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% THE RESTRICTED GEOMETRIC LANGLANDS EQUIVALENCE IN POSITIVE CHARACTERISTIC
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%
% This is a self-contained rewrite: same theorem as v1, tighter narrative,
% clearer routing of the three regimes and their inputs. Read the "Roadmap"
% at the end of Section 1 before diving into Sections 2-7.
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# The Restricted Geometric Langlands Equivalence in Positive Characteristic
maketitle
Abstract:
We prove the mathbb Qbar_bbQbar_ell-linear restricted geometric Langlands
equivalence for a smooth projective connected curve and a connected reductive
group over a field of characteristic p>0, with ellne p, in two regimes.
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
onto the entire spectral stack.
We then record three consequences: the local category on each spectral
component is nonzero and corresponds to its own summand of automorphic
sheaves; in the finite-field regime the arithmetic trace formula holds
with the full derived Frobenius fixed-point stack; and every parameter
of the spectral stack admits a nonzero Hecke eigenobject. A final section
handles rational coefficients by coefficient comparison.
tableofcontents
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# Introduction
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## The question
Geometric Langlands predicts that automorphic objects on the moduli
Bun_G(X) of G-bundles on a smooth projective curve X are matched
with spectral objects depending on local systems on X for the dual
group check G. The modern formulation compares whole sheaf categories,
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.
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
stack of local systems with restricted variation introduced by
Arinkin--Gaitsgory--Kazhdan--Raskin--Rozenblyum--Varshavsky [cite].
Positive characteristic is different: the theory has additional
arithmetic inputs, and the comparison one can construct naturally
only reaches an open-and-closed union of spectral components. The
question left open by [cite] is whether any component can be missing.
We prove that none is, in the two regimes stated in the abstract.
Our proof is geometric. A missing component C would give a nonzero
constructible object F on Bun_G over the geometric generic fibre
of a lift mathscr X/SscrX/S of X to the Witt trait, satisfying Psi(F)=0
for geometric nearby cycles Psi. We then construct a small, finite
slice transverse to the support of F on which Psi(F) cannot vanish,
provided the slice suitably avoids the nilpotent cone. That contradiction
finishes the proof.
## The spectral stack
Let k be an algebraically closed field of characteristic p>0,
let ellne p be a prime, and set E=mathbb Qbar_bbQbar_ell. Let
X/k be a smooth projective connected curve and G/k a connected
reductive group; such a group is split over k. Write
check G/E for the Langlands dual group and
mathfrak g=frakg=Lie(G).
We work with the E-linear ind-constructible sheaf categories of
[cite]. In particular Shv(X) is ind-constructible, and
QLisse_E(X)subsetShv(X) is the full subcategory whose ordinary
cohomology sheaves are filtered colimits of finite-rank lisse
E-sheaves cite[Def.~1.2.6]{AGKRRV}. The restricted spectral prestack
Y=LS^{restr}_{check G}(X) assigns to a derived affine
E-scheme T the groupoid
Y(T)=Fun^{ ⊗ ,r.t.ex}!(Rep_E(check G),,
QCoh(T) ⊗ _EQLisse_E(X)),
of right-t-exact symmetric monoidal functors. An E-parameter
sigmain Y(E) is such a functor with T=Spec E: concretely, an
assignment of a lisse sheaf sigma(V) to each finite-dimensional
representation of check G, monoidal in V.
The reason for the restricted variation in (eq:Y-def) is the same
as in characteristic zero. Ind-coherent sheaves alone with no support
condition do not capture the Arthur-type data that automorphic sheaves
must carry. On the automorphic side one restricts to sheaves with
nilpotent singular support, i.e. those whose wavefront set is contained
in the global nilpotent cone mathcal NcalNsubset T^*Bun_G; write this
subcategory Shv_{Nilp}(Bun_G(X)) (see Ssec:intro-cat below
for the precise definition). On the spectral side one imposes an
analogous condition and uses IndCoh_{Nilp}(Y). These categories
already exist in [cite]; our new assertion is about the
spectral components on which they are supported.
Both sides carry a coherent action of QCoh(Y), coming from the
universal family of local systems on Ytimes X and the construction
of Hecke operators. This makes the comparison functor
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
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*)]
There is a nondegenerate G-invariant symmetric bilinear form
kappa on mathfrak gfrakg whose restriction to Lie Z(M) is
nondegenerate for every Levi subgroup M.
Chevalley restriction k[Lie M]^Mto k[Lie T_M]^{W_M} is an
isomorphism for every Levi M and every maximal torus T_Msubset M.
The scheme-theoretic centraliser in G of every semisimple
element of mathfrak gfrakg is a Levi subgroup.
Every nilpotent element of mathfrak gfrakg lies in Lie R_u(P) for
some parabolic Psubset G, after any field extension.
Conditions (i)--(iii) are cite[S14.4.1]{AGKRRV}; condition (iv) is
cite[S D.1.1]{AGKRRV}. Both hypotheses below keep the notion of
semisimple and nilpotent as in [cite]: semisimple elements of
mathfrak gfrakg 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,bbF_q, and a split connected reductive group
G_0/mathbb F_qbbF_q with k=mathbb Fbar_q,bbFbar_q,
X=X_0 ⊗ _{mathbb F_q}bbF_qk, G=G_0 ⊗ _{mathbb F_q}bbF_qk.
Hypothesis~hyp:common holds. No restriction on |W_G| is imposed.
[Regime B: arbitrary algebraically closed base]
k is any algebraically closed field of characteristic p>0.
Hypothesis~hyp:common holds, p is very good for G, and
pnmid|W_G|. No finite-field descent is assumed.
``Very good'' is the usual root-system condition. For type A_n it
means pnmid n+1; for B_n,C_n,D_n it means pne2; for
G_2,F_4,E_6,E_7 that pne2,3; for E_8 that pne2,3,5; applied on
each irreducible factor. A torus has no such restriction, and every
prime is very good for a torus.
The two regimes differ only in one Lie-theoretic input
(Section~sec:lie): how to produce a central cocharacter with a
nonvanishing pairing with the constant term of a Higgs field. Regime A
solves this with a lattice argument that uses only kappa and
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
Proposition~lie:input, and everything downstream is common to
the two regimes.
## Nilpotent supports and the relevant categories
The subcategories we use are the standard ones from [cite], but
it is worth pinning them down once, because several comparisons later
depend on the precise definition.
On the automorphic side, the global nilpotent cone
mathcal NcalNsubset T^*Bun_G(X) consists of Higgs pairs (P,A) with
Ain H^0(X,Ad P ⊗ omega_X) pointwise nilpotent. A bounded
constructible E-sheaf on Bun_G(X) has nilpotent singular support
if its wavefront set lies in mathcal N;calN; the full subcategory
Shv_{Nilp}(Bun_G(X))subsetShv(Bun_G(X)) consists of the
ind-objects whose every cohomology sheaf has this property. This
definition uses Beilinson's theory of singular support for constructible
sheaves in arbitrary characteristic [cite];
it is not the coherent singular-support theory of [cite],
which lives on the spectral side.
On the spectral side, ind-coherent sheaves on Y with nilpotent singular
support, denoted IndCoh_{Nilp}(Y), are defined in cite[SS21.2.1--21.2.5]{AGKRRV}.
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
compatibility with geometric Satake are all proved in [cite].
The two categories play different roles: the automorphic nilpotent cone is
geometric, the spectral nilpotent support is Arthur-theoretic. The
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
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
inclusion is the natural fully faithful functor, and its image is a
summand.
What is missing from (eq:primed) is the assertion Y'=Y: all
components, not just some, support a nonzero summand of automorphic
sheaves. This is the content of the following theorem.
[Full support]
Under either Hypothesis~hyp:A or Hypothesis~hyp:B, the
inclusion Y'subset Y in (eq:primed) is equality of spectral
prestacks, and the natural functor
[
mathbb L_G^bbL_G^{restr}colon
Shv_{Nilp}(Bun_G(X))xrightarrow{ sim }
IndCoh_{Nilp}(Y)
]
is a QCoh(Y)-linear equivalence.
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.
## Consequences and rational coefficients
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
IndCoh_{Nilp}(C);
in regime A, the arithmetic trace formula holds with the full
derived fixed-point stack Y^{Frob} rather than the primed
(Y')^{Frob};
every E-parameter sigmain Y(E) admits a nonzero Hecke
eigenobject with coherent tensor and fusion structure.
None of these requires any additional hypothesis beyond
Theorem~thm:full-support.
Section~sec:coefficients proves the same support statement for
rational coefficients mathbb Q_bbQ_ell. The theorem there is conditional
on a single input: the existence of the primed trace formula over
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
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
% =========================================================================
Throughout this section we assume Hypothesis~hyp:A or
Hypothesis~hyp:B; we never assume Theorem~thm:full-support
itself. The goal is a single statement: if Y'ne Y, then there exists
a nonzero sheaf F on the geometric generic fibre for which all the
nearby-cycle vanishings we need later do actually hold. The
construction has three parts. First we place everything over the Witt
trait. Second we extract such an F from the missing component.
Third we propagate its vanishing along smooth pullbacks and Hecke
transforms.
## The Witt trait and geometric specialisation
Let
[
R=W(k), S=Spec(R), K=Frac(R)bar,
]
where W(k) is the ring of Witt vectors of k. Since k is perfect,
R is a complete Noetherian discrete valuation ring with uniformizer
p and residue field k. It is strictly henselian, and it is
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.
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
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
to a finite field.
The root datum of the split G gives a split reductive group over
mathbb Z,bbZ, and base change to S gives a lift, which we also denote
G cite[Thms.~6.1.16(2) and~6.1.17]{Conrad}. Write
[
mathcal B=calB=Bun_G(mathscr X/S),scrX/S),
s=Spec(k), eta=Spec(K).
]
Then mathcal B=calB=Hom_S(mathscr X,B_SG)scrX,B_SG) is an algebraic stack locally
of finite type over S with affine diagonal
cite[Thm.~1.2(i)--(iii)]{HallRydh}: mathscr X/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.
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
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).
]
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[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
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
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)
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
D in the indicated ind-category.
## The spectral summand attached to a missing component
Set
[
mathcal C_s=calC_s=Shv_{Nilp}(mathcal B_s),calB_s),
mathcal C_K=calC_K=Shv_{Nilp}(mathcal B_K),calB_K),
Y_K=LS^{restr}_{check G}(mathscr X_K).scrX_K).
]
Restriction to the closed fibre gives an identification
QLisse(mathscr X)scrX)simeqQLisse(X) cite[S3.1.3]{GR}. Its inverse,
composed with restriction to the geometric generic fibre, gives a
symmetric monoidal functor
[
rhocolonQLisse(X)toQLisse(mathscr X_K).scrX_K).
]
The corresponding map
[
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
mathcal D_Y:=calD_Y:=QCoh(Y)underset{QCoh(Y_K)}{ ⊗ }mathcal C_KcalC_K
is a direct summand of mathcal C_K.calC_K.
We need the actual specialisation functor to commute with this summand,
not merely a functor obtained by transporting mathbb L_GbbL_G across the
specialisation. Corollary~4.3.6 and S4.3.9 of [cite] show that the
chartwise geometric nearby-cycle functor we built above restricts to
[
Psi_{mathcal B}calBcolon mathcal D_Y→calD_Y→ mathcal C_s,calC_s,
]
and it is QCoh(Y)-linear by cite[Thm.~3.1.6(B) and S4.3.10]{GR}.
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).
In positive characteristic we use only the primed version:
mathcal C_scalC_ssimeqIndCoh_{Nilp}(Y') for some open-and-closed
Y'subset Y.
If a connected component Csubset Y is disjoint from Y', then the
summand
[
mathcal D_C:=calD_C:=QCoh(C)underset{QCoh(Y)}{ ⊗ }mathcal D_YcalD_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.
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
the same object acts by zero, because Ccap Y'=∅ and the
primed equivalence is spectral-linear. Therefore for every
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_CcalD_C is
IndCoh_{Nilp}(C), which is nonzero: the fully faithful
QCoh(C)↪IndCoh_{Nilp}(C) sends mathcal O_CcalO_Cne0 to
a nonzero object cite[SS1.3.1--1.3.4]{GR}.
The category mathcal D_YcalD_Y is compactly generated
cite[S3.1.7]{GR}. In the binary decomposition
mathcal D_YcalD_Ysimeqmathcal D_CcalD_Ctimesmathcal D_calD_{Y∖ C} both
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
projection is nonzero, since otherwise the colimit closure would give
mathcal D_C=0.calD_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
of a summand preserves compactness. The inclusion
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}.
The argument uses only one missing component and binary decomposition.
It does not require finitely many components of Y or compactness of
e_C. If Y'ne Y then such a C exists because Y' is a union of
open-and-closed components of Y.
For the remainder of the geometric argument we fix a missing component
C and the associated F as in Proposition~spec:missing. If
Y'=Y there is nothing to prove.
## Hecke transforms of a specialising object
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_calE_{V,s}inQCoh(Y) ⊗ QLisse(X),
mathcal E_calE_{V,K}inQCoh(Y_K) ⊗ QLisse(mathscr X_K).scrX_K).
]
On an affine test Tto Y the restriction of mathcal E_VcalE_V is the
family of local systems obtained by applying the corresponding tensor
functor to V. Acting by its first tensor factor and tensoring with
the exterior product gives mathrm H_VrmH_V on the spectral side
cite[S14.3.3]{AGKRRV}.
By definition of i,
(i^* ⊗ id)(mathcal E_calE_{V,K})
simeq(id ⊗ rho)(mathcal E_calE_{V,s}).
On every affine test the two sides are the same local system; the
passage from these identities to (eq:spec-universal-family) uses
dualizability of QLisse(X). Concretely, QLisse(X) for a smooth
projective curve is compactly generated
cite[S1.2.11,~SS E.2.6--E.2.7]{AGKRRV}, hence dualizable as a
presentable dg category, and tensoring with a dualizable category
commutes with the descent limit defining QCoh(Y).
The second, distinct input is cite[Rem.~14.3.9]{AGKRRV}:
mathcal E_calE_{V,s}inQCoh(Y) ⊗ IndLisse(X),
where IndLisse(X)=Ind(Lisse(X)) and
Lisse(X) consists of bounded constructible complexes
with finite-rank lisse cohomology sheaves. We do not identify
IndLisse(X) with QLisse(X); that identification fails for
X=mathbb P^1.bbP^1. The two properties used above (dualizability for
descent, membership in IndLisse for propagation) are separate.
Now apply the QCoh(Y)-action in (eq:spec-IndLisse) to F. The
result is an object of mathcal D_CcalD_C ⊗ IndLisse(X). After applying
id ⊗ rho and the exterior-product functor, it is precisely
mathrm H_V(F)rmH_V(F) by (eq:spec-universal-family). Elementary tensors
generate mathcal D_CcalD_C ⊗ IndLisse(X) under colimits, and bounded
truncations build every object of Lisse(X) from
finite-rank lisse sheaves. Therefore mathrm H_V(F)rmH_V(F) lies in the stable
colimit closure of objects
Dboxtimesmathcal L_K,calL_K, Dinmathcal D_C,calD_C,
where mathcal L_KcalL_K is the restriction of a finite-rank lisse sheaf
mathcal L_ScalL_S on mathscr X.scrX.
Let acolon Ttomathcal BcalB be smooth, with T a finite-type S-scheme
or algebraic space, and let ell_Tcolon Ttomathscr XscrX be any
S-morphism. Then for F as in Proposition~spec:missing and
every finite-dimensional V,
[
Psi_T(a_K^*F)=0,
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}calB(F)=0 and smooth base
change along a. For an exterior generator in
(eq:spec-exterior-generators), ordinary pullback gives
[
(a,ell_T)_K^*(Dboxtimesmathcal L_K)calL_K)
=a_K^*D ⊗ ell_{T,K}^*mathcal L_K.calL_K.
]
The sheaf ell_T^*mathcal L_ScalL_S is lisse on T, regardless of whether
ell_T is smooth. Formula~(eq:spec-lisse-tensor) and
Psi_{mathcal B}calB(D)=0 therefore give
[
Psi_T(a_K^*D ⊗ ell_{T,K}^*mathcal L_K)calL_K)
simeq a_s^*Psi_{mathcal B}calB(D) ⊗ ell_{T,s}^*mathcal L_s=0.calL_s=0.
]
Both pullback and specialisation are continuous, so the same vanishing
holds for the stable colimit closure of these exterior generators. That
closure contains mathrm H_V(F).rmH_V(F). Finite trait replacement preserves
geometric specialisation, so no new hypothesis is needed there.
[What is and is not used]
The proof uses smoothness only of acolon Ttomathcal B;calB; the leg
map ell_T may be arbitrary. It makes no smooth-base-change assertion
for the combined map (a,ell_T). This is exactly the freedom needed
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
whether the cut is smooth in the leg direction.
Sections~sec:lie and~sec:hecke convert a non-nilpotent
cotangent direction into a modification that exhibits this vanishing,
nonzero incoming covector and both transverse derivatives.
% =========================================================================
# A Levi reduction and a central cocharacter
% =========================================================================
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
of Ad P ⊗ omega_X whose values are not all nilpotent. We want to
find a Hecke modification of P whose moving-point covector is nonzero,
and whose fixed-input / fixed-output cotangent sections have invertible
derivatives. Both properties will come from a central cocharacter of a
Levi subgroup.
The Lie-theoretic input is the following. Choose a point xin X(k)
where A is non-nilpotent. In a formal coordinate t and bundle frame
at x, write A=a(t),dt with a(t)inmathfrak g[[t]].frakg[[t]]. Chevalley
restriction identifies the nilpotent cone in mathfrak gfrakg with
chi^{-1}(chi(0)), where chicolonmathfrak gfrakgtomathfrak g/frakg/!/G is
the adjoint quotient; so the requirement that A be non-nilpotent at x
becomes chi(a(0))nechi(0).
[Lie-theoretic reduction]
Let k and G satisfy Hypothesis~hyp:common and either
Hypothesis~hyp:A or Hypothesis~hyp:B, and let kappa be the
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
that
a(t)inmathfrak m[[t]],frakm[[t]], kappa(a(0),dlambda)ne0,
ad_{a(0)}colonmathfrak g/frakg/mathfrak mfrakm
xrightarrow{ sim }mathfrak g/frakg/mathfrak m,frakm,
where mathfrak m=frakm=Lie M and dlambda=dlambda(1)inmathfrak gfrakg is the
image of 1inLie(mathbb G_m)=k.bbG_m)=k. The subgroup M may equal G; the
invertibility includes the zero vector space.
The proof has three layers. First, root-pairing and Jordan facts
(Ssec:lie-basic): kappa pairs root vectors with their opposite
partners, mathfrak tfrakt is nondegenerate, and every element is conjugate
into the Lie algebra of a Borel subgroup. Second, one of two
regime-specific arguments produces a central cocharacter
(Ssec:lie-A and~Ssec:lie-B). Third, a formal smoothing
argument lifts the constant-term reduction to the full formal arc
(Ssec:lie-formal).
## Root pairings and constant terms
Fix a maximal torus Tsubset G. Set
Λ=X_*(T), Λ^vee=X^*(T), mathfrak t=frakt=Lie T=Λ ⊗ k.
For a root alpha write dalphacolonmathfrak tfraktto k for the
differential of the character alpha and dalpha^veeinmathfrak tfrakt
for the differential of the coroot.
The form kappa is nondegenerate on mathfrak t.frakt. For each root alpha
there is a scalar c_alphain k^times such that
kappa(dalpha^vee,h)=c_alpha,dalpha(h)(hinmathfrak t).frakt).
In particular neither dalpha nor dalpha^vee vanishes. Every element
of mathfrak gfrakg is conjugate into the Lie algebra of a Borel subgroup
containing T, and every semisimple element is conjugate into
mathfrak t.frakt.
Proof.
Under the T-weight decomposition
mathfrak g=frakg=mathfrak tfrakt ⊕ ⊕_alphamathfrak g_frakg_alpha,
invariance of kappa makes mathfrak tfrakt orthogonal to each root space
and makes mathfrak g_frakg_alpha pair only with mathfrak g_frakg_{-alpha}.
Nondegeneracy of kappa on mathfrak gfrakg then gives nondegeneracy on
mathfrak tfrakt and on each root pairing. Choose compatible root vectors
e_alpha,e_{-alpha} with [e_alpha,e_{-alpha}]=dalpha^vee; this
follows from the SL_2-root homomorphism and holds for every
isogeny type cite[SS8.1.1--8.1.4]{Springer}, cite[Cor.~5.1.12, Eq.~(5.1.4)]{Conrad}.
Invariance of kappa gives (eq:root-coroot-pairing) with
c_alpha=kappa(e_alpha,e_{-alpha})ne0. Hence one of
dalpha,dalpha^vee vanishes iff the other does. Simultaneous
vanishing would put alphain pΛ^vee and alpha^veein pΛ,
so <alpha,alpha^vee>=2 would be divisible by p^2,
impossible. This argument uses characters, not just their differentials,
and works in characteristic two.
For the Borel-conjugation statement: consider the incidence morphism
[
Gtimes^Bmathfrak b→frakb→mathfrak g,frakg,
[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 tfrakt outside the root differential hyperplanes.
At [1,c], the differential of the incidence morphism is onto (the
negative-root tangent directions map isomorphically to the negative root
spaces, and mathfrak bfrakb supplies the rest), so the morphism is dominant,
hence surjective.
For the semisimple-conjugation statement: write a semisimple element of
mathfrak bfrakb as c+u with cinmathfrak tfrakt and u in the positive
root Lie algebra. Conjugate by positive-root groups in increasing height.
If dalpha(c)ne0, conjugation by the alpha-root group changes the
alpha coefficient by -z,dalpha(c) and only touch higher-height
coefficients; it therefore kills that coefficient. After all steps we
are at c+u_0 with [c,u_0]=0. In a faithful representation c is
semisimple, u_0 is nilpotent, and their sum is semisimple, so the
(semisimple+nilpotent) uniqueness forces u_0=0.
Choose a closed faithful representation G↪GL(V).
Its differential realises mathfrak gfrakg as a restricted Lie subalgebra of
End(V), with xmapsto x^{[p]} equal to matrix pth power. We use
the ordinary matrix Jordan decomposition; the two lemmas below prove
its two parts lie in mathfrak gfrakg in each regime; compare
cite[S4.4, Thm.~4.4.20]{Springer}.
Suppose a_0=s+ninmathfrak gfrakg where s,ninmathfrak g,frakg, s is
semisimple, n is nilpotent in a faithful representation, and
[s,n]=0. Suppose also chi(a_0)nechi(0). After conjugation, s is
a nonzero element of mathfrak t,frakt, M=Z_G(s) is a Levi subgroup,
a_0inmathfrak m,frakm, and ad_{a_0} is invertible on
mathfrak g/frakg/mathfrak m.frakm. Moreover for every lambdacolonmathbb G_mbbG_m
to Z(M),
kappa(a_0,dlambda)=kappa(s,dlambda).
Proof.
The root-pairing lemma conjugates s into mathfrak t.frakt. If s=0 then
n=a_0 is nilpotent. The projection mathfrak bfrakbtomathfrak tfrakt preserves
restricted p-operations; since n is killed by some iterated
p-operation and that operation is injective on mathfrak t,frakt, its
toral projection vanishes. A strictly dominant cocharacter contracts
the positive-root Lie algebra to zero. Therefore every invariant
polynomial has the same value on n as on 0, contradiction with
chi(a_0)nechi(0). Thus sne0.
By Hypothesis~hyp:common(iii), M=Z_G(s) is a Levi subgroup. Its
Lie algebra is kerad_s, which contains a_0. On
mathfrak g/frakg/mathfrak mfrakm the operator ad_s is diagonalisable with
nonzero eigenvalues, while ad_n is nilpotent (indeed
ad_n^{p^e}=ad_{n^{[p^e]}}=0 for large e) and commutes with
ad_s. Therefore on each ad_s-eigenspace the sum
ad_{a_0}=ad_s+ad_n is invertible.
For (eq:nilpotent-orthogonality): conjugate n inside M into
the Lie algebra of a Borel of M containing T; the incidence
argument of the previous lemma applies to M too, and its roots have
nonzero differentials and kappa is nondegenerate on mathfrak mfrakm by
the same weight decomposition. Restricted nilpotence again kills the
toral projection, so the conjugated n lies in the positive-root Lie
algebra of M. That Lie algebra is orthogonal to mathfrak t,frakt, so
kappa(n,dlambda)=0 for dlambdainmathfrak tfrakt fixed by M; by
invariance the same holds before conjugation.
The remaining work is choosing the central cocharacter with
nonvanishing pairing against s. This is where the two regimes
diverge; everything else is common.
## Regime A: a lattice argument for actual central cocharacters
Assume Hypothesis~hyp:A, so k=mathbb Fbar_q.bbFbar_q. Two facts
are needed. First, the Jordan parts s,n of a_0 constructed in
Lemma~lie:constant-parts actually lie in mathfrak g;frakg; they start
life only in an ambient End(V). Second, some lambdain
X_*(Z(M)) has kappa(s,dlambda)ne0. Under Hypothesis~hyp:A,
both are integral.
The Jordan parts first. In a faithful representation write a_0=s+n for
the matrix Jordan decomposition. The eigenvalues of s lie in a finite
field mathbb F_bbF_{p^h}; choose e divisible by h and large enough
to have n^{p^e}=0. Then a_0^{[p^e]}=a_0^{p^e}=s^{p^e}+n^{p^e}=s^{p^e}
so s=a_0^{[p^e]}, hence sinmathfrak gfrakg as the p^eth restricted
power of an element of mathfrak g.frakg. Consequently ninmathfrak gfrakg too.
Lemma~lie:constant-parts then supplies a nonzero sinmathfrak t,frakt,
M=Z_G(s) and the quotient invertibility.
For the cocharacter, we use the following lattice statement. It uses
only Hypothesis~hyp:common(i) and~(iii); no Weyl-order assumption.
Assume kappa and Hypothesis~hyp:common(i),~(iii). Then every Levi
Cartan determinant is a p-adic unit. Moreover, for every Levi
Msupset T,
span_k{dlambda: lambdain X_*(Z(M))}
=mathfrak z(frakz(mathfrak m),frakm),
where the dlambdainmathfrak tfrakt are differentials of actual
central cocharacters of Z(M).
Proof.
Fix a positive system and let I={alpha_1,...,alpha_r} be the
simple roots of M. Set
J_I=span_k{dalpha_i^vee}subsetmathfrak t.frakt. A
central element of mathfrak mfrakm has no root-space component because all
root differentials are nonzero (Lemma~lie:root-pairing), so
(eq:root-coroot-pairing) gives
[
mathfrak z(frakz(mathfrak m)=∩_ifrakm)=∩_iker dalpha_i=J_I^perp.
]
The form on this center is nondegenerate by hypothesis, and on
mathfrak tfrakt by Lemma~lie:root-pairing; nondegeneracy on J_I
follows.
Define integral maps UcolonΛtomathbb Z^r,bbZ^r,
ymapsto(<alpha_i,y>)_i, and Vcolonmathbb Z^rbbZ^rto
Λ, e_jmapstoalpha_j^vee. Their composite is the Cartan matrix
C_I=UV. Reduction to k is denoted by a subscript; by
(eq:root-coroot-pairing) we have V_k^*kappa=D U_k for D a
diagonal invertible matrix. The radical of V_k^*kappa V_k is exactly
ker V_k, again by nondegeneracy on J_I=im V_k. Hence
rank(C_{I,k})=rank(V_k)
=rank(U_k).
If some Levi Cartan determinant were divisible by p, the standard
determinant list
cite[Thm.~C.56,~S24(c)]{Milne}
[
{c|ccccccccc}
type&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2
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 rtimes N_T
and N_Ttimes r. By (eq:cartan-ranks) both reductions have rank
<r, so every maximal minor of either matrix is divisible by p.
Cauchy--Binet gives
[
det C_I
=sum_{substack{Jsubset{1,...,N_T}|J|=r}}
det(U_{:,J}),det(V_{J,:}),
]
each summand divisible by p^2, contradicting det C_I=p. This uses
the full lattices, with no isogeny-type assumption.
For an arbitrary Levi set d_I=det C_I and define
Pi_I=d_Iid_Λ-Vadj(C_I)UinEnd(Λ). Then
UPi_I=0; the lattice ker U is exactly the set of integral
cocharacters into Z(M). For zinker U_k, Pi_{I,k}(z)=d_I z; since
d_I is invertible in k, the differentials of ker U span
ker U_k=mathfrak z(frakz(mathfrak m),frakm), giving
(eq:central-span). For r=0 take d_I=1 and Pi_I=id; the
same proof covers tori.
Now return to the regime-A story. Under Hypothesis~hyp:A, the
Jordan parts s,ninmathfrak gfrakg have been constructed. Lemma~lie:constant-parts
gives sinmathfrak z(frakz(mathfrak m)∖frakm)∖{0} and the quotient
invertibility. Since kappa is nondegenerate on mathfrak z(frakz(mathfrak
m)frakm) by Hypothesis~hyp:common(i),
Lemma~lie:lattice produces an actual lambdain X_*(Z(M))
with kappa(s,dlambda)ne0. Orthogonality
(eq:nilpotent-orthogonality) gives the required
kappa(a_0,dlambda)ne0. This used no Weyl-order assumption.
## Regime B: Weyl-order averaging
Now assume Hypothesis~hyp:B. Eigenvalues of a matrix over k need
not be fixed by any positive power of Frobenius, so the restricted-power
trick of regime A does not apply. Instead we interpolate the inverse of
Frobenius using an mathbb F_p-linearbbF_p-linear additive polynomial.
In the faithful representation write a_0=s+n and choose e with
n^{p^e}=0, so a_0^{p^e}=s^{p^e}. Let V_0subset k be the
finite-dimensional mathbb F_p-spanbbF_p-span of the eigenvalues of s, and let
V_e={v^{p^e}:vin V_0}. The inverse of Frobenius V_eto V_0 is
mathbb F_p-linear.bbF_p-linear. If V_0=0 then s=0 and a_0=ninmathfrak g,frakg,
contradicting chi(a_0)nechi(0)
(Lemma~lie:constant-parts). Hence r=dim_{mathbb F_p}bbF_pV_ege1 and
we may assume V_0ne0.
Take a basis beta_1,...,beta_r of V_e over mathbb F_p.bbF_p. The
Moore matrix (beta_i^{p^j})_{1le ile r, 0le j<r} is invertible:
if not, some nonzero polynomial sum_{j=0}^{r-1}c_j z^{p^j} of degree
le p^{r-1} would vanish on all p^r elements of V_e, contradicting
|V_e|=p^r. Therefore there is an additive polynomial
Q(z)=sum_{j=0}^{r-1}c_j z^{p^j} agreeing with inverse-Frobenius on
V_e. Applying it to the diagonalisable matrix s^{p^e} gives
[
s=Q(a_0^{p^e})=sum_{j=0}^{r-1}c_j a_0^{[p^{e+j}]}inmathfrak g.frakg.
]
Hence ninmathfrak gfrakg as well. Lemma~lie:constant-parts now
supplies sinmathfrak t∖frakt∖{0}, M=Z_G(s) and the quotient
invertibility.
To find lambda: let W_M=N_M(T)/T be the Weyl group of M and set
h_M=|W_M|. Since W_Msubset W_G, Hypothesis~hyp:B implies
h_M is invertible in k. The averaging operator
[
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_{win W_M}wmu_iinΛ^{W_M}. Their
differentials span mathfrak t^frakt^{W_M} because dnu_i=h_M e_M(dmu_i).
The integral reflection formula s_alpha(lambda)=lambda-
<alpha,lambda>alpha^vee and torsion-freeness of
Λ show that Λ^{W_M}=X_*(Z(M)): a W_M-invariant
cocharacter has zero pairing with every root of M, and conversely. No
division of a cocharacter by h_M is made. Nondegeneracy of kappa on
mathfrak t^frakt^{W_M} now gives an integral lambdain X_*(Z(M)) with
kappa(s,dlambda)ne0, and (eq:nilpotent-orthogonality) gives
kappa(a_0,dlambda)ne0.
This argument applies to any algebraically closed k. Its use of
pnmid|W_G| is specific to Hypothesis~hyp:B.
## Formal smoothing: from constant term to whole arc
Both regimes have now produced the same constant-term data: a Levi M,
a central cocharacter lambda, and a_0=a(0)inmathfrak mfrakm with
kappa(a_0,dlambda)ne0 and ad_{a_0} invertible on
mathfrak g/frakg/mathfrak m.frakm. The remaining step moves the entire formal
arc a(t) into mathfrak m[[t]].frakm[[t]].
Let G be a connected reductive group over an algebraically closed
field k, Msubset G a Levi subgroup, a(t)inmathfrak g[[t]]frakg[[t]] with
a_0=a(0)inmathfrak m.frakm. If ad_{a_0} is invertible on
mathfrak g/frakg/mathfrak m,frakm, then there is g(t)in G(k[[t]]) with
g(0)=1 and Ad(g(t))a(t)inmathfrak m[[t]].frakm[[t]].
Proof.
Proceed by successive approximation. Suppose the arc already lies in
mathfrak m[[t]]frakm[[t]] modulo t^r, rge1, and let c_rin
mathfrak g/frakg/mathfrak mfrakm be the class of its t^r-coefficient. Choose
Dinmathfrak gfrakg with c_r+[D,a_0]=0 in the quotient (possible by
invertibility). Smoothness of G gives g_r(t)in G(k[[t]]) with
g_r(t)equiv1+t^rDpmod{t^{r+1}}; formally, this is a lift of a point
in the tangent kernel of the square-zero ideal (t^r)/(t^{r+1}) to
G(k[t]/(t^{r+1})), obtained by iterating formal smoothness at G. Then
conjugation by g_r(t) changes the t^r-coefficient by [D,a_0] and
leaves all lower coefficients fixed, so it places the arc in
mathfrak m[[t]]frakm[[t]] modulo t^{r+1}.
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]
Under Hypothesis~hyp:A, use the restricted-power construction of
Jordan parts together with Lemma~lie:lattice. Under
Hypothesis~hyp:B, use the Moore-matrix construction together with
integral Weyl averaging. In both cases Lemma~lie:constant-parts
gives the Levi M, the quotient invertibility and
kappa(a_0,dlambda)ne0. Apply Lemma~lie:formal after the
initial constant conjugation; the formal change of frame fixes a_0 by
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 gfrakg injects into
mathfrak g/frakg/mathfrak m,frakm, so the adjoint action is invertible on the
truncated nonzero-weight modules used below. We do not need
Z_G(lambda)=M. If lambda is central in G the Hecke orbit has
dimension zero, but the nonzero pairing in (eq:lie-output) still
supplies the moving-point covector. Tori are therefore included in every
allowed characteristic.
% =========================================================================
# Hecke modifications and transverse cotangent sections
% =========================================================================
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
xin X(k), a cocharacter lambda, and a modification
(P,P',x)in H^lambda_s(k) such that:
[label=(roman*)]
the moving-point displacement (dlambda/t)|_x pairs nonzero
against A to give a leg covector xine0;
the fixed-input cotangent section S_A and the fixed-output one
S_{A'} both vanish at the modification and both have invertible
derivatives on their respective tangent spaces.
These are the "transversality" and "isolation" conditions.
## Correspondence and kernels
We keep the trait S, the curve mathscr X/SscrX/S and the bundle stack
mathcal BcalB from Section~sec:specialization. A one-point Hecke
modification is a quadruple (P,P',x,alpha) where xinmathscr XscrX
and alpha identifies P' and P away from the graph of x. Write
[
p(P,P',x,alpha)=P, o(P,P',x,alpha)=P',
ell_H(P,P',x,alpha)=x.
]
Convention: in a formal coordinate t at x, modification by
lambda means the output frame is lambda(t) times the input frame.
Relative position depends only on the Weyl orbit of lambda. Let
mu=lambda^+ be its dominant conjugate, m=< 2rho,mu>
where 2rho is the sum of positive roots, mu^vee=-w_0mu, and
mathcal Z=calZ=mathcal BcalBtimes_Smathscr X.scrX. Denote the exact-position
correspondence by H=H^lambda, and set
[
widetilde p=(p,ell_H)colon Htomathcal Z,calZ,
q=(o,ell_H)colon Htomathcal Z.calZ.
]
Both widetilde p and q forget some of the modification data.
[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
are representable and proper. On the output side the bound on
overlineoverline HH has dominant label mu^vee.
Let V be the irreducible representation of check G corresponding
to this bound, with the indicated input/output convention. Then the
generic Hecke functor has the form
mathrm H_V(F)=rmH_V(F)=overlineoverline q_q_{K,*}mathcal I_K,calI_K,
with mathcal I_KcalI_K on H_K equal to p_K^*F tensored with a fixed
invertible constant complex (the perverse intersection-cohomology
normalisation E[m], plus a Tate line if weight-normalised).
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[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
by Beauville--Laszlo descent [cite]. More explicitly,
the graph is an effective Cartier divisor. Gluing applies to the
coordinate algebra, multiplication and coaction of an affine G-torsor
(as it does to modules); the torsor identity is checked on the two
pieces. A disc frame exists locally by lifting a frame on the graph
successively through its infinitesimal neighbourhoods using smoothness
of G. Changes of frame and of coordinate preserve the orbit and
its bound, so properness, smoothness and dimension all descend to the
input-and-leg projection.
Swapping the two bundles and inverting alpha is an involution of the
Hecke correspondence. On double cosets it sends mu to -mu, whose
dominant representative is mu^vee, and it carries the bound to the
oppositely-labelled bound. This gives the assertions for q, and
< 2rho,mu^vee>=< 2rho,mu>=m. Separatedness
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.
The fibres here are relative-position spaces: an automorphism of the
moving bundle compatible with its off-graph identification is the
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
with p_K^*F and pushes forward along the proper output-and-leg
projection cite[SS4.2.1--4.2.4]{GR}. Relabelling by mu^vee if
needed matches these projections with the chosen V. A fixed overall
shift or Tate line changes neither the description on the open orbit nor
any vanishing assertion below. Only the open-orbit restriction of the
kernel is identified here; on the boundary the kernel remains its
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
element of H^0 of the cotangent-complex fibre. Deformation theory and
Serre duality cite[Tag~0BS4]{Stacks}, together with kappa, identify
this space as
[
T_P^*mathcal B_s=H^1(X,calB_s=H^1(X,mathfrak g_P)^*=H^0(X,frakg_P)^*=H^0(X,mathfrak g_Pfrakg_P ⊗ omega_X),
]
where mathfrak g_Pfrakg_P is the adjoint bundle. We call such a vector A a
Higgs field. For a representable smooth map, the cotangent
pullback is injective and its image is the kernel of passage to the
relative cotangent; we use this in the form supplied by the cotangent
triangle.
[Transverse modification]
Let Pinmathcal B_s(k)calB_s(k) and let Ain T^*_Pmathcal B_scalB_s be
non-nilpotent. There exist a point xin X(k), a cocharacter lambda,
a modification hin H^lambda_s(k) from P to a bundle P', and a
covector
[
theta=(A',xi)in T^*_{(P',x)}(mathcal B_scalB_stimes X), xine0,
]
with A'=A away from x, satisfying
dp_h^*A=dq_h^*theta.
Moreover, on the fixed-side fibres
[
H^{in}_{P,x}=widetilde p^{-1}(P,x),
H^{out}_{P',x}=q^{-1}(P',x),
]
with fixed-bundle identifications retained, the following hold. The
restriction of p^*A to the q-relative tangent bundle defines a
section S_A of Ω_{H_s/mathcal Z_s,q}calZ_s,q; the restriction of
o^*A' to the widetilde p-relative tangent bundle defines a section
S_{A'}. Both sections vanish at h, and both derivatives
(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
zero at h.
Proof.
Choice of lambda. Since A is non-nilpotent, choose
xin X(k) where its value is non-nilpotent. Choose a formal coordinate
t, a bundle frame at x, and write A=a(t),dt. Chevalley restriction
identifies the nilpotent fibre with chi^{-1}(chi(0)), so
chi(a(0))nechi(0). Proposition~lie:input then supplies a
formal change of frame, a Levi M, and a cocharacter
lambdacolonmathbb G_mbbG_mto Z(M) with
a(t)inmathfrak m[[t]],frakm[[t]],
ad_{a(0)} invertible on mathfrak g/frakg/mathfrak m,frakm,
kappa(a(0),dlambda)ne0.
After constant conjugation we regard lambda as an element of the
fixed cocharacter lattice, so it labels an integral model of
Lemma~hecke:models. Modify by lambda(t). Since lambda
centralises M, the transported field in the output frame is again
a(t),dt; combined with A away from x, this defines the global
Higgs field A' on P'. This central-in-a-Levi modification and the
resulting leg covector originate in
cite[SS20.4--20.6]{AGKRRV}. What follows makes both derivative
assertions explicit.
Tangent spaces. In the input frame, put
L=mathfrak g[[t]],frakg[[t]], L'=Ad(lambda(t))L, I=Lcap L'. Let
mathfrak g_jfrakg_j be the weight-j space of lambda; the integers j
are genuine characters of mathbb G_m,bbG_m, not reduced mod p. Then
L'=⊕_j t^jmathfrak g_j[[t]],frakg_j[[t]],
P_1:=L/I=⊕_{j>0}mathfrak g_jfrakg_j ⊗ k[[t]]/(t^j),
P_2:=L'/I=⊕_{j<0}mathfrak g_jfrakg_j ⊗ (t^jk[[t]]/k[[t]]).
An infinitesimal disc automorphism 1+ε D, Din L, acts on
the output lattice while fixing (P,x), and its stabiliser tangent is
I. Its orbit is smooth of dimension dim_k(L/I)=m by the root
formula, so this computes the full tangent space:
T_hH^{in}_{P,x}=P_1,
T_hH^{out}_{P',x}=P_2.
The same argument with output fixed gives the second identity. The
kernel over dual numbers is identified with the Lie algebra via the
canonical isomorphism, no exponential map needed.
There is a perfect pairing
betacolon P_1times P_2to k,
beta(D,C)=Res_{t=0}kappa(D,C),dt.
Indeed, L is its own residue-annihilator on mathfrak g((t)),frakg((t)), and
invariance of kappa gives the same statement for L'. So beta
descends to the quotients. Concretely, kappa pairs mathfrak g_jfrakg_j
perfectly with mathfrak g_frakg_{-j} and the coefficient of t^a in the
first space, for 0le a<j, pairs with the coefficient of t^{-a-1}
in the second. This is perfect, including when j is divisible by p.
Cotangent identity and moving leg. Take the gluing sign
convention where an infinitesimal transition 1+ε C gives the
boundary class of the principal part C. Serre duality pairs it with
A by Reskappa(a,C),dt. For output-fixed tangent
Cin L' the residue is zero because ain L' and L' is self-annihilating.
Therefore dp_h^*A vanishes in the q-relative cotangent, so it is
dq_h^*theta for a unique covector theta.
The bundle component of theta is A'. Indeed, choose x'ne x and
vary both bundle transitions only at x', leaving the modification at
x unchanged. Principal parts at x' surject to
H^1(X,mathfrak g_frakg_{P'}) (Serre vanishing for Ngg0), and the
pairings agree on these directions since A=A' near x'. Write
theta=(A',xi).
Now keep the input fixed and move the leg from t=0 to t=ε.
The output transition is g_ε=lambda(t-ε) and to first
order in ε,
(delta g)g^{-1}=-dlambda/t. Evaluating the cotangent identity on
this motion gives
[
0=-kappa(a(0),dlambda)+xi(∂_t),
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 Din L; a test
vector Cin L' extends as C_ε=Ad(1+ε D)C. The input
field remains a(t),dt. Differentiating the residue pairing gives
bigl<(dS_A)_h(Dbmod I),Cbmod Ibigr>
=Reskappa(a,[D,C]),dt
=Reskappa([a,D],C),dt
=beta(ad(a)D,C).
Since S_A(h)=0 the derivative is well-defined, and since ad(a)
preserves L,L',I, the formula is independent of representatives.
For invertibility: ad(a(t)) preserves every weight summand in
(hecke:weight-quotients). Because lambda is central in M,
each nonzero weight space injects as an invariant summand into
mathfrak g/frakg/mathfrak m;frakm; its endomorphism from ad(a(0)) is therefore
invertible by (hecke:lie-data). The corresponding k[[t]]-linear
block has unit determinant, so ad(a(t)) is invertible on every
truncated summand of P_1 and P_2. Combined with
(hecke:derivative) and (hecke:residue-pairing), this shows
(dS_A)_hcolon P_1to P_2^* is an isomorphism. Zero-weight directions
do not appear in these quotients, so no extra regularity of lambda
outside M is needed.
The reverse derivative. The value of S_{A'} at h is zero
because ain L and L is self-annihilating. Keeping the output fixed,
vary the input lattice by Cin L' and extend Din L to
Ad(1+ε C)D. The same calculation, with the same gluing
convention, gives
[
bigl<(dS_{A'})_h(Cbmod I),Dbmod Ibigr>
=Reskappa(a,[C,D]),dt
=Reskappa([a,C],D),dt,
]
and invertibility on P_2 with residue duality proves the second
assertion. Both sections are sections of rank-m vector bundles on
smooth m-dimensional fibres; invertible derivatives make their zeros
reduced and isolated.
When m=0, both tangent spaces vanish and both derivatives are
isomorphisms between zero spaces. The smooth zero-dimensional fibres
already isolate h, and the moving-leg computation still applies.
Modifications of this form, and tori, are thus included.
The proposition gives both properties we need. The nonvanishing
xine0 allows a suitable hypersurface cut of the Hecke correspondence
to be smooth over mathcal B,calB, while the two relative derivatives
isolate h and set up the quadratic coordinates around it. In
Section~sec:twohecke, these two facts are used in reverse order:
the isolating one first, and the nonvanishing one later.
% =========================================================================
# Two Hecke correspondences and a quadratic test
% =========================================================================
We now convert Lemma~spec:vanishing into a local test for a
non-nilpotent cotangent direction, in one big step. The first Hecke
correspondence supplies a proper pushforward whose fibre is a smooth
curve; the second makes the pushed-forward sheaf vanish everywhere
except at a single point; the two transverse sections of
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
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
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,
]
where u_i,v_i are affine coordinates (distinct symbols from the point
u) and f is pulled back from U. Then at the point
nu=(u,x,0,0),
(Psi_{Q^a}(tau_K^*F_T))_nu=0.
When m=0, this is the vanishing on Q^a=V(f)times_Smathscr X.scrX.
Proof.
Apply Proposition~hecke:transverse to A to obtain a
modification h, its leg x, its orbit dimension m, and a covector
theta=(A',xi) with xine0, satisfying
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
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_2times_{p_2,mathcal B,b}calB,bU,
W=H_2times_{mathcal B}calBDsubset W_0.
]
Both map to mathcal Z=calZ=mathcal BcalBtimes_Smathscr XscrX by
q_2=(o_2,ell_{H_2}). The map W_0tomathcal ZcalZ is smooth: its
projection to H_2 is a base change of the smooth chart b, and q_2
is smooth. At w=(h,u), the differential of the equation cutting out
W from W_0 is the pullback of theta by (eq:two-cotangent).
Its relative differential over the output bundle stack is nonzero,
because the leg component is xine0. The smooth-hypersurface
criterion therefore makes Wtomathcal BcalB (via the output bundle)
smooth near w. The arbitrary-leg assertion of
Lemma~spec:vanishing gives
(Psi_W(q_{2,K}^*mathrm H_V(F)))_w=0.rmH_V(F)))_w=0.
Note: smoothness is used only for the projection to mathcal B,calB, matching
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.
]
The following squares are cartesian; in the second the vertical map
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 ZcalZ
[column sep=large,row sep=large]
overlineoverline P_DP_D arrow[r] arrow[d,"pi"']
& overlineoverline H_1H_1 arrow[d,"overlineoverline q_1"]q_1"]
W arrow[r,"q_2"'] & mathcal Z.calZ.
]
Write ccolon Pto U for the map through W_0, and
acolon Ptomathcal BcalB for the input map p_1. So a records the
input from which the sheaf is pulled back, and c records the second
input in U on which we impose f=0. The distinguished point is
z=(h,h,u)in P_D(k).
Let mathcal Ibar_KcalIbar_K be the pullback of the Satake integrand
to (overlineoverline P_D)_K.P_D)_K. Proper base change on the generic fibre gives
Rpi_{K,*}mathcal Ibar_KcalIbar_Ksimeq q_{2,K}^*mathrm H_V(F).rmH_V(F).
Apply proper compatibility of specialisation, then proper base change
at w, to (eq:two-hecke-zero). With
[
Z_w=pi^{-1}(w),
mathcal K=(calK=(Psi_{overlineoverline P_D}P_Dmathcal Ibar_K)|_calIbar_K)|_{Z_w},
]
we get
RGamma(Z_w,mathcal K)=0.calK)=0.
Here Z_w is the proper bounded Hecke fibre with output bundle and leg
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
exact-position part of Z_w, the differential of f(c) is q_1^*theta
pulled back. If the relative differential over mathcal BcalB of f(c)
were zero, cotangent injectivity for the smooth map Pto H_1 would
imply that q_1^*theta comes from the input cotangent under p_1.
Then passing to the relative cotangent of widetilde p_1=(p_1,ell_{H_1}),
the leg term drops out, and the necessary condition becomes
S_{A'}(h_1)=0. The second half of
Proposition~hecke:transverse says this section has an isolated
zero at h on the fixed-output-and-leg fibre. Hence P_Dtomathcal BcalB
via a is smooth at every point of a punctured neighbourhood of z
in Z_w. On the exact-position open, the generic integrand equals
a_K^*F. Lemma~spec:vanishing therefore gives a neighbourhood
O of z in Z_w with
[
mathcal K|_calK|_{O∖{z}}=0.
]
This local vanishing prevents any cancellation with the boundary of the
Hecke bound. Put C_w=Z_w∖ O (reduced structure). The closed
subsets {z} and C_w are disjoint, so open-closed localization
identifies
[
mathcal KcalKsimeq i_{z,*}mathcal K_zcalK_z ⊕ i_{C_w,*}(i_{C_w}^*mathcal K),calK),
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.
No boundary behaviour of the Satake kernel is used.
Step 3: chart lift along the whole diagonal.:
We must now translate (eq:two-isolated-zero) into coordinates.
First, the distinguished diagonal Deltasimeq W_0subset P is a
section of the smooth separated map Pto W_0 of relative dimension
m, hence a closed regular immersion near z. On Delta there is a
specified isomorphism bcsimeq a, coming from the two identical
modifications.
We claim that after an 'etale replacement of P near z, there
is a map rcolon Pto U with an isomorphism
brsimeq a, r|_Delta=c|_Delta,
whose restriction to Delta is the specified isomorphism. For this,
form the smooth space E_P=Ptimes_{a,mathcal B,b}calB,bUto P. The
diagonal data give a section over Delta. First take an 'etale
scheme neighbourhood in P. Since mathcal BcalB has affine diagonal
(Section~sec:specialization), E_Pto Ptimes_S U is affine, so
E_P is a scheme as well. Near the section point, smooth coordinates
give an 'etale map E_Ptomathbb A_P^bbA_P^{e}
cite[Tag~039P]{Stacks}. After shrinking, the section over Delta
lands in this coordinate neighbourhood. Extend its coordinate functions
Deltato P, then pull back the 'etale map along the extended tuple.
This gives an 'etale neighbourhood of P with a map to E_P. Its
inverse image of Delta contains the open branch specified by the
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.
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),
]
is smooth of relative dimension m. So T,Delta,P are smooth over
S of relative dimensions
[
N+1, N+1+m, 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
the exact identity
f(c)-f(r)=sum_{i=1}^m u_iv_i.
The restrictions u_i|_Delta are determined by the conormal class of
the left side.
Consider the whole local diagonal fibre Delta_{(u,x)} on the special
fibre. This parametrises modifications with input bundle b(u), its
identification, and leg x fixed. The relative differential of f(c)
for Pto W_0 is zero. On this diagonal fibre, the relative
differential of f(r) is induced by df_u=b^*A: the restriction
r|_{Delta_{(u,x)}} is constant at value u, and the bundle
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)}.
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
Pto Ttimes_Smathbb A_S^bbA_S^{2m}.
]
Its relative differential at z is an isomorphism: a tangent vector in
its kernel is first killed by every dv_i, so lies in T_zDelta; its
(dr,dleg) then places it in T_zDelta_{(u,x)}; then the
derivative isomorphism in (eq:two-coefficients) forces the vector
to vanish. Source and target are both smooth over S with the same
relative dimension N+1+2m, so this injective differential is an
isomorphism, and the relative Jacobian criterion makes Phi 'etale
at z.
By (eq:two-exact-quadratic), the cut P_D=V(f(c)) is exactly
Phi^{-1}(Q^a). So the induced map Phi_Dcolon P_Dto Q^a is
'etale near z, sending z to nu. Moreover (eq:two-chart-lift)
identifies
[
a_K^*Fsimeq r_K^*F_UsimeqPhi_{D,K}^*tau_K^*F_T
quadon (P_D)_K,
]
so Phi_D identifies the actual generic sheaf and the equation
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=Dtimes_Smathscr X.scrX. Then smooth base change along Dtimes_S
mathscr XscrXto D yields (Psi_D(F_U|_{D_K}))_u=0 directly.
The argument used no division by 2 and no singular-support claim in
mixed characteristic. Its output for m>0 is a vanishing on the quadric
Q^a; the next section recovers the corresponding vanishing on the
hypersurface V(f) itself.
% =========================================================================
# From quadratic models to transverse slices
% =========================================================================
Section~sec:twohecke gave a vanishing on the affine quadric Q^a
at its vertex. To get a test on the hypersurface D=V(f) itself, we
compactify the quadric to a projective family and use the relation
between hyperplane powers and the closed support D. Concretely, the
cone of the hyperplane-class map is a shifted constant Tate line
supported on f=0; after tensoring with the ambient sheaf, the
hyperplane terms have vanishing nearby cycles from the smooth-chart
result of Lemma~spec:vanishing, and the whole proper quadric
vanishing then forces the hypersurface vanishing. A second proper
comparison, using projective incidence, handles an arbitrary collection
of equations simultaneously.
## The projective quadric comparison
[Quadric comparison]
Let K_0 be an algebraically closed field of characteristic neell,
E=mathbb Qbar_bbQbar_ell, and B a finite-type K_0-scheme. For
finGamma(B,mathcal O_B)calO_B) put icolon D=V(f)↪ B. If
mge1, form
[
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]
→ Rg_*E_{Q_f}
→ i_*E_D(-m)[-2m]
→ (⊕_{j=0}^{2m-1}E_B(-j)[-2j])[1].
The identification of the third term with the Tate line requires only a
fixed one-dimensional generator. Neither B nor D needs to be smooth
or reduced.
Proof.
Write h=c_1(mathcal O_calO_{Q_f}(1))in H^2(Q_f,E(1)). Adjunction on each
h^j gives the morphism E_B(-j)[-2j]to Rg_*E_{Q_f}. Let
mathcal KcalK be the cone of their direct sum, and let
j_Bcolon B∖ D↪ B.
We recall the cohomology of a split smooth projective quadric of
dimension n: no odd cohomology, one E(-r) in degree 2r for
0le rle n, and one additional copy when n is even and r=n/2
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 (congmathbb P^1)bbP^1) and
the split zero-dimensional quadric (two points). Localization plus
RGamma_c(mathbb A^r,E)=E(-r)[-2r]bbA^r,E)=E(-r)[-2r] gives the cohomology. On a smooth
quadric of positive dimension the top hyperplane power has degree 2,
hence is nonzero in E; all lower powers are nonzero as well.
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.
Over D the family, with its hyperplane line bundle, is the constant
family with fibre
[
V_m=Bigl{sum_{a=1}^m u_av_a=0Bigr}subsetmathbb P_bbP_{K_0}^{2m}.
]
Let e=[0:...:0:1] be its vertex. Projection away from e identifies
V_m∖{e} with the total space of a line bundle over the split
smooth quadric
[
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, H^1(V_m,E)=0,
H^r(V_m,E)=H^{r-2}(C_m,E)(-1)(rge2).
So V_m has the cohomology of mathbb P^bbP^{2m-1} plus one extra
E(-m) in degree 2m.
The hyperplane powers on V_m remain nonzero. To check the top one
without requiring smoothness of V_m, resolve the vertex by the
projective bundle of lines
[
picolonwidetilde V_m=mathbb P_bbP_{lines,C_m}
(mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m})
→ C_m.
]
The inclusion mathcal O_calO_{C_m}(-1) ⊕ mathcal O_calO_{C_m}↪
(trivial rank-(2m+1)) defines widetilde V_mto V_m; its
fibre over a point of C_m is the projective line joining that point
to e. If zeta is the pullback of the hyperplane class and
h_C=c_1(mathcal O_calO_{C_m}(1)), the projective-bundle relation is
zeta^2=pi^*h_C,zeta cite[S23]{MilneEtale}, hence
[
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 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 KcalK with the constant
pullback of this vector-space cone (both the family over D and the
hyperplane-power morphism are pulled back from V_m). Open-closed
localization gives
[
(j_B)_!j_B^*mathcal KcalKtomathcal KcalKto i_*i^*mathcal KcalKtocdot,
]
whose first term vanishes, so mathcal KcalKsimeq i_*E_D(-m)[-2m]. In
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
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
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.
## The hypersurface test
Work over the trait S, with mathcal BcalB and Finmathcal D_CcalD_C as in
Proposition~spec:missing. For a smooth finite-type chart
bcolon Utomathcal B,calB, write F_U=b_K^*F, and let
mathcal N_UcalN_Usubset T^*U_s be the image of the global nilpotent cone
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
[
d(f_s)_unotin(mathcal N_U)_u,calN_U)_u,
]
then
[
(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,calB_s,
its image in the relative cotangent of b is nonzero, so Dtomathcal BcalB
is smooth near u; Lemma~spec:vanishing gives the claim.
Otherwise d(f_s)_u=b^*A for some non-nilpotent
Ain T^*_{b(u)}mathcal B_s.calB_s. Proposition~two:vertex then provides
xinmathscr X_s(k)scrX_s(k) and mge0 such that nearby cycles of the
pullback of F_U to
Q^a={sum_{a=1}^m u_av_a+f=0} vanish at (t_0,0,0), where
T=Utimes_Smathscr XscrX and t_0=(u,x). If m=0 this affine model is
D_T and smooth compatibility for D_Tto D gives the result.
Assume mge1. Compactify to
[
gcolon Q_f=Bigl{sum_{a=1}^m u_av_a+fw^2=0Bigr}
subsetmathbb P_T^bbP_T^{2m}to T.
]
The fibre over t_0 is the split cone of Lemma~slice:quadric. At
every point of this fibre except its vertex, some u_a or v_a is
nonzero: in the chart u_a=1 its equation solves uniquely for v_a
and g is smooth there (works also in characteristic 2). Lemma~spec:vanishing
gives Psi_TF_T=0, and smooth compatibility gives vanishing of nearby
cycles of g_K^*F_T at all those nonvertex points. The vertex is in
the chart w=1, where Proposition~two:vertex gives the vanishing.
So the nearby-cycle complex restricts to zero on the entire proper
fibre. Proper compatibility plus the projection formula give
(Psi_T(F_T ⊗ Rg_{K*}E))_{t_0}=0.
Apply Lemma~slice:quadric on T_K and tensor with F_T: the
projection formula identifies the third term with
i_{K,*}(F_T|_{(D_T)_K})(-m)[-2m] for
icolon D_T↪ T. After Psi_T and stalk at t_0,
the first term vanishes (Psi_TF_T=0) and the second by
(eq:slice-quadric-pushforward-zero); proper compatibility for i
identifies the third with
(Psi_{D_T}(F_T|_{(D_T)_K}))_{t_0}(-m)[-2m], which therefore also
vanishes. Smooth compatibility for D_Tto D finishes the proof.
The passage through the projective quadric uses no purity theorem for
the possibly singular hypersurface D; it uses the actual closed-support
term of the comparison triangle. The same principle handles several
equations simultaneously.
## Projective incidence and higher codimension
[Incidence comparison]
Let B be a finite-type scheme over an algebraically closed field of
characteristic neell, let f_1,...,f_dinGamma(B,mathcal O_B)calO_B)
with dge1, and set icolon J=V(f_1,...,f_d)↪ B.
For the incidence family
[
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]
→ Rr_*E_I
→ i_*E_J(-(d-1))[-2(d-1)]
→ (⊕_{j=0}^{d-2}E_B(-j)[-2j])[1].
For d=1 the first term is zero and I=J.
Proof.
If d=1, the equation in Btimesmathbb P^0bbP^0 is f_1=0 and the
statement is immediate. Suppose dge2. Off J, the map
mathcal O_B^dcalO_B^dtomathcal O_BcalO_B with coefficients (f_1,...,f_d) is
surjective, so its kernel is a vector bundle of rank d-1 and I is
its projective bundle of lines. The restricted hyperplane powers identify
its direct image with the first term of
(eq:slice-incidence-triangle) (check on geometric fibres, using
proper base change and the cohomology of mathbb P^bbP^{d-2}).
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
this restricted cone.
[Higher-codimension test]
Let bcolon Utomathcal BcalB be smooth with U finite type over S,
and let F_U=b_K^*F. Suppose uin U_s(k) and f_1,...,f_d are
regular on U vanishing at u. Suppose their special-fibre
differentials at u are linearly independent and
[
span_k{d(f_{1,s})_u,...,d(f_{d,s})_u}
cap(mathcal N_U)_u=calN_U)_u={0}.
]
Then for J=V(f_1,...,f_d),
[
(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 dge1 form the
proper incidence morphism
[
rcolon I=Bigl{sum_{a=1}^d c_af_a=0Bigr}
subset Utimes_Smathbb P_S^bbP_S^{d-1}→ U.
]
Each affine projective-space chart in Utimes_Smathbb P_S^bbP_S^{d-1} is
still smooth over mathcal B.calB. On the chart c_a=1, the hypersurface
equation is f_a+sum_{bne a}c_bf_b=0, with special-fibre differential
at (u,[c]) equal to sum_{b=1}^dc_b,d(f_{b,s})_u (the projective
parameter term is zero since every f_b(u)=0). This combination is
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
Proposition~slice:hypersurface gives vanishing of nearby cycles of
r_K^*F_U at every closed point over u.
The restriction of this nearby-cycle complex to the projective fibre is
constructible; a nonzero constructible complex on a finite-type scheme
over k has a nonzero stalk at some closed k-point, so it must
vanish on the entire fibre. Proper compatibility gives
[
(Psi_U Rr_{K*}r_K^*F_U)_u=0.
]
Tensor the triangle of Lemma~slice:incidence on U_K with F_U
and use the projection formula: the first term is a finite sum of shifts
and twists of F_U, the second is Rr_{K*}r_K^*F_U, and the third is
i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)] for icolon J↪ U.
After Psi_U, the first two terms vanish at u, and proper
compatibility for i shows that the third is a shift and twist of
(Psi_J(F_U|_{J_K}))_u. Thus the third vanishes. For d=1 the first
sum is empty and the same argument applies.
We have thus obtained all finite transverse-slice tests, while applying
the hypersurface argument only to schemes smooth over mathcal B.calB. It
remains to find one such slice that meets Psi on the support of F
in a nonzero way.
% =========================================================================
# A finite slice of the support
% =========================================================================
We finish the proof of Theorem~thm:full-support by applying the
slicing theorem to the actual support of F. The structure of the
argument: we first arrange that a nonzero generic stalk specializes to
the closed fibre; then we cut the support down to a finite scheme over
the trait; on a finite scheme nearby cycles are a finite direct sum of
stalks, so they cannot vanish if any single one is nonzero.
## Extending a generic bundle
Let mathscr XscrXto S be the smooth projective curve from
Section~sec:specialization, and let K be the fixed algebraic
closure of Frac(R). Every G-bundle on mathscr X_KscrX_K
extends to a G-bundle on mathscr X_scrX_{R'} for some finite extension
Frac(R')/Frac(R) inside K, where
R' is the integral closure of R in that extension.
Proof.
Let mathcal P_KcalP_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
reductive groups over any field). Choose a trivialization away from
distinct points x_1,...,x_rinmathscr X(K).scrX(K). Gluing at those points
gives a chain of one-point modifications
[
mathcal P_0calP_0dashrightarrowmathcal P_1calP_1dashrightarrow...
dashrightarrowmathcal P_r=calP_r=mathcal P_K,calP_K,
mathcal P_0calP_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
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
chains is a proper algebraic space of finite type over S. It carries
the universal final bundle and contains the chosen chain as a
K-point. This point descends to a finite extension of
Frac(R) (the space is finite type and K is algebraic).
The valuative criterion extends the descended point over R', and the
universal final bundle gives the extension. The construction allows
legs to coincide on the closed fibre, since it uses successive one-point
modifications.
Now suppose a spectral component is missing and let F be the nonzero
compact object from Proposition~spec:missing. It is constructible
on finite-type smooth charts, so there is a K-valued bundle at which
its stalk is nonzero (a nonzero constructible complex on a finite-type
scheme has a nonzero stalk at some closed K-point). Apply
Lemma~support:extension to that bundle and replace R by the
resulting R'; all specialization statements from
Section~sec:specialization remain valid with the same geometric
generic field K.
Choose a finite-type smooth affine chart bcolon Utomathcal BcalB
through the special value of the extended bundle. The fibre product of
this chart with its trait section is smooth over S and has a
k-point. Formal smoothness lifts that k-point to compatible points
modulo every power of the uniformizer, and completeness of R then
gives an honest R-point of an affine 'etale neighbourhood; thus the
trait section lifts to U. After shrinking around its closed point, U/S
has constant relative dimension N. The generic stalk of F_U=b_K^*F
on the lifted section is nonzero, so the support closure of F_U meets
U_s.
## Finding a slice with nonzero nearby cycles
The next lemma is purely about the dimension of a closed cone in the
special-fibre cotangent bundle. We state it separately to make clear
that only the support strata, not the sheaf itself, are descended
to a finite extension.
Let R be a complete discrete valuation ring with algebraically closed
residue field k of characteristic neell, S=Spec R, and K an
algebraic closure of its fraction field. Let U/S be a smooth affine
scheme of constant relative dimension N, mathcal FcalF a bounded
constructible E-complex on U_K, and
Λsubset T^*U_s a closed conical subset with dimΛle N.
Suppose the closure in U of the nonzero-stalk locus of mathcal FcalF
meets U_s. Then after a finite extension of the trait and shrinking
U, there exist uin U_s(k), 0le dle N and functions
f_1,...,f_d vanishing at u whose special-fibre differentials are
independent, such that
[
span{d(f_{1,s})_u,...,d(f_{d,s})_u}
capΛ_u⊆{0},
]
and for J=V(f_1,...,f_d),
(Psi_J(mathcal F|_calF|_{J_K}))_une0.
For d=0 the list is empty and J=U.
Proof.
Step 1: descend the finite support data.
Let A_1,...,A_t be the irreducible components of the closure of the
nonzero-stalk locus in U_K. Bounded constructibility gives a proper
closed subset B_i⊊ A_i such that every stalk on
A_i∖ B_i is nonzero. Give these closed sets reduced
structures. Their finitely generated ideals descend to a common finite
extension of Frac(R); enlarge that extension to descend
all containments too. Replace the trait accordingly. The new support
closure still meets the special fibre: the finite map from the new
model to the old is closed and agrees with the geometric generic
support, so it maps the new closure onto the old. This step does not
require mathcal FcalF 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.
Every nonempty special fibre (overlineoverline A_i)_sA_i)_s is pure of dimension
e_i cite[Tag~0B2J]{Stacks}. Applying this to the irreducible
components of B_i shows that whenever
(overlineoverline B_i)_sB_i)_sne∅,
[
dim(overlineoverline B_i)_s<e_i.B_i)_s<e_i.
]
The trait is excellent, hence universally catenary (a complete Noetherian
local ring is excellent cite[Tag~07QW]{Stacks}), so at any closed
vin(overlineoverline A_i)_s,A_i)_s,
dimmathcal O_calO_{overlineoverline A_i,v}A_i,v=e_i+1
qquadcite[Tag~02JT]{Stacks}.
Step 2: choose the special point.
Let d be the maximum of the e_i such that (overlineoverline A_i)_sA_i)_sne
∅; such exists by the support assumption. Choose a reduced
irreducible component Tsubset(overlineoverline A_i)_sA_i)_s of dimension d for
some such i. At a general closed point of T the reduced structure is
smooth, no other distinct reduced component of ∪_i(overlineoverline A_i)_sA_i)_s
passes through the point, and no overlineoverline B_iB_i passes through it: an
i with empty special fibre contributes nothing, while every remaining
bad special locus has dimension <d. Different horizontal components
may share the same reduced T; we keep them all.
We may also arrange
dimΛ_ule N-d.
Indeed, restrict Λ to T. Its components that do not dominate
T can be excluded over a proper closed subset of T. The generic
fibre of each remaining component has dimension le N-d because
dimΛle N; the same bound holds over a nonempty open subset of
T. Choose uin T(k) satisfying all requirements.
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
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=Jcap∪_ioverlineoverline A_i,A_i, the point
u is isolated in Z_s after shrinking; the reduced special fibre of
the union of all support closures near u is precisely T, even if
several horizontal components specialize to T.
Step 4: a generic generization in the slice.
Choose i with e_i=d and T a component of (overlineoverline A_i)_s.A_i)_s. By
(eq:support-local-dimension), dimmathcal O_calO_{overlineoverline A_i,u}A_i,u
=d+1. Quotienting by f_1,...,f_d leaves dimension ge1; then
quotienting by a uniformizer pi gives dimension zero (isolation). So
some prime of the cut local ring does not contain pi -- otherwise
pi would be nilpotent and the ring would be dimension zero. This gives
a generic-fibre point z specializing to u.
The morphism Zto S is quasi-finite at u (closed point isolated in
its fibre cite[Tag~01TH]{Stacks}); its quasi-finite locus is open, so
it contains z. Thus z has residue field finite over
Frac(R). Its geometric lifts to K avoid B_i: a
point in the closed set overlineoverline B_iB_i cannot specialize to
unotinoverlineoverline B_i.B_i. Each such lift therefore has a nonzero stalk of
mathcal F|_calF|_{J_K}.
Step 5: nearby cycles on a finite neighbourhood.
Take an affine neighbourhood of u inside the quasi-finite locus of
Zto S. The henselian decomposition of a finite-type algebra
cite[Tag~04GJ]{Stacks} gives a finite local factor containing u: an
open neighbourhood Q of u in Z, finite over S, whose special
fibre has only the point u (equivalently, Zariski's main theorem
embeds a quasi-finite neighbourhood into a finite S-scheme; the
henselian local factor at u lies in the original open because an open
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
finite neighbourhood, remove Z∖ Q from J: this complement is
closed in J hence in U, so remove it from U and pass to an affine
neighbourhood of u. That neighbourhood contains Q because every
point of the finite local Q specializes to u. The new J is still
cut out by the same functions in a smooth affine chart, and the change
leaves its nearby-cycle stalk at u unchanged while making Q closed
in J. If iotacolon Q↪ J is this closed immersion
and mathcal H=calH=mathcal F|_calF|_{Q_K}, localization gives
mathcal F|_calF|_{J_K}simeqiota_{K,*}mathcal H.calH. Proper compatibility for
iota reduces (eq:support-nonzero) to the stalk of Psi_Q
mathcal HcalH at u.
Let acolon Qto S be the finite structural map. Its special fibre
has only u, so proper compatibility gives
[
(Psi_Qmathcal H)_ucalH)_usimeqPsi_S(RGamma(Q_K,mathcal H))calH))
simeqRGamma(Q_K,mathcal H).calH).
]
The second identification uses geometric specialization on the trait,
which is the identity on coefficient complexes. The finite K-scheme
Q_K has the 'etale topos of a finite discrete set, so
RGamma(Q_K,mathcal H)calH)simeq⊕_{xin|Q_K|}mathcal H_x.calH_x. At
least one summand is nonzero (Step 4), so this complex is nonzero,
proving (eq:support-nonzero).
These are geometric nearby cycles with monodromy forgotten; no inertia
invariants. The finite direct-sum calculation is compatible with finite
coefficients, their ell-adic limit, inversion of ell, and
extension to E, so it applies to the specialization functor of
Section~sec:specialization without requiring a field of descent
for mathcal F.calF.
## Completion of the full-support proof
Proof. [Proof of Theorem~thm:full-support]
If Y'ne Y, Proposition~spec:missing supplies the nonzero
compact F used above. Lemma~support:extension and the
subsequent chart construction give a smooth affine Utomathcal BcalB of
constant relative dimension N for which the closure of the nonzero-stalk
locus of F_U meets U_s.
Let mathcal N_UcalN_Usubset T^*U_s be the pullback of the global nilpotent
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.
Apply Lemma~support:finite-slice with mathcal F=F_UcalF=F_U and
Λ=mathcal N_U.calN_U. It gives a slice J=V(f_1,...,f_d) and
uin J_s(k) such that
[
(Psi_J(F_U|_{J_K}))_une0
]
while the independent differentials of its equations span a plane
meeting (mathcal N_U)_ucalN_U)_u only at zero. The higher-codimension vanishing
of Proposition~slice:higher applies to the same chart and slice
and says that this stalk is zero. Contradiction. This also handles d=0,
where the test is the original smooth-chart vanishing.
Thus the complementary union is empty. The primed equivalence recalled
in Section~sec:specialization is consequently the full restricted
equivalence asserted in Theorem~thm:full-support.
% =========================================================================
# Component categories and consequences
% =========================================================================
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),
]
and write mathbb L_G^bbL_G^{restr}colonmathcal C_scalC_stomathcal DcalD
for the restricted Langlands equivalence. Its mathcal A-linearcalA-linear
structure is the one furnished by cite[Lemma~1.3.7]{GR}.
## The category attached to one component
For an open-and-closed component icolon Z↪ Y, define
[
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_ZcalO_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).
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.
Every component contains a semisimple E-point
cite[Prop.~3.7.2,~Cor.~3.7.4]{AGKRRV}. Pullback to this point shows
mathcal O_ZcalO_Zne0, and the fully faithful
QCoh(Z)↪IndCoh_{Nilp}(Z) then gives nonvanishing.
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
of eigenobjects below uses the actual parameter, including its
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
stack
[
Y^{arithm}:=Y^{Frob}=Yunderset{Ytimes_EY}{overset{mathrm R}rmRtimes}Y,
]
where the two maps to Ytimes_EY are the diagonal and the graph of
Frob. An E-point of Y^{arithm} is a geometric local
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)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
derived fixed-point stack.
Proof.
The primed union Y' is Frobenius-stable. The spectral trace
calculation of Beraldo--Lin--Reeves
cite[SS6.4.12--6.4.13]{BeraldoLinReeves} enters the known primed
formula cite[SS1.5.3--1.5.5,~Cor.~1.5.6]{GR}:
[
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
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 ΥcolonQCohtoIndCoh. We
neither truncate Y^{arithm} classically nor replace its
dualizing object by its structure sheaf.
## Constructing Hecke eigenobjects
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,
E-linear symmetric monoidal category whose compact objects are
dualizable, and suppose its tensor product preserves colimits
separately. Let acolonmathcal A_0calA_0toVect_E be a continuous
E-linear symmetric monoidal functor, and give Vect_E the
mathcal A_0-actioncalA_0-action through a. Then the right adjoint R_a of a
is continuous, carries a coherent mathcal A_0-linearcalA_0-linear structure, and
R_a(E)ne0.
Proof.
Presentability gives R_a. Every compact object of mathcal A_0calA_0 is
dualizable, so its image under a is a perfect complex; thus a
preserves compact objects. Testing against compact generators and using
adjunction shows R_a preserves filtered colimits. Both categories are
stable, so R_a is exact, and exact + colimit-preserving implies
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).
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.
Finally R_a(E)ne0 because
RHom_{mathcal A_0}calA_0(mathbf 1,R_a(E))bf1,R_a(E))
simeqRHom_E(E,E)simeq E; this last argument does not require a
compact unit.
For our spectral stack, mathcal A=calA=QCoh(Y) satisfies the
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
[
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=calQ_I=QLisse(X)^{ ⊗ I}. Given
VinRep(check G)^{ ⊗ I}, let H_I(V,-) denote the multi-leg
Hecke functor and let
mathcal E_V^IcalE_V^Iinmathcal AcalA ⊗ mathcal Q_IcalQ_I be the universal
evaluation object. For a parameter sigmacolonSpec Eto Y define
sigma^I(V):=(sigma^* ⊗ id)(mathcal E_V^I)calE_V^I)inmathcal Q_I.calQ_I.
A Hecke eigenobject of eigenvalue sigma with coherent structure is an
Minmathcal C_scalC_s equipped, for all finite sets I, with isomorphisms
H_I(V,M)simeq Mboxtimessigma^I(V), natural in Vin
Rep(check G)^{ ⊗ I} and compatible with units, tensor products,
permutations of legs, collisions of legs, and all homotopy-coherent
compatibilities cite[S15.2]{AGKRRV}.
Under either hypothesis regime, every parameter sigmacolonSpec Eto
Y admits a nonzero object M_sigmainShv_{Nilp}(Bun_G) with a
coherent Hecke eigenstructure of eigenvalue sigma. If sigma lies
in a component Z, then M_sigmainmathcal C_calC_{s,Z}. The assertion
is in the ind-constructible dg category; no boundedness, perversity
or Weil structure is included.
Proof.
Apply Lemma~lem:parameter-adjoint to a=sigma^*colonmathcal AcalA
toVect_E, and write sigma_* for its continuous right adjoint.
The composite
[
Phi_sigmacolonVect_sigmaxrightarrow{sigma_*}mathcal AcalA
xrightarrow{jmath_Y}mathcal DcalD
xrightarrow{(mathbb L_G^bbL_G^{restr})^{-1}}mathcal C_scalC_s
]
is continuous mathcal A-linear,calA-linear, where Vect_sigma denotes
Vect_E with the action through sigma^*. Set
M_sigma=Phi_sigma(E). It is nonzero by the lemma, full faithfulness
of jmath_Y, and the Langlands equivalence. If sigma lies in Z,
the idempotent e_Z pulls back to E, while its complementary
idempotent pulls back to zero; linearity puts M_sigma in
mathcal C_calC_{s,Z}.
For the eigenstructure, fix a finite set I. The action of
mathcal E_V^IcalE_V^I on mathcal C_scalC_s is the multi-leg Hecke functor
H_I(V,-) cite[SS14.2--14.3]{AGKRRV}. Extend Phi_sigma by the
identity on mathcal Q_I.calQ_I. Its module structure gives
[
H_I(V,M_sigma)simeq(Phi_sigma ⊗ id)(sigma^I(V))
simeq M_sigmaboxtimessigma^I(V),
]
the second equivalence using that a continuous E-linear functor from
Vect_E is just tensoring with its value on E. Naturality, tensor
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,
semisimple or not.
% =========================================================================
# Rational coefficients and spectral support
% =========================================================================
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,
and assume the curve and group satisfy the rational hypotheses in the
Introduction (Hypothesis~hyp:A plus a split dual group over
E_0). Put mathcal C_E=calC_E=Shv_{Nilp,E}(Bun_G), retain
mathcal C_0=calC_0=Shv_{Nilp,E_0}(Bun_G) and Y_0 from the Introduction,
and let Y denote the spectral prestack over E. All categorical tensor
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
multi-leg Hecke system with its tensor, permutation and collision
compatibilities. We call this the rational spectral-action datum.
[Rational spectral support]
For the rational datum above, every nonempty open-and-closed substack
Usubset Y_0 satisfies
[
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.
The plan is: for Xnemathbb P^1bbP^1 we compare the sheaf categories,
prestacks and actions so that Theorem~thm:full-support in regime A
rules out the vanishing of any nonempty idempotent; faithful flatness of
E/E_0 then descends the absence of a missing summand. For
X=mathbb P^1bbP^1 connectedness of the spectral prestack gives the result
directly.
## Coefficient categories, including unbounded objects
On a finite-type T/k, Shv_L(T) is the ind-completion of bounded
constructible L-adic complexes, Lin{E_0,E}. For a smooth T and a
closed conical Nsubset T^*T, let mathcal P_calP_{N,L}(T) be the category
of constructible perverse sheaves with singular support in N; the
category Shv_{N,L}(T) consists of objects whose perverse cohomology
belongs in Indmathcal P_calP_{N,L}(T) in every degree. This
is the cohomological support definition
cite[SS E.5.1--E.5.8]{AGKRRV}; it is not necessarily the ind-completion
of support-bounded complexes. On a smooth scheme the zero-cone case is
QLisse_L, with its equivalent ordinary-cohomology definition
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}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
descent. The second is symmetric monoidal and respects the t-structures
used in the tensor-functor definition of Y_0,Y. On smooth charts the
third respects cohomological nilpotent support. Free and forgetful
functors here are t-exact for the ordinary and perverse t-structures
whenever these are used.
Proof.
Ambient statement. For a finite extension L/E_0, coefficient
extension and restriction preserve bounded constructibles, and on
objects extended from E_0 the mapping complexes are obtained by
tensoring the originals with L. Every L-constructible M is a
retract of L ⊗ _{E_0}Res_{L/E_0}M, the retraction
being given by the separability idempotent of L/E_0. So the induced
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
chart over the separably closed k (either algebraic
mathbb Q_bbQ_ell-coefficient field), so its compactness statements apply.
Passing to ind-completions proves (eq:coeff-ambient).
Support condition. View (eq:coeff-ambient) as internal
E-modules in Shv_{E_0}(T). An internal E-module has the E-support
condition iff its underlying E_0-object has the E_0-condition. Write
F for the free and U for the forgetful functor; both are t-exact for
ordinary and perverse cohomology. For U, using that a constructible
E-object over a finite extension L/E_0 has underlying
[
U(M_L ⊗ _L E)=
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.
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 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
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
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
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.
Finite restriction preserves the same bound. The same arguments apply
to lissity.
It follows that F and U preserve the relevant ind-perverse hearts;
for U use the colimit formula and filtered colimits, never treating a
finite E-rank as a finite E_0-rank. For the converse, let P be a
perverse internal E-module with U(P) in the E_0-ind-heart. The
counit F(U(P))to P is an epimorphism in the module heart because its
underlying map splits as umapsto1 ⊗ u. Its source is in the
E_0-ind-heart, which is closed under quotients by the Serre property
of the support heart; hence P is as well. So the cohomological support
condition for an E-module coincides with the E_0-condition on its
underlying E_0-object, degree by degree.
Scalar extension and smooth descent. We now realise these
subcategories as scalar extensions. The perverse Serre condition gives
closure under finite cones; since perverse cohomology commutes with
filtered colimits, these subcategories are closed under filtered
colimits and arbitrary direct sums, hence under all colimits including
bar realisations, and closed under scalar tensors. Internal modules in
each such subcategory are exactly ambient modules whose underlying
object belongs to it, and the free-forgetful bar resolution
constructs every such module from free modules inside the subcategory.
This proves (eq:coeff-qlisse) and the corresponding support
comparison on smooth charts. Ordinary tensor products and structural
maps commute with coefficient extension, giving the monoidal assertion.
Connective free modules generate the connective module category under
connective colimits, so the induced tensor t-structures agree.
Finally, take the smooth-chart limit defining sheaves on a stack.
Internal modules commute with these limits. Equivalently, Vect_E is
the module category of the continuous monad E ⊗ _{E_0}(-) on
Vect_{E_0} and is dualizable as a presentable E_0-linear category
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).
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).
## Affine families and coherent Hecke action
The sheaf comparison controls the affine families defining the spectral
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),
with the second symmetric monoidal. These identify the universal
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 ⊗ _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
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
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,~SS22.3--22.5,~Thm.~22.42]{Milne}. The universal
property of scalar extension therefore identifies tensor functors into
the displayed target, and right t-exactness is equivalent before and
after extension.
Present Y_0 as a colimit of affine test schemes. Base change presents
the left side of (eq:coeff-spectral) by base-changed affines, and
quasi-coherent categories form the corresponding limit. On an affine,
coefficient comparison is
Vect_E ⊗ _{E_0}Mod_Bsimeq
Mod_{B ⊗ _{E_0}E}. Affine pullbacks in this diagram
are continuous, and Vect_E is categorically dualizable, so tensoring
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
by these identifications; the assertion is natural in representations,
affine tests and maps of finite sets. It therefore identifies the whole
universal evaluation system, including tensor products and collisions
of legs.
For Xnemathbb P^1bbP^1 we can go further and compare the given rational
spectral action with the geometric one. This is the only case needed
below; for mathbb P^1bbP^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
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.
Proof.
First, QLisse_E(X)=IndLisse_E(X) for this curve
cite[S1.3.3,~S E.2.8]{AGKRRV}. The same equality holds over E_0:
extend an object of QLisse_{E_0}(X) and use the constructible lisse
generators over E; forgetting coefficients takes those generators
into the colimit closure of E_0-constructible lisse objects, and the
original object is a retract of its extension-then-forgetting, using an
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
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,
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
highest weights. On each bounded Schubert support, coefficient extension
commutes with exterior products, pullbacks, descent and proper
convolution pushforwards. It is perverse t-exact, so it also commutes
with intermediate extension (written as the image of the map from
perverse extension by zero to perverse direct image). So it takes the
Satake IC objects, their convolution maps, and the fusion diagrams with
unit and symmetry constraints, to the corresponding diagrams over E.
The fiber and weight functors in geometric Satake identify the same
split dual group under this operation
cite[SS4--6,~Thm.~14.1]{MirkovicVilonen}.
Derived functors here are derived extensions of naive geometric Satake.
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[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
through the fully faithful exterior-product functors
cite[S14.2.5]{AGKRRV}; Lemma~lem:coeff-sheaves compares these
subcategories, so the comparison holds after factorisation.
Lemma~lem:coeff-spectral identifies universal evaluation. By
assumption, restricting the extended spectral action along evaluation
gives exactly the established coherent Hecke system over E. The
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}.
No Frobenius structure or normalised rational Langlands functor was
chosen. What this comparison establishes is the identification of the
given rational spectral action with the geometric action used in
Theorem~thm:full-support.
## Descent of support and the arithmetic qualification
For Xnemathbb P^1,bbP^1, the sheaf, prestack and coherent-action
comparisons are now in place.
[Detection of a clopen support]
Suppose the coefficient comparisons (eq:coeff-nilp)
and~(eq:coeff-spectral) identify the two spectral actions. If an
open-and-closed substack Usubset Y_0 acts by zero on mathcal C_0,calC_0,
then U=∅.
Proof.
Let e_U=mathcal O_UcalO_U extended by zero in QCoh(Y_0). Its extension
is the analogous idempotent e_{U_E}, where U_E=Utimes_{E_0}E.
Compatibility of the actions and generation by scalar-extended objects
show e_{U_E} acts by zero on mathcal C_E.calC_E.
The curve and split group descend to a finite field, so
Theorem~thm:full-support in regime A identifies this action with
the action on IndCoh_{Nilp}(Y). Apply e_{U_E} to
jmath_Y(mathcal O_Y);calO_Y); its value is jmath_Y(e_{U_E}). Full
faithfulness of jmath_Y implies e_{U_E}=0, hence U_E=∅.
To descend, take any affine test T=Spec Ato Y_0. The inverse image
of U is an open-and-closed affine summand of T, cut out by an
idempotent of pi_0(A). Its base change to A ⊗ _{E_0}E is
empty, and faithful flatness of E/E_0 forces the idempotent to
vanish. This holds on every affine test, so U=∅. No
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
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
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
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
dualizing object has shifted constant restrictions on smooth charts and
zero singular support), so its only nonempty open-and-closed
localization is nonzero.
Finally, the stipulated complement of Y'_0 acts by zero, so it is
empty. In particular a given spectral-linear primed equivalence has
this property by linearity, and its specified inclusion is an equality
of prestacks.
[Conditional rational unpriming]
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)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}.
Proof.
Theorem~thm:rational-support identifies the actual inclusion
Y'_0subset Y_0 with equality, so the derived fixed-point stacks,
dualizing objects and ind-coherent global-sections functors agree.
Substitute.
The extra comparison map in Corollary~cor:rational-arithmetic
is not constructed here. The primed arithmetic theorem cited in
Corollary~cor:arithmetic has coefficients E; its existence does
not by itself produce a canonical map over E_0. An unconditional
rational arithmetic formula still requires that map with its Frobenius,
dualizing-object, ind-coherent global-sections and excursion
compatibilities. Equality of derived fixed-point prestacks alone
supplies none of these additional data.
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