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Lifting the Cartier transform
- f Ogus and Vologodsky
Contents X a smooth scheme over a perfect field k of
- charc. p > 0.
Lifting the Cartier transform of Ogus and Vologodsky modulo p n - - PDF document
Lifting the Cartier transform of Ogus and Vologodsky modulo p n - - PDF document
Lifting the Cartier transform of Ogus and Vologodsky modulo p n [following H. Oyama, A. Shiho and D. Xu] Ahmed Abbes (CNRS & IHS) Schloss Elmau, 7-13 May 2017 Contents X a smooth scheme over a perfect field k of charc. p > 0 . 1-
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f∶X → S a morphisme of schemes, M an OX- module, λ ∈ Γ(S,OS). A λ-connection on M relatively to S is an OS-linear morphism ∇∶M → Ω1
X/S ⊗OX M
such that ∀t ∈ OX and ∀u ∈ M, ∇(tu) = λd(t) ⊗ u + t∇(u). We say that ∇ is integrable if ∇ ○ ∇ = 0. 1-connections are called connections. Integrable 0-connections are called Higgs fields. λ-MIC(X/S) the category of OX-modules with integrable λ-connection,
MIC(X/S) the category of OX-modules with
integrable connection.
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k a perfect field of characteristic p, W = W(k),
S = Spf(W), X a smooth formal S -scheme,
X′ = X ×S ,σ S , Xn = (X,OX/pn) (∀n ≥ 1), X = X1 X
FX/k
- FX
- X′
- ◻
X
- Spec(k) Fk
Spec(k)
Given an Sn+1-morphism Fn+1∶Xn+1 → X′
n+1
lifting FX/k, dFn+1 induces an OXn-linear mor- phism dFn+1 p ∶F∗
n(Ω1 X′
n/Sn) → Ω1
Xn/Sn
that fits into a commutative diagram F∗
n(Ω1 X′
n/Sn) dFn+1 p
Ω1
Xn/Sn p
- F∗
n+1(Ω1 X′
n+1/Sn+1) dFn+1
- Ω1
Xn+1/Sn+1
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Shiho defined the functor Φn∶ p-MIC(X′
n/Sn)
→ MIC(Xn/Sn) (M′,∇′) ↦ (F∗
n(M′),∇),
where ∇ is defined ∀t ∈ OXn and ∀x ∈ M′, by ∇(tF∗
n(x)) = t⋅(dFn+1
p ⊗id)(F∗
n(∇′(x)))+dt⊗F∗ n(x),
[ F∗
n(Ω1 X′
n/Sn ⊗OX′ n M′) dFn+1 p
⊗id
Ω1
Xn/Sn ⊗OXn F∗ n(M′)]
For n = 1, Φ1 was considered first by Ogus- Vologodsky who proved that it is compatible with their Cartier transform C−1
X′
2
. Proposition 1 (Shiho) Φn induces an equiv.
- f cat.
Φn∶p-MICqn(X′
n/Sn) ∼
→ MICqn(Xn/Sn).
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The quasi-nilpotence is a local condition. If ∃ an étale map Xn → Spec(Wn[T1,...,Td]). (M,∇) an OX-module with (p-)connection /Sn. ∃ OS-linear endomorphisms ∇∂1,...,∇∂d of M such that ∀u ∈ M, ∇(u) = ∑
1≤i≤d
dti ⊗ ∇∂i(u). ∇ is integrable ⇔ ∇∂i ○ ∇∂j = ∇∂j ○ ∇∂i ∀i,j. So we can define the endomorphism ∇∂n = ∏1≤i≤d(∇∂i)ni of M ∀n = (n1,...,nd) ∈ Nd. We say that (M,∇) is quasi-nilpotent (rela- tively to f) if for any local section u of M, there exists N ≥ 1 such that for any n ∈ Nd with ∣n∣ ≥ N, ∇∂n(u) = 0. It is well known that quasi-nilpotent integrable connections can be described in term of strat- ifications. There is a similar description for p-connections that requires a slightly gener- alized notion of stratifications.
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A Hopf algebra of a (commutative) ringed topos (T ,A) is the data of a (commutative) ring B of T and five homomorphisms A
d1 d0
B ,
B δ
B ⊗A B ,
B π A , B σ B , where the tensor product is taken on the left (resp. right) for the A-algebra structure de- fined by d1 (resp. d0), satisfying the usual compatibility conditions. A B-stratification on an A-module M is a B- linear isomorphism ε∶B ⊗A M ∼ → M ⊗A B, satisfying π∗(ε) = idM and the cocycle condi- tion B ⊗A B ⊗A M
idB⊗ε
- ε⊗B,δB⊗AB
M ⊗A B ⊗A B
B ⊗A M ⊗A B
ε⊗idB
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X a smooth formal S -scheme. PXn(r) the PD envelop of the diagonal immer- sion Xn → Xr+1
n
compatible with the canonical PD structure on Wn (n,r ≥ 1). PX(r) = (PXn(r))n≥1 is an adic formal S -scheme. PX ∶= PX(1) has a natural structure of a formal X-groupoid over S (⇒ PX = OPX is a formal Hopf OX-algebra of Xzar). X
- X
- PX(2)
X3
q1,3
X2
⇒ PX ×X PX
∼
→ PX(2) → PX. There exists a canonical equiv. of cat. {
OXn-mod.
+ PX-strat. } ∼ → MICqn(Xn/Sn).
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U an open of X2 such that the diagonal X =
X1 → X2 factors through a closed immersion X → U , I ⊂ OU the associated coherent open ideal, Z the admissible blow-up of I in U , RX the maximal open subscheme of Z where (I OZ)∣RX = (pOZ)∣RX.
- RX,1 → X2 factors through X → X2.
- Universal property.
Let Y be a flat adic formal S -scheme, f∶Y → X2 an S -morphism that fits into a commutative diagram Y1
- g
- Y
f f′
- X
X2
RX
- Then, there exists a unique map f′ lifting f.
RX has a natural structure of a formal X- groupoid over S (⇒ RX = ORX is a formal Hopf OX-algebra of Xzar).
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TXn the PD envelop of the closed immer- sion Xn → RX,n lifting the diagonal, compati- ble with the canonical PD-structure on Wn. TX ∶= (TXn)n≥1 is an adic formal S -scheme. It has a natural structure of a formal X-groupoid
- ver S which lifts that of RX (⇒ TX = OTX is
a formal Hopf OX-algebra of Xzar). [Shiho] There exists a can. equiv. of cat. { OXn-mod. + TX-strat. } ∼ → p-MICqn(Xn/Sn). Example : If ∃ X → Spf(W{T1,...,Td}) an étale morphism, ti ∈ OX the image of Ti and ξi = 1 ⊗ ti − ti ⊗ 1. The ideal I is generated by p,ξ1,...,ξd on an open neighborhood of the diagonal X in X2. We have isomorphisms of
OX-algebras OX{ζ1,...,ζd}
∼
→ q1∗(RX),
OX{ζ1,...,ζd}
∼
→ q2∗(RX), that map ζi to ξi
p .
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X a smooth formal S -scheme. Given an S - morphism F∶X → X′ = X ×S ,σ S lifting FX/k. There exists a unique morphism g∶PX1 → X′ which fits into the commutative diagram PX1
- g
- PX
- φ
- X2
F2
- X′ ∆ X′2
RX′
- φ is morphism of groupoids above F.
X
F ιP
PX
φ
- ϕ
- X′
ιR
RX′
TX′
- ϕ∶PX → TX′ a morphism of formal groupoids
above F. Shiho proved that the functor ϕ∗
n∶ { OX′
n-mod.
+TX′-strat. }
- →
{ OXn-mod. +PX-strat. } (M,ε)
- →
(F∗
n(M),ϕ∗(ε))
coincides with his functor Φn
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For any k-scheme Y , we denote by Y the schematic image of FY , i.e. the closed sub- scheme of Y defined by the ideal of OY con- sisting of sections with vanishing p-th power. Y
fY /k
- FY /k
- Y ′
Y ′
We construct by dilatation a canonical adic formal X2-scheme QX satisfying the following properties:
- QX,1 → QX,1 → X2 factors through X → X2.
- Let Y be flat adic formal S -scheme, f∶Y →
X2 an S -morphism that fits into a commu- tative diagram Y1
- g
- Y
f f′
- X
X2
QX
- Then, there exists a unique map f′ lifting f.
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QX has a natural structure of a formal X- groupoid over S (⇒ QX = OQX is a formal Hopf OX-algebra of Xzar). Example : X → Spf(W{T1,...,Td}) an étale
- morphism. We have canonical isomorphisms
- f OX-algebras
OX{ζ1,...,ζd}
∼
→ q1∗(QX),
OX{ζ1,...,ζd}
∼
→ q2∗(QX), that map ζi to ξp
i
p .
QX fits into a commutative diagram of formal X-groupoids over S PX
ϕ λ φ
- TX′
̟
- QX ψ
RX′
where λ,̟ are canonical but ψ,ϕ,φ depend
- n the lifting F∶X → X′ of FX/k.
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{ OX′
n-mod.
+RX′-strat. }
ψ∗
n
- ̟∗
n
{ OXn-mod. +QX-strat. }
λ∗
n
- { OX′
n-mod.
+TX′-strat. }
ϕ∗
n
{ OXn-mod.
+PX-strat. } p-MICqn(X′
n/Sn) Φn
MICqn(Xn/Sn)
Theorem 2 (Oyama, Xu) There exists an
- equiv. of cat. that depends only on X
C∗
X/W∶{ OX′
n-mod.
+RX′-strat. } → { OXn-mod. +QX-strat. } Moreover, given a lifting F∶X → X′ of FX/k, there is a canonical isomorphism ηF∶ψ∗
n(M,ε) ∼
→ C∗
X/W(M,ε).
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Oyama topoi. X a k-scheme.
- E : the cat. of triples (U,T ,u) where U ⊂ X
- pen, T is a flat adic formal S -scheme and
u∶T ∶= T1 → U is an affine k-morphism.
- E : the cat. of triples (U,T ,u) where U ⊂ X
- pen, T is a flat adic formal S -scheme and
u∶T → U is an affine k-morphism. (U,T ,u) ∈ E U
FU/k
T
FT/k
- u
- T
FT/k
- fT/k
- U′
T ′
u′
- T ′
⇒ (U′,T ,u′○fT/k) ∈ E ′ ∶= E (X′/S ), X′ = X⊗k,σk. ⇒ functor ρ∶
E
→
E ′
(U,T ,u) ↦ (U′,T ,u′ ○ fT/k)
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- A morphism f∶(U1,T1) → (U2,T2) of E is
Cartesian if T1 → U1 ×U2 T2 is an isom.
- A morphism f of E is Cartesian if ρ(f) is
Cartesian. The functor π∶E (resp. E ) → Zar/X, (U,T ,u) ↦ U, is a fibered functor and the Cartesian mor- phisms of E (resp. E ) are precisely the Carte- sian morphisms for π. For (U,T ) ∈ E (resp.
E ), let Cov(U,T ) be
the set of families of Cartesian morphisms {(Ui,Ti) → (U,T )}i∈I of E (resp. E ) such that U = ∪i∈IUi. The Zariski topology on E (resp. E ) is the topology generated by the pretopology de- fined by the Cov(U,T )’s.
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̃
E (resp.
̃
E ) is the topos of sheaves of sets
- n E (resp. E ) for the Zariski topology.
̃
E ′ ∶= ̃ E (X′/S ).
Proposition 3 The functor ρ∶E → E ′ is fully faithful, continuous and cocontinuous. Hence, ρ induces a morphism of topoi CX/W∶ ̃
E → ̃ E ′
such that the pull-back functor is induced by the composition with ρ. The functor on E (resp. E ) (U,T ) ↦ Γ(T ,OTn) defines a sheaf of rings OE ,n (resp. OE ,n). For any sheaf F on E (resp.
E ) and any
- bject (U,T ,u) of E (resp. E ), the functor
V ∈ Zar/U ↦ F(V,TV )
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defines a sheaf F(U,T ,u) of Uzar. (OE ,n)(U,T ,u) = u∗(OTn) (OE ,n)(U,T ,u) = u∗(OTn) An OE ,n-module F of ̃
E (resp. OE ,n-module F of ̃ E ) ⇔ the following data:
(i) ∀(U,T ) ∈ E (resp. E ), an u∗(OTn)-module
F(U,T ) of Uzar;
(ii) ∀ map f∶(U1,T1) → (U2,T2) of E (resp.
E ), an u∗(OT1,n)-linear morphism
̃ cf∶u∗(OT1,n) ⊗u∗(OT2,n)∣U1 F(U2,T2)∣U1 → F(U1,T1) subject to the following conditions: (a) if f is Cartesian, ̃ cf is an isomorphism; (b) cocycle condition.
- F is quasi-coherent if ∀(U,T ) ∈ E (resp. E ),
F(U,T ) is a quasi-coherent u∗(OTn)-module.
- F is a crystal if for any map f of E (resp.
E ), ̃
cf is an isom.
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X a smooth formal S -scheme, X = X1. (X,RX) ∈ E , (X,QX) ∈ E . The are can. equiv. between the following: (i) the cat. of quasi-coherent crystals of OE ,n- modules of ̃
E (resp. OE ,n-modules of ̃ E );
(ii) the cat.
- f quasi-coherent OXn-modules
with RX-strat. (resp. QX-strat.); (iii) the cat.
- f data {F(U,T ),cf} consisting
∀(U,T ) ∈ E (resp. E ) of a quasi-coherent OTn- module F(U,T ) of Tzar, and ∀ map f∶(U1,T1) → (U2,T2) of an OT1,n-linear isomorphism cf∶f∗
n(F(U2,T2)) ∼
→ F(U1,T1), satisfying a cocycle condition. (i)⇒(ii) F ↦ F(X,X) and the RX-strat. comes from the can. proj. RX ⇉ X
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Theorem 4 (Oyama, Xu) For any smooth k-scheme X, the pull-back and push-forward functors of the morphism CX/W∶ ̃
E → ̃ E ′ induce
- equiv. of cat. quasi-inverse to each other
C qcoh(OE ′,n)
C qcoh(OE ,n)
- .
We call the Cartier transform the functor C∗
X/W∶C qcoh(OE ′,n) ∼
→ C qcoh(OE ,n). Comparison with Shiho’s construction X a smooth formal S -scheme, X = X1. Assume given a lifting F∶X → X′ of FX/k. It induces a map ψ∶QX → RX′ of X-groupoids
- ver F2.
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We can consider F and ψ as maps of E ′ that fit into a commutative diagram ρ(X,QX) ψ
qi
(X′,RX′)
qi
- ρ(X,X)
F
(X′,X′)
[ρ(X,X) = (X′,X,FX/k)]. Proposition 5 (Xu) The diagram
C qcoh(OE ′,n)
C∗
X/W
- C qcoh(OE ,n)
- { OX′
n-mod.
+RX′-strat. }
ψ∗
n
{ OXn-mod.
+QX-strat. } is commutative up to a canonical functorial isomorphism of OXn-modules with QX-strat., ∀F ∈ C qcoh(OE ′,n), ηF∶F∗(F(X′,X′)) ∼ → (C∗
X/WF)(X,X).
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Comparison with Ogus-Vologodsky X a smooth formal S -scheme, X = X1. Proposition 6 (Oyama) The Hopf algebra
RX,1 is canonically isomorphic to S(Ω1
X/k) and
the OX-algebra H omOX(RX,1,OX) is canoni- cally isomorphic to ̂ Γ(TX/k). Proposition 7 (Oyama) There is a canoni- cal isomorphism of F∗
X/k(̂
Γ(TX′/k))-algebras
Dγ
X/k ∶= ̂
D(0)
X/k ⊗̂ S(TX′/k) ̂
Γ(TX′/k)
∼
H omOX(QX,1,OX).
⇒ the cat.
- f OX-modules with QX-strat.
(resp.
RX′-strat.)
is equiv. to the cat.
- f
quasi-nilpotent Dγ
X/k-modules (resp. ̂
Γ(TX′/k)- modules).
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Oyama compared his transform C∗
X/W with
O-V’s transform C−1
X′
2
. Xu gave another proof using an admissibility relation à la Fontaine in ̃
E .
Cartier transform and cohomology Theorem 8 (O-V) X a smooth k-scheme, ̃ X′ a smooth lifting of X′ over W2, 0 ≤ ℓ ≤ p−1. (i) The Cartier transform C−1
̃ X′ induces an equiv.
- f cat.
C−1
̃ X′∶HIGℓ(X′/k) ∼
→ MICℓ(X/k). (ii) ∀(M′,θ′) ∈ HIGℓ(X′/k) and (M,∇) = C−1
̃ X′(M′,θ′),
there exists a can. isom. in D(OX′) τ<p−ℓ(M′ ⊗ Ω●
X′/k) ∼
→ τ<p−ℓ(FX/k∗(M ⊗ Ω●
X/k)).
[∃ an increasing θ′-stable filtration M′
- of M′
- f length ≤ ℓ + 1 such that GrM′
- (θ′) = 0]
Oyama gave another proof of this result using his topoi.
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Fontaine modules X a smooth formal S -scheme, X = X1, n ≥ 1.
MICF(Xn/Sn) the cat. of triples (M,∇,M●),
where (M,∇) in an OXn-module with a quasi- nilpotent integrable connection and (Mi)i∈Z is a decreasing filtration of M such that Mi = M ∀i ≤ 0 and satisfying Griffiths’ transversality ∇(Mi) ⊂ Mi−1 ⊗OX Ω1
Xn/Sn,
∀i ≥ 0. Let (M,∇,M●) be an object of MICF(Xn/Sn)
- f level ≤ ℓ (i.e., Mℓ+1 = 0).
̃ M = coker(⊕ℓ
i=1Mi g
- → ⊕ℓ
i=0Mi),
where g is defined for mi ∈ Mi, by g(mi) = (mi,−pmi) ∈ Mi ⊕ Mi−1.
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Consider the Wn-linear map h∶⊕ℓ
i=0Mi → (⊕ℓ i=0Mi) ⊗OX Ω1 Xn/Sn
defined by h∣Mi = ∇∶Mi → Mi−1 ⊗OX Ω1
Xn/Sn,
∀1 ≤ i ≤ ℓ, h∣M0 = p∇∶M0 → M0 ⊗OX Ω1
Xn/Sn.
Then h induces a quasi-nilpotent integrable p-connection ̃ ∇ on ̃ M, and hence a TX-strat. [If pM = 0, then ̃ M = ⊕iMi/Mi+1 and ̃ ∇ is the Higgs field induced by ∇.] If ℓ ≤ p−1, ̃ ∇ descends to an RX-strat. ̃ ε on ̃ M. (̃ M, ̃ ε) defines a crystal ̃
M of OE ,n-modules.
Let ̃
M ′ be the crystal of OE ′,n-modules as-
sociated to the OX′
n-module with RX′-strat.
(π∗(̃ M),π∗
R(̃
ε)), where π∶X′ → X and πR∶RX′ = RX×S ,σS → RX are the canonical projections.
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If the Mi’s are quasi-coh. OXn-modules, then C∗
X/W( ̃
M ′) is a quasi-coh.
crystal of OE ,n-
- modules. It defines an OXn-module with QX-
- strat. and hence an OXn-module with quasi-
nilpotent integrable connection ν(C∗
X/W( ̃
M ′)).
A (pn-torsion) Fontaine module over X is an
- bject (M,∇,M●) of MICp−1
F
(Xn/Sn) such that the Mi’s are quasi-coh., equipped with a mor- phism of MICqn(Xn/Sn) ϕ∶ν(C∗
X/W( ̃
M ′)) → (M,∇).
We say that it is strongly divisible is ϕ is an isomorphism. Given an S -morphism F∶X → X′ lifting FX/S. Put FX = π ○ F∶X → X which is a lifting of FX. If (M,∇,M●,ϕ) is a Fontaine module, F in- duces a functorial isom. of MICqn(Xn/Sn) ηF∶Φn(π∗(̃ M, ̃ ∇)) ∼ → ν(C∗
X/W( ̃
M ′)),
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where Φn(π∗(̃ M, ̃ ∇)) = (F∗
X(̃
M),∇F) is Shiho’s
- functor. We deduce a horizontal map
ϕF = ϕ ○ ηF∶(F∗
X(̃
M),∇F) → (M,∇) and ∀0 ≤ i ≤ p − 1, a σ-linear map ϕi
F∶Mi
→ ̃ M
ϕF
- → M
We have ϕi
F∣Mi+1 = pϕi+1 F
and Mi ∇
ϕi
F
- Mi−1 ⊗OXn Ω1
Xn/Sn ϕi−1
F ⊗dFX p
- M
∇
M ⊗OXn Ω1
Xn/Sn
We get a (filtered) Fontaine module in the sense of Fontaine-Faltings. The two definitions are equivalent (given F), and the notions of strong divisibility corre- spond. Cartier transform provides one way to glue Faltings’ local definition of Fontaine modules.
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Faltings’ original global definition uses a Tay- lor formula to glue the local definitions rela- tively to different Frob. liftings. Such a for- mula is in fact encoded in Oyama topoi. Inspired by results of Fontaine-Messing, Falt- ings proved the following theorem that Xu adapted to his context. Theorem 9 (Faltings) Let (M,∇,M●,ϕ) be a strongly divisible Fontaine module of level ≤ ℓ (i.e. Mℓ+1 = 0). The de Rham complex
C = M ⊗OXn Ω●
OXn/Sn is filtered by
C i = Mi−● ⊗OXn Ω●
OXn/Sn.
Put d = dim(X). (i) For all m ≥ 0 such that min(m,d) + ℓ ≤ p − 1 and i ≤ p − 1, the map Hm(C i) → Hm(C ) is injective.
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(ii) For all m ≥ 0 such that min(m,d − 1) + ℓ ≤ p − 2, ϕ induces a natural strongly divisible Fontaine module over W (Hm(C ),Hm(C i)0≤i≤p−1,(ϕi
H)0≤i≤p−1).
(iii) The differentials dr,s
1
- f the spectral se-