mathcal N_U)_u,calN_U)_u,
+ d(f_s)_unotin(N_U)_u,
]
then
[
@@ -1799,22 +1736,22 @@ then
Proof.
-If d(f_s)_u does not lie in the image of T^*_{b(u)}mathcal B_s,calB_s,
-its image in the relative cotangent of b is nonzero, so Dtomathcal BcalB
+If d(f_s)_u does not lie in the image of T^*_{b(u)}B_s,
+its image in the relative cotangent of b is nonzero, so DtoB
is smooth near u; Lemma~spec:vanishing gives the claim.
Otherwise d(f_s)_u=b^*A for some non-nilpotent
-Ain T^*_{b(u)}mathcal B_s.calB_s. Proposition~two:vertex then provides
-xinmathscr X_s(k)scrX_s(k) and mge0 such that nearby cycles of the
+Ain T^*_{b(u)}B_s. Proposition~two:vertex then provides
+xinX_s(k) and mge0 such that nearby cycles of the
pullback of F_U to
Q^a={sum_{a=1}^m u_av_a+f=0} vanish at (t_0,0,0), where
-T=Utimes_Smathscr XscrX and t_0=(u,x). If m=0 this affine model is
+T=Utimes_SX 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.
+ g: Q_f=Bigl{sum_{a=1}^m u_av_a+fw^2=0Bigr}
+ subsetP_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
@@ -1832,7 +1769,7 @@ fibre. Proper compatibility plus the projection formula give
Apply Lemma~slice:quadric on T_K and tensor with F_T: the
projection formula identifies the third term with
i_{K,*}(F_T|_{(D_T)_K})(-m)[-2m] for
-icolon D_T↪ T. After Psi_T and stalk at t_0,
+i: 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
@@ -1849,12 +1786,12 @@ equations simultaneously.
[Incidence comparison]
Let B be a finite-type scheme over an algebraically closed field of
-characteristic neell, let f_1,...,f_dinGamma(B,mathcal O_B)calO_B)
-with dge1, and set icolon J=V(f_1,...,f_d)↪ B.
+characteristic neell, let f_1,...,f_dinGamma(B,O_B)
+with dge1, and set i: J=V(f_1,...,f_d)↪ B.
For the incidence family
[
- rcolon I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
- subset Btimesmathbb P^bbP^{d-1}→ B,
+ r: I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
+ subset BtimesP^{d-1}→ B,
]
the hyperplane powers give a distinguished triangle
@@ -1867,16 +1804,16 @@ For d=1 the first term is zero and I=J.
Proof.
-If d=1, the equation in Btimesmathbb P^0bbP^0 is f_1=0 and the
+If d=1, the equation in BtimesP^0 is f_1=0 and the
statement is immediate. Suppose dge2. Off J, the map
-mathcal O_B^dcalO_B^dtomathcal O_BcalO_B with coefficients (f_1,...,f_d) is
+O_B^d O_BcalO_B with coefficients (f_1,...,f_d) is
surjective, so its kernel is a vector bundle of rank d-1 and I is
its projective bundle of lines. The restricted hyperplane powers identify
its direct image with the first term of
(eq:slice-incidence-triangle) (check on geometric fibres, using
-proper base change and the cohomology of mathbb P^bbP^{d-2}).
+proper base change and the cohomology of P^{d-2}).
-On J, the family is exactly Jtimesmathbb P^bbP^{d-1} and its hyperplane
+On J, the family is exactly JtimesP^{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
@@ -1884,13 +1821,13 @@ this restricted cone.
[Higher-codimension test]
-Let bcolon Utomathcal BcalB be smooth with U finite type over S,
+Let b: UtoB be smooth with U finite type over S,
and let F_U=b_K^*F. Suppose uin U_s(k) and f_1,...,f_d are
regular on U vanishing at u. Suppose their special-fibre
differentials at u are linearly independent and
[
span_k{d(f_{1,s})_u,...,d(f_{d,s})_u}
- cap(mathcal N_U)_u=calN_U)_u={0}.
+ cap(N_U)_u={0}.
]
Then for J=V(f_1,...,f_d),
[
@@ -1903,17 +1840,17 @@ Proof.
For d=0 this is Lemma~spec:vanishing. For dge1 form the
proper incidence morphism
[
- rcolon I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
- subset Utimes_Smathbb P_S^bbP_S^{d-1}→ U.
+ r: I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
+ subset Utimes_SP_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
+Each affine projective-space chart in Utimes_SP_S^{d-1} is
+still smooth over B. On the chart c_a=1, the hypersurface
equation is f_a+sum_{bne a}c_bf_b=0, with special-fibre differential
at (u,[c]) equal to sum_{b=1}^dc_b,d(f_{b,s})_u (the projective
parameter term is zero since every f_b(u)=0). This combination is
nonzero by linear independence and, by hypothesis, lies outside
-(mathcal N_U)_u.calN_U)_u. The nilpotent cone on the product chart is the
-pullback of mathcal N_UcalN_U with zero parameter component, so
+(N_U)_u. The nilpotent cone on the product chart is the
+pullback of N_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.
@@ -1927,7 +1864,7 @@ vanish on the entire fibre. Proper compatibility gives
Tensor the triangle of Lemma~slice:incidence on U_K with F_U
and use the projection formula: the first term is a finite sum of shifts
and twists of F_U, the second is Rr_{K*}r_K^*F_U, and the third is
-i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)] for icolon J↪ U.
+i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)] for i: 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
@@ -1935,7 +1872,7 @@ sum is empty and the same argument applies.
We have thus obtained all finite transverse-slice tests, while applying
-the hypersurface argument only to schemes smooth over mathcal B.calB. It
+the hypersurface argument only to schemes smooth over B. It
remains to find one such slice that meets Psi on the support of F
in a nonzero way.
@@ -1954,33 +1891,33 @@ stalks, so they cannot vanish if any single one is nonzero.
## Extending a generic bundle
-Let mathscr XscrXto S be the smooth projective curve from
+Let X S be the smooth projective curve from
Section~sec:specialization, and let K be the fixed algebraic
-closure of Frac(R). Every G-bundle on mathscr X_KscrX_K
-extends to a G-bundle on mathscr X_scrX_{R'} for some finite extension
+closure of Frac(R). Every G-bundle on X_K
+extends to a G-bundle on X_{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_KcalP_K be the bundle. By
+Let P_K be the bundle. By
cite[Thm.~2,~Rem.~2(b)]{DrinfeldSimpson} it is trivial on some
-nonempty open subset of mathscr X_KscrX_K (this theorem applies to connected
+nonempty open subset of X_K (this theorem applies to connected
reductive groups over any field). Choose a trivialization away from
-distinct points x_1,...,x_rinmathscr X(K).scrX(K). Gluing at those points
+distinct points x_1,...,x_rinX(K). Gluing at those points
gives a chain of one-point modifications
[
- mathcal P_0calP_0dashrightarrowmathcal P_1calP_1dashrightarrow...
- dashrightarrowmathcal P_r=calP_r=mathcal P_K,calP_K,
- mathcal P_0calP_0 trivial,
+ P_0 P_1calP_1dashrightarrow...
+ dashrightarrowP_r=,
+ P_0 trivial,
]
whose ith modification has leg x_i. Choose a Schubert bound for each
of the finitely many relative positions.
Start with the trivial bundle over S=Spec R and form the space of
-bounded chains (legs varying on mathscr X).scrX). At each stage the bounded
+bounded chains (legs varying on X). At each stage the bounded
Hecke space is representable and proper over the preceding bundle and
-leg. Since mathscr X/SscrX/S is proper, induction shows that the space of
+leg. Since X/SscrX/S is proper, induction shows that the space of
chains is a proper algebraic space of finite type over S. It carries
the universal final bundle and contains the chosen chain as a
K-point. This point descends to a finite extension of
@@ -2001,7 +1938,7 @@ 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 bcolon Utomathcal BcalB
+Choose a finite-type smooth affine chart b: UtoB
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
@@ -2023,10 +1960,10 @@ to a finite extension.
Let R be a complete discrete valuation ring with algebraically closed
residue field k of characteristic neell, S=Spec R, and K an
algebraic closure of its fraction field. Let U/S be a smooth affine
-scheme of constant relative dimension N, mathcal FcalF a bounded
+scheme of constant relative dimension N, F a bounded
constructible E-complex on U_K, and
Λsubset T^*U_s a closed conical subset with dimΛle N.
-Suppose the closure in U of the nonzero-stalk locus of mathcal FcalF
+Suppose the closure in U of the nonzero-stalk locus of F
meets U_s. Then after a finite extension of the trait and shrinking
U, there exist uin U_s(k), 0le dle N and functions
f_1,...,f_d vanishing at u whose special-fibre differentials are
@@ -2037,7 +1974,7 @@ independent, such that
]
and for J=V(f_1,...,f_d),
- (Psi_J(mathcal F|_calF|_{J_K}))_une0.
+ (Psi_J(F|_calF|_{J_K}))_une0.
For d=0 the list is empty and J=U.
@@ -2054,7 +1991,7 @@ 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 FcalF to descend.
+require F to descend.
Write overlineoverline A_i,A_i,overlineoverline B_iB_i for the horizontal closures in U,
and e_i=dim A_i. Each overlineoverline A_iA_i is integral and dominates S.
@@ -2069,8 +2006,8 @@ The trait is excellent, hence universally catenary (a complete Noetherian
local ring is excellent cite[Tag~07QW]{Stacks}), so at any closed
vin(overlineoverline A_i)_s,A_i)_s,
- dimmathcal O_calO_{overlineoverline A_i,v}A_i,v=e_i+1
- qquadcite[Tag~02JT]{Stacks}.
+ dimO_{overlineoverline A_i,v}A_i,v=e_i+1
+ cite[Tag~02JT]{Stacks}.
Step 2: choose the special point.
@@ -2098,7 +2035,7 @@ Step 3: choose transverse equations.
Let V=T_u^*U_s, so dim V=N. If d>0, a general d-plane
Lsubset V satisfies LcapΛ_u⊆{0}: from
(eq:support-cone-fiber),
-dimmathbb P(Λ_u)bbP(Λ_u)le N-d-1, and the incidence of d-planes
+dimP(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
@@ -2114,7 +2051,7 @@ several horizontal components specialize to T.
Step 4: a generic generization in the slice.
Choose i with e_i=d and T a component of (overlineoverline A_i)_s.A_i)_s. By
-(eq:support-local-dimension), dimmathcal O_calO_{overlineoverline A_i,u}A_i,u
+(eq:support-local-dimension), dimO_{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
@@ -2127,7 +2064,7 @@ it contains z. Thus z has residue field finite over
Frac(R). Its geometric lifts to K avoid B_i: a
point in the closed set overlineoverline B_iB_i cannot specialize to
unotinoverlineoverline B_i.B_i. Each such lift therefore has a nonzero stalk of
-mathcal F|_calF|_{J_K}.
+F|_calF|_{J_K}.
Step 5: nearby cycles on a finite neighbourhood.
Take an affine neighbourhood of u inside the quasi-finite locus of
@@ -2140,29 +2077,29 @@ 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|_calF|_{J_K} is supported on Z_K. To use the
+The restriction F|_calF|_{J_K} is supported on Z_K. To use the
finite neighbourhood, remove Z∖ Q from J: this complement is
closed in J hence in U, so remove it from U and pass to an affine
neighbourhood of u. That neighbourhood contains Q because every
point of the finite local Q specializes to u. The new J is still
cut out by the same functions in a smooth affine chart, and the change
leaves its nearby-cycle stalk at u unchanged while making Q closed
-in J. If iotacolon Q↪ J is this closed immersion
-and mathcal H=calH=mathcal F|_calF|_{Q_K}, localization gives
-mathcal F|_calF|_{J_K}simeqiota_{K,*}mathcal H.calH. Proper compatibility for
+in J. If iota: Q↪ J is this closed immersion
+and H= F|_calF|_{Q_K}, localization gives
+F|_calF|_{J_K}~=iota_{K,*}H. Proper compatibility for
iota reduces (eq:support-nonzero) to the stalk of Psi_Q
-mathcal HcalH at u.
+H at u.
-Let acolon Qto S be the finite structural map. Its special fibre
+Let a: Qto S be the finite structural map. Its special fibre
has only u, so proper compatibility gives
[
- (Psi_Qmathcal H)_ucalH)_usimeqPsi_S(RGamma(Q_K,mathcal H))calH))
- simeqRGamma(Q_K,mathcal H).calH).
+ (Psi_QH)_u~=Psi_S(RGamma(Q_K,H))
+ ~=RGamma(Q_K,H).
]
The second identification uses geometric specialization on the trait,
which is the identity on coefficient complexes. The finite K-scheme
Q_K has the 'etale topos of a finite discrete set, so
-RGamma(Q_K,mathcal H)calH)simeq⊕_{xin|Q_K|}mathcal H_x.calH_x. At
+RGamma(Q_K,H)~=⊕_{xin|Q_K|}H_x. At
least one summand is nonzero (Step 4), so this complex is nonzero,
proving (eq:support-nonzero).
@@ -2171,7 +2108,7 @@ invariants. The finite direct-sum calculation is compatible with finite
coefficients, their ell-adic limit, inversion of ell, and
extension to E, so it applies to the specialization functor of
Section~sec:specialization without requiring a field of descent
-for mathcal F.calF.
+for F.
## Completion of the full-support proof
@@ -2179,24 +2116,24 @@ for mathcal F.calF.
Proof. [Proof of Theorem~thm:full-support]
If Y'ne Y, Proposition~spec:missing supplies the nonzero
compact F used above. Lemma~support:extension and the
-subsequent chart construction give a smooth affine Utomathcal BcalB of
+subsequent chart construction give a smooth affine UtoB of
constant relative dimension N for which the closure of the nonzero-stalk
locus of F_U meets U_s.
-Let mathcal N_UcalN_Usubset T^*U_s be the pullback of the global nilpotent
+Let N_U T^*U_s be the pullback of the global nilpotent
cone. Under either hypothesis regime, the nilpotent-parabolic
assumption and invariant form give the half-dimensionality theorem of
cite[App.~D.1.1--D.1.8]{AGKRRV}. In its smooth-chart formulation
-cite[App.~D.1.2]{AGKRRV}, this says dimmathcal N_UcalN_Uledim U_s=N.
+cite[App.~D.1.2]{AGKRRV}, this says dimN_U U_s=N.
-Apply Lemma~support:finite-slice with mathcal F=F_UcalF=F_U and
-Λ=mathcal N_U.calN_U. It gives a slice J=V(f_1,...,f_d) and
+Apply Lemma~support:finite-slice with F=F_U and
+Λ=N_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)_ucalN_U)_u only at zero. The higher-codimension vanishing
+meeting (N_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.
@@ -2215,41 +2152,41 @@ Throughout this section assume Hypothesis~hyp:A or
Hypothesis~hyp:B, so Theorem~thm:full-support holds.
Put
[
- mathcal C_s=calC_s=Shv_{Nilp}(Bun_G),
- mathcal A=calA=QCoh(Y),
- mathcal D=calD=IndCoh_{Nilp}(Y),
+ C_s={Nilp}(Bun_G),
+ A=,
+ D={Nilp}(Y),
]
-and write mathbb L_G^bbL_G^{restr}colonmathcal C_scalC_stomathcal DcalD
-for the restricted Langlands equivalence. Its mathcal A-linearcalA-linear
+and write L_G^{restr}colonC_s DcalD
+for the restricted Langlands equivalence. Its A-linearcalA-linear
structure is the one furnished by cite[Lemma~1.3.7]{GR}.
## The category attached to one component
-For an open-and-closed component icolon Z↪ Y, define
+For an open-and-closed component i: Z↪ Y, define
[
- mathcal C_calC_{s,Z}:=QCoh(Z)underset{QCoh(Y)} ⊗ mathcal C_s,calC_s,
+ C_{s,Z}:=QCoh(Z)underset{QCoh(Y)} ⊗ C_s,
]
the tensor product being that of presentable dg categories. The
-idempotent quasi-coherent mathcal O_ZcalO_Z (extended by zero) acts as the
+idempotent quasi-coherent O_Z (extended by zero) acts as the
projection to this summand.
Under either hypothesis regime, every connected component Zsubset Y
has nonzero localized automorphic category, and the restriction of
-mathbb L_G^bbL_G^{restr} induces an equivalence
-mathcal C_calC_{s,Z}xrightarrow{sim}IndCoh_{Nilp}(Z).
+L_G^{restr} induces an equivalence
+C_{s,Z}xrightarrow{sim}IndCoh_{Nilp}(Z).
Proof.
The decomposition Y=Zsqcup(Y∖ Z) decomposes quasi-coherent
and nilpotent ind-coherent sheaves into two summands, so
-QCoh(Z) ⊗ _{mathcal A}calAmathcal DcalDsimeqIndCoh_{Nilp}(Z). Tensor
-the mathcal A-linearcalA-linear equivalence of Theorem~thm:full-support
-with QCoh(Z) over mathcal A,calA, and the assertion follows.
+QCoh(Z) ⊗ _{A} DcalD~=IndCoh_{Nilp}(Z). Tensor
+the A-linearcalA-linear equivalence of Theorem~thm:full-support
+with QCoh(Z) over A, 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_ZcalO_Zne0, and the fully faithful
+O_Z, and the fully faithful
QCoh(Z)↪IndCoh_{Nilp}(Z) then gives nonvanishing.
@@ -2279,8 +2216,8 @@ 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)bbF_q),E)
- simeqGamma^{IndCoh}(Y^{arithm},omega_{Y^{arithm}}).
+ Funct_c(Bun_{G_0}(F_q),E)bbF_q),E)
+ ~=Gamma^{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
@@ -2294,8 +2231,8 @@ calculation of Beraldo--Lin--Reeves
cite[SS6.4.12--6.4.13]{BeraldoLinReeves} enters the known primed
formula cite[SS1.5.3--1.5.5,~Cor.~1.5.6]{GR}:
[
- Funct_c(Bun_{G_0}(mathbb F_q),E)bbF_q),E)
- simeqGamma^{IndCoh}((Y')^{Frob},omega_{(Y')^{Frob}}).
+ Funct_c(Bun_{G_0}(F_q),E)bbF_q),E)
+ ~=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
@@ -2319,18 +2256,18 @@ We now prove that every parameter has a nonzero eigenobject. The
categorical input is simple.
-Let mathcal A_0calA_0 be a compactly generated, presentable, stable,
+Let A_0 be a compactly generated, presentable, stable,
E-linear symmetric monoidal category whose compact objects are
dualizable, and suppose its tensor product preserves colimits
-separately. Let acolonmathcal A_0calA_0toVect_E be a continuous
+separately. Let acolonA_0 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
+A_0-actioncalA_0-action through a. Then the right adjoint R_a of a
+is continuous, carries a coherent A_0-linearcalA_0-linear structure, and
R_a(E)ne0.
Proof.
-Presentability gives R_a. Every compact object of mathcal A_0calA_0 is
+Presentability gives R_a. Every compact object of A_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
@@ -2340,28 +2277,28 @@ colimit-preserving.
The canonical projection b ⊗ R_a(V)to R_a(a(b) ⊗ V) is
invertible for dualizable b: for any c,
- RHom_{mathcal A_0}calA_0(c,b ⊗ R_a(V))
- &simeqRHom_{mathcal A_0}calA_0(b^vee ⊗ c,R_a(V))
- &simeqRHom_E(a(b)^vee ⊗ a(c),V)
- &simeqRHom_E(a(c),a(b) ⊗ V).
+ RHom_{A_0},b ⊗ R_a(V))
+ &~=RHom_{A_0} ⊗ c,R_a(V))
+ &~=RHom_E(a(b)^vee ⊗ a(c),V)
+ &~=RHom_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-linearcalA_0-linear structure on R_a.
+coherent A_0-linearcalA_0-linear structure on R_a.
Finally R_a(E)ne0 because
-RHom_{mathcal A_0}calA_0(mathbf 1,R_a(E))bf1,R_a(E))
-simeqRHom_E(E,E)simeq E; this last argument does not require a
+RHom_{A_0} 1,R_a(E))bf1,R_a(E))
+~=RHom_E(E,E)~= E; this last argument does not require a
compact unit.
-For our spectral stack, mathcal A=calA=QCoh(Y) satisfies the
+For our spectral stack, A= satisfies the
hypotheses of Lemma~lem:parameter-adjoint by
cite[S7.9,~Cor.~C.4.4]{AGKRRV}. On a disjoint union of components the
compact generators have finite component support. We use the continuous,
-fully faithful mathcal A-linearcalA-linear functor
+fully faithful A-linearcalA-linear functor
[
jmath_YcolonQCoh(Y)xrightarrow[sim]{Υ_Y}
IndCoh_{{0}}(Y)↪IndCoh_{Nilp}(Y),
@@ -2370,17 +2307,17 @@ where Υ_Y(Q)=Q ⊗ omega_Y
cite[S1.3.1,~Eq.~(1.2)]{GR}. Its essential image has zero singular
support, hence lies in the nilpotent category.
-For a finite set I, put mathcal Q_I=calQ_I=QLisse(X)^{ ⊗ I}. Given
+For a finite set I, put Q_I={ ⊗ I}. Given
VinRep(check G)^{ ⊗ I}, let H_I(V,-) denote the multi-leg
Hecke functor and let
-mathcal E_V^IcalE_V^Iinmathcal AcalA ⊗ mathcal Q_IcalQ_I be the universal
+E_V^I AcalA ⊗ Q_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.
+sigma^I(V):=(sigma^* ⊗ id)(E_V^I) Q_I.calQ_I.
A Hecke eigenobject of eigenvalue sigma with coherent structure is an
-Minmathcal C_scalC_s equipped, for all finite sets I, with isomorphisms
-H_I(V,M)simeq Mboxtimessigma^I(V), natural in Vin
+MinC_s equipped, for all finite sets I, with isomorphisms
+H_I(V,M)~= 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}.
@@ -2390,35 +2327,35 @@ compatibilities cite[S15.2]{AGKRRV}.
Under either hypothesis regime, every parameter sigmacolonSpec Eto
Y admits a nonzero object M_sigmainShv_{Nilp}(Bun_G) with a
coherent Hecke eigenstructure of eigenvalue sigma. If sigma lies
-in a component Z, then M_sigmainmathcal C_calC_{s,Z}. The assertion
+in a component Z, then M_sigmainC_{s,Z}. The assertion
is in the ind-constructible dg category; no boundedness, perversity
or Weil structure is included.
Proof.
-Apply Lemma~lem:parameter-adjoint to a=sigma^*colonmathcal AcalA
+Apply Lemma~lem:parameter-adjoint to a=sigma^*colonA
toVect_E, and write sigma_* for its continuous right adjoint.
The composite
[
- Phi_sigmacolonVect_sigmaxrightarrow{sigma_*}mathcal AcalA
- xrightarrow{jmath_Y}mathcal DcalD
- xrightarrow{(mathbb L_G^bbL_G^{restr})^{-1}}mathcal C_scalC_s
+ Phi_sigmacolonVect_sigmaxrightarrow{sigma_*}A
+ xrightarrow{jmath_Y}D
+ xrightarrow{(L_G^{restr})^{-1}}C_s
]
-is continuous mathcal A-linear,calA-linear, where Vect_sigma denotes
+is continuous A-linear,calA-linear, where Vect_sigma denotes
Vect_E with the action through sigma^*. Set
M_sigma=Phi_sigma(E). It is nonzero by the lemma, full faithfulness
of jmath_Y, and the Langlands equivalence. If sigma lies in Z,
the idempotent e_Z pulls back to E, while its complementary
idempotent pulls back to zero; linearity puts M_sigma in
-mathcal C_calC_{s,Z}.
+C_{s,Z}.
For the eigenstructure, fix a finite set I. The action of
-mathcal E_V^IcalE_V^I on mathcal C_scalC_s is the multi-leg Hecke functor
+E_V^I on C_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
+identity on Q_I. Its module structure gives
[
- H_I(V,M_sigma)simeq(Phi_sigma ⊗ id)(sigma^I(V))
- simeq M_sigmaboxtimessigma^I(V),
+ H_I(V,M_sigma)~=(Phi_sigma ⊗ id)(sigma^I(V))
+ ~= 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
@@ -2435,18 +2372,18 @@ semisimple or not.
% =========================================================================
We now prove the rational-coefficient form of full support. Put
-E_0=mathbb Q_bbQ_ell and E=mathbb Qbar_bbQbar_ell, keep k=mathbb Fbar_q,bbFbar_q,
+E_0=Q_ and E=Qbar_, keep k=Fbar_q,
and assume the curve and group satisfy the rational hypotheses in the
Introduction (Hypothesis~hyp:A plus a split dual group over
-E_0). Put mathcal C_E=calC_E=Shv_{Nilp,E}(Bun_G), retain
-mathcal C_0=calC_0=Shv_{Nilp,E_0}(Bun_G) and Y_0 from the Introduction,
+E_0). Put C_E={Nilp,E}(Bun_G), retain
+C_0={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 the geometric field.
Recall the standing datum: a continuous QCoh(Y_0)-action on
-mathcal C_0calC_0 whose restriction along universal evaluation is the usual
+C_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.
@@ -2454,29 +2391,29 @@ compatibilities. We call this the rational spectral-action datum.
For the rational datum above, every nonempty open-and-closed substack
Usubset Y_0 satisfies
[
- QCoh(U)underset{QCoh(Y_0)} ⊗ mathcal C_0calC_0ne0.
+ QCoh(U)underset{QCoh(Y_0)} ⊗ C_0
]
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,
+whose complementary structure idempotent acts by zero on C_0,
then Y'_0=Y_0 as spectral prestacks. In particular this applies to the
support of any QCoh(Y_0)-linear primed equivalence.
-The plan is: for Xnemathbb P^1bbP^1 we compare the sheaf categories,
+The plan is: for XneP^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^1bbP^1 connectedness of the spectral prestack gives the result
+X=P^1 connectedness of the spectral prestack gives the result
directly.
## Coefficient categories, including unbounded objects
On a finite-type T/k, Shv_L(T) is the ind-completion of bounded
constructible L-adic complexes, Lin{E_0,E}. For a smooth T and a
-closed conical Nsubset T^*T, let mathcal P_calP_{N,L}(T) be the category
+closed conical Nsubset T^*T, let P_{N,L}(T) be the category
of constructible perverse sheaves with singular support in N; the
category Shv_{N,L}(T) consists of objects whose perverse cohomology
-belongs in Indmathcal P_calP_{N,L}(T) in every degree. This
+belongs in IndP_{N,L}(T) in every degree. This
is the cohomological support definition
cite[SS E.5.1--E.5.8]{AGKRRV}; it is not necessarily the ind-completion
of support-bounded complexes. On a smooth scheme the zero-cone case is
@@ -2486,11 +2423,11 @@ cite[SS1.2.5--1.2.11]{AGKRRV}.
[Coefficient comparison]
Extension of coefficients induces equivalences
- Vect_E ⊗ _{E_0}Shv_{E_0}(T)&simeqShv_E(T),
+ Vect_E ⊗ _{E_0}Shv_{E_0}(T)&~=Shv_E(T),
- Vect_E ⊗ _{E_0}QLisse_{E_0}(X)&simeqQLisse_E(X),
+ Vect_E ⊗ _{E_0}QLisse_{E_0}(X)&~=QLisse_E(X),
- Vect_E ⊗ _{E_0}mathcal C_0&calC_0&simeqmathcal C_E.calC_E.
+ Vect_E ⊗ _{E_0}C_0&calC_0&~=C_E.
The first extends to locally finite-type algebraic stacks by smooth
@@ -2516,7 +2453,7 @@ cite[Prop.~5.2,~Thm.~7.7,~Lemma~7.9,~Prop.~8.2,~Cor.~8.3]{HemoRicharzScholbach};
see also cite[S2]{HansenScholze}. Cohomological finiteness
cite[Lemma~8.6(1)]{HemoRicharzScholbach} holds on every finite-type
chart over the separably closed k (either algebraic
-mathbb Q_bbQ_ell-coefficient field), so its compactness statements apply.
+Q_-coefficient field), so its compactness statements apply.
Passing to ind-completions proves (eq:coeff-ambient).
Support condition. View (eq:coeff-ambient) as internal
@@ -2537,7 +2474,7 @@ ambient ind-categories.
For bounded constructibles, coefficient extension preserves and
reflects singular-support bounds. The defining test pairs express the
bound by universal local acyclicity; use the dualizability criterion in
-the algebraic mathbb Q_bbQ_ell-setting cite[Thm.~1.6]{HansenScholze}.
+the algebraic Q_-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
@@ -2560,7 +2497,7 @@ for U use the colimit formula and filtered colimits, never treating a
finite E-rank as a finite E_0-rank. For the converse, let P be a
perverse internal E-module with U(P) in the E_0-ind-heart. The
counit F(U(P))to P is an epimorphism in the module heart because its
-underlying map splits as umapsto1 ⊗ u. Its source is in the
+underlying map splits as u|->1 ⊗ 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
@@ -2593,8 +2530,8 @@ chartwise support comparison therefore proves (eq:coeff-nilp).
The lemma exchanges scalar extension with the actual cohomological
categories, not with any unspecified left completion. It applies to
-QLisse(mathbb P^1)bbP^1) without identifying that category with
-IndLisse(mathbb P^1).bbP^1).
+QLisse(P^1) without identifying that category with
+IndLisse(P^1).
## Affine families and coherent Hecke action
@@ -2605,8 +2542,8 @@ prestacks. We retain the right-t-exactness in their tensor functors.
[Spectral coefficient comparison]
There are natural identifications
- Y_0times_{Spec E_0}Spec Esimeq Y,
- Vect_E ⊗ _{E_0}QCoh(Y_0)simeqQCoh(Y),
+ Y_0times_{Spec E_0}Spec E~= Y,
+ Vect_E ⊗ _{E_0}QCoh(Y_0)~=QCoh(Y),
with the second symmetric monoidal. These identify the universal
evaluation systems for every finite set of legs.
@@ -2617,7 +2554,7 @@ Let A be a connective commutative E-algebra, Mod_A
=QCoh(Spec A). Lemma~lem:coeff-sheaves and associativity give
[
Mod_A ⊗ _{E_0}QLisse_{E_0}(X)
- simeqMod_A ⊗ _EQLisse_E(X).
+ ~=Mod_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
@@ -2641,7 +2578,7 @@ Present Y_0 as a colimit of affine test schemes. Base change presents
the left side of (eq:coeff-spectral) by base-changed affines, and
quasi-coherent categories form the corresponding limit. On an affine,
coefficient comparison is
-Vect_E ⊗ _{E_0}Mod_Bsimeq
+Vect_E ⊗ _{E_0}Mod_B~=
Mod_{B ⊗ _{E_0}E}. Affine pullbacks in this diagram
are continuous, and Vect_E is categorically dualizable, so tensoring
commutes with this limit and gives the second identification, monoidal
@@ -2654,16 +2591,16 @@ universal evaluation system, including tensor products and collisions
of legs.
-For Xnemathbb P^1bbP^1 we can go further and compare the given rational
+For XneP^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^1bbP^1 we deduce support from connectedness of the
+below; for P^1 we deduce support from connectedness of the
spectral prestack.
[Comparison of spectral actions]
-Suppose Xnemathbb P^1bbP^1 and equip mathcal C_0calC_0 with the rational
+Suppose XneP^1 and equip C_0 with the rational
spectral-action datum of Theorem~thm:rational-support. Under
Lemmas~lem:coeff-sheaves and~lem:coeff-spectral, its scalar
-extension is the established QCoh(Y)-action on mathcal C_E.calC_E.
+extension is the established QCoh(Y)-action on C_E.
Proof.
@@ -2726,26 +2663,26 @@ Theorem~thm:full-support.
## Descent of support and the arithmetic qualification
-For Xnemathbb P^1,bbP^1, the sheaf, prestack and coherent-action
+For XneP^1, the sheaf, prestack and coherent-action
comparisons are now in place.
[Detection of a clopen support]
Suppose the coefficient comparisons (eq:coeff-nilp)
and~(eq:coeff-spectral) identify the two spectral actions. If an
-open-and-closed substack Usubset Y_0 acts by zero on mathcal C_0,calC_0,
+open-and-closed substack Usubset Y_0 acts by zero on C_0,
then U=∅.
Proof.
-Let e_U=mathcal O_UcalO_U extended by zero in QCoh(Y_0). Its extension
+Let e_U=O_U extended by zero in QCoh(Y_0). Its extension
is the analogous idempotent e_{U_E}, where U_E=Utimes_{E_0}E.
Compatibility of the actions and generation by scalar-extended objects
-show e_{U_E} acts by zero on mathcal C_E.calC_E.
+show e_{U_E} acts by zero on C_E.
The curve and split group descend to a finite field, so
Theorem~thm:full-support in regime A identifies this action with
the action on IndCoh_{Nilp}(Y). Apply e_{U_E} to
-jmath_Y(mathcal O_Y);calO_Y); its value is jmath_Y(e_{U_E}). Full
+jmath_Y(O_Y); its value is jmath_Y(e_{U_E}). Full
faithfulness of jmath_Y implies e_{U_E}=0, hence U_E=∅.
To descend, take any affine test T=Spec Ato Y_0. The inverse image
@@ -2757,21 +2694,21 @@ rational-point description of components is used.
Proof. [Proof of Theorem~thm:rational-support]
-For Xnemathbb P^1,bbP^1, Lemma~lem:coeff-action supplies the
+For XneP^1, Lemma~lem:coeff-action supplies the
hypothesis of Lemma~lem:coeff-idempotent. An open-and-closed
localization is the image of its idempotent projection, so vanishing of
the localized category is exactly zero action of that idempotent. The
lemma gives the claim.
-If X=mathbb P^1,bbP^1, the geometric spectral prestack Y has one
+If X=P^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^1bbP^1 is trivial (a connected finite
-'etale cover Ctomathbb P^1bbP^1 of degree n satisfies
+fundamental group of P^1 is trivial (a connected finite
+'etale cover CtoP^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_0calC_0 is nonzero (its
+open-and-closed substack. The category C_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.
@@ -2789,7 +2726,7 @@ 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)bbF_q),E_0)
+ Funct_c(Bun_{G_0}(F_q),E_0)bbF_q),E_0)
xrightarrow{ sim }
Gamma^{IndCoh}((Y'_0)^{Frob},omega_{(Y'_0)^{Frob}}),
]