Simpson correspondence in characteristic p > 0 and splittings of - - PowerPoint PPT Presentation

Simpson correspondence in characteristic p > 0 and splittings of the algebra of PD-differential

• perators

Michel Gros (CNRS &University of Rennes I) Simons Symposium, Schloss Elmau, 7-13 May 2017

̃ S a flat scheme over Z/p2Z, S = ̃ S ⊗Zp Fp, X a smooth S-scheme, X FX/S

• FX
• X′
• ◻

X

• S FS

S

Theorem 1 (Deligne-Illusie) (i) Given smooth ̃ S-schemes ̃ X and ̃ X′ and an ̃ S-morphism ̃ F∶ ̃ X → ̃ X′ lifting FX/S, there exists a canonical quasi- isomorphism ⊕i≥0Ωi

X′/S[−i] → FX/S∗(Ω● X/S),

inducing the Cartier operator C−1.

(ii) If we are given only a smooth lifting ̃ X′ of X′ over ̃ S, there exists a canonical isomorphism in Db(OX′) ⊕p−1

i=0 Ωi X′/S[−i] ∼

→ τ<pFX/S∗(Ω●

X/S).

X → S a smooth separated morphism,

I ⊂ OX×SX the ideal of the diagonal, Pn

X/S = OX×SX/I n+1

(n ≥ 0);

DX/S,n = H omOX(Pn

X/S,OX)

PX/S,(0) the PD-envelop of I , I ⊂ PX/S,(0) the PD-ideal generated by I , Pn

X/S,(0) = PX/S,(0)/I [n+1]

(n ≥ 0).

D(0)

X/S,n = H omOX(Pn X/S,(0),OX)

[OX-algebras via the first projection.] The OX-modules

DX/S

= ⋃

n≥0

DX/S,n D(0)

X/S

= ⋃

n≥0

D(0)

X/S,n,

are naturally equipped with ring structures.

D(0)

X/S is the sheaf of PD-differential operators.

Assume in the following that pOS = 0 and X → S smooth. Proposition 2 The image of the composed homomor- phism

D(0)

X/S → DX/S ↪ E ndOS(OX)

is the ring of OX′-linear endomorphisms of OX, and its kernel is the bilateral ideal K of D(0)

X/S locally generated

by the operators ∂p

i (for 1 ≤ i ≤ d ∶= dim(X/S)); the latter

are elements of the center ZD(0)

X/S of D(0) X/S.

D(0)

X/S

= D(0)

X/S/K ,

̂

D(0)

X/S

= lim

←

• n≥0

D(0)

X/S/K n.

⇒ ∃ a canonical isomorphism of OX′-algebras FX/S∗(D(0)

X/S) ∼

→ E ndOX′(FX/S∗OX). General result of linear algebra (example of Morita equiv- alence)

A a ring, M a non-trivial locally free A -module of finite

type, B = E ndA (M ) The functors: ψ ∶ Mod(B) → Mod(A ),

E

↦ H omB(M ,E ), φ ∶ Mod(A ) → Mod(B),

F

↦ M ⊗A F, are equivalences of categories quasi-inverse to each other (φ ○ ψ

∼

→

ev Id, Id ∼

• →

can ψ ○ φ).

By Morita equivalence, the functors:

Mod(D(0)

X/S)

→ Mod(OX′),

E

↦ FX/S∗(H om

D(0)

X/S

(OX,E )),

Mod(OX′)

→ Mod(D(0)

X/S),

F

↦ F∗

X/S(F) = F ⊗OX′ OX,

are equivalences of categories quasi-inverse to each other (Cartier’s Frobenius descent). ∃ an OX-linear morphism c∶F∗

X(TX/S) → ZD(0) X/S,

D ↦ Dp − D(p). inducing an isomorphism of OX′-algebras S(TX′/S) ∼

• →

c FX/S∗(ZD(0) X/S).

A left D(0)

X/S-module ⇔ an OX-module E with an inte-

grable connection ∇∶E → E ⊗OX Ω1

X/S.

Then, c induces an OX-linear morphism ψ∶E → E ⊗OX F∗

X/S(Ω1 X′/S),

satisfying ψ ∧ ψ = 0, called the p-curvature.

• ψ vanishes ⇔ the action of D(0)

X/S factors through

D(0)

X/S.

• ψ is nilpotent ⇔ the action of D(0)

X/S factors through

D(0)

X/S/K N for N ≫ 0.

• ψ is quasi-nilpotent ⇔ ∀x ∈ E , ∃N ≥ 0 such that

K N ⋅ x = 0 (⇒ the action of D(0)

X/S on E extends to an

action of ̂

D(0)

X/S).

Want to extend Cartier’s Frobenius descent to quasi- nilpotent objects. Theorem 3 (Berthelot, Ogus-Vologodsky) Given smooth ̃ S-schemes ̃ X and ̃ X′ and an ̃ S-morphism ̃ F∶ ̃ X → ̃ X′ lifting FX/S, there exists a canonical isomorphism of ̂ S(TX′/S)-algebras λ̃

F∶FX/S∗( ̂

D(0)

X/S) ∼

→ FX/S∗(D(0)

X/S) ⊗OX′ ̂

S(TX′/S) compatible with the natural augmentations to FX/S∗(D(0)

X/S).

We deduce an isomorphism ̂

D(0)

X/S ∼

→ E nd̂

S(TX′/S)(OX ⊗OX′ ̂

S(TX′/S)).

The lifting ̃ F equip F∗

X/S(̂

S(TX′/S)) with a left ̂

D(0)

X/S-

module structure and induces equiv.

• f cat.

quasi- inverse to each other :

Mod( ̂ D(0)

X/S) → Mod(̂

S(TX′/S)),

E ↦ H om ̂

D(0)

X/S

(F∗

X/S(̂

S(TX′/S)),E ),

Mod(̂

S(TX′/S)) → Mod( ̂

D(0)

X/S),

F ↦ F∗

X/S(F) = F ⊗̂ S(TX′/S) F∗ X/S(̂

S(TX′/S)).

F an ̂

S(TX′/S)-module ⇒ TX′/S ⊗OX′ F → F ⇒ θ∶F →

F ⊗OX′ Ω1

X′/S OX′-linear; θ ∧ θ = 0,

[θ is a called a Higgs field]. This is Simpson correspondance in characteristic p > 0.

Proposition 4 The direct image by FX/S∗ of the de Rham complex of E is quasi-isomorphic to the Higgs complex of F. Sketch of proof : Koszul resolution of OX′ : [... S(TX′/S) ⊗OX′ ⋀2 TX′/S S(TX′/S) ⊗OX′ TX′/S S(TX′/S)]

OX′ F ⊗OX′ Ω●

X′ ∼

→

H om̂

S(TX′/S)(̂

S(TX′/S) ⊗OX′ ⋀●TX′/S,F)

∼

• → H om ̂

D(0)

X/S

(F∗

X/S(̂

S(TX′/S) ⊗OX′ ⋀●TX′/S),E )

Spencer resolution of OX : [...

D(0)

X/S ⊗OX ⋀2 TX/S

D(0)

X/S ⊗OX TX/S

D(0)

X/S]

OX

Reduced to compare two resolutions of OX. ⋯

̂

D(0)

X/S ⊗OX TX/S

• ̂

D(0)

X/S]

• P↦P.1
• OX

⋯

F∗

X/Ŝ

S(TX′/S) ⊗OX F∗

X/STX′/S

F∗

X/Ŝ

S(TX′/S)]

OX

commutes with left vertical arrow induced by the dual

• f 1

p!d( ̃

F) ∶ F∗

X/SΩ1 X′/S → Ω1 X/S.

Sketch of proof of thm. 3 Will proceed by duality using Lemma 5 The ring ̂

D(0)

X/S is isomorphic to the ring (of

hyper-PD-differential operators) H omOX(PX/S,(0),OX). Local coordinates t1,...,td on X, τi = 1 ⊗ ti − ti ⊗ 1 ∈ I . Locally PX/S,(0) = ⊕k OX×SX/(τp

1,...,τp d).τ[p.k] is filtered

increasingly and exhaustively by FilnPX/S,(0) = ⊕∣k∣≤n OX×SX/(τp

1,...,τp d).τ[p.k]

• n which K n+1 acts trivially, hence

H omOX(FilnPX/S,(0)PX/S,(0),OX) ∼

→ D(0)

X/S/K n+1.

Lemma 6 The map 1

p!(F∗ ×F∗) ∶ I → PX/S,(0) ; f → f[p]

composed with the projection PX/S,(0) → PX/S,(0)/I .PX/S,(0) is an F∗

X-linear map that is zero on I 2.

Linearization gives an OX-linear map : F∗

X/SΩ1 X′/S → PX/S,(0)/I .PX/S,(0)

called divided Frobenius. Let Γ(Ω1

X′/S) the PD-algebra of the OX′-module Ω1 X′/S.

Proposition 7 The divided Frobenius map extends uniquely to an isomorphism of PD-OX-algebras : F∗

X/SΓ(Ω1 X′/S) ∼

• → PX/S,(0)/I .PX/S,(0).

Description in local coordinates : t′

1,....,t′ d the pull-back

• f the ti’s.

OX < dt′

1,....,dt′ d >

→ OX < τi,....,τd > /(τ1,...,τd) dt′

i

↦ τ[p]

i

Remark : The composite S(TX′/S) ∼

• →

c FX/S∗(ZD(0) X/S) ↪ FX/S∗D(0) X/S.

can be obtained by duality from the composite PX/S,(0) →

PX/S,(0)/I .PX/S,(0)

∼

← F∗

X/SΓ(Ω1 X′/S).

Next step : show that the data of thm. 3 (we fix such smooth ̃ S-schemes ̃ X and ̃ X′ and an ̃ S-morphism ̃ F∶ ̃ X → ̃ X′ lifting FX/S in the sequel) allows to canonically lift the latter isomorphism F∗

X/SΓ(Ω1 X′/S) ∼

• → PX/S,(0)/I .PX/S,(0)

to a morphism F∗

X/SΓ(Ω1 X′/S) → PX/S,(0).

Proposition 8 There exists a well-defined map 1 p!(̃ F∗ × ̃ F∗) ∶ ˜

I ′ → pP ̃

X/̃ S,(0) ∼

←

.p!

P ̃

X/̃ S,(0)/pP ̃ X/̃ S,(0) ∼

• →

PX/S,(0)

that factors through Ω1

X′/S such that the induced map

Ω1

X′/S → PX/S,(0) is a lifting of the divided Frobenius :

Ω1

X′/S

PX/S,(0)

• Ω1

X′/S

PX/S,(0)/I .PX/S,(0)

commutes.

Sketch of proof : want to understand (̃ F× ̃ F)∗ ∶ ˜

I ′ → ˜ I .

Take x ∈ OX, x′ ∶= 1 ⊗ x ∈ OX′ with lifts ˜ x ∈ O ˜

X, ˜

x′ ∈ O ˜

X′,

̃ F∗(˜ x′) = ˜ xp + py. ˜ ξ ∶= 1 ⊗ ˜ x − ˜ x ⊗ 1, ˜ ξ′ ∶= 1 ⊗ ˜ x′ − ˜ x′ ⊗ 1 (̃ F × ̃ F)∗(˜ ξ′) = 1 ⊗ ˜ xp − ˜ xp ⊗ 1 + p.(1 ⊗ y − y ⊗ 1) = ˜ ξp +

p−1

∑

i=1

( p i )(˜ xp−i ⊗ 1)˜ ξi + p(1 ⊗ y − y ⊗ 1) ≡ ˜ ξp mod p ˜

I

Hence (̃ F × ̃ F)∗(˜ ξ′) = p!.˜ ξ[p] + pζ ∈ P ̃

X/̃ S,(0)

with ζ ∈ ˜

I .

Then divides by p!.

Proposition 9 . The divided Frobenius extends canon- ically to a morphism F∗

X/SΓ(Ω1 X′/S) → PX/S,(0)

that, by duality, induces a morphism of OX-modules : Φ ̃

F ∶ ̂

D(0)

X/S → F∗ X/S(̂

S(TX′/S)) ↪ ̂

D(0)

X/S

Warning : Φ ̃

F is not a morphism of rings but its re-

striction to the center is one.

Explicit formulas Proposition 10 . Given local coordinates t1,...,td on X, then Φ ̃

F(∂n) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 if ∣n∣ = 0, if 0 < ∣n∣ < p,

1 p! ∑d j=1 ∂i( ˜

F ∗

̃ X(˜

t′

j))∂p j

if n = p.1i. If ˜ F ∗

̃ X(˜

t′

j) = ˜

t′p

j +p˜

gj , the last expression can be rewritten −tp−1

i

∂p

i − d

∑

j=1

∂i(gj)∂p

j

Theorem 11 . The divided Frobenius map 1

p!(̃

F∗ × ̃ F∗) extends canonically to a morphism σ̃

F ∶ OX×X′X ⊗OX′ Γ(Ω1 X′/S) ∼

→ PX/S,(0) which is a PD-isomorphism of OX×X′X-algebras. Theorem 3 follow by duality. Note that OX×X′X = OX×SX/I (p). The verification is the local : PX/S,(0) is filtered by the FilnPX/S,(0)’s and Γ(Ω1

X′/S) is graded. The application

σ̃

F is a PD-morphism and gr(σ̃ F) send 1 ⊗ dt′ i to τ[p] i

.

The ̂ S(TX′/S)-algebra nature of the morphism λ̃

F comes

from the commutative diagram

OX×X′X ⊗OX′ Γ(Ω1

X′/S)

• dt′

i↦τ[p] i

+ζi

PX/S,(0)

• OX ⊗OX′ Γ(Ω1

X′/S) dt′

i↦τ[p] i

PX/S,(0)/I .PX/S,(0)

and the compatibility of σ̃

F with comultiplications :

Γ(Ω1

X′/S)

→

∆ Γ(Ω1 X′/S ⊕ Ω1 X′/S) ∼

• → Γ(Ω1

X′/S) ⊗OX′ Γ(Ω1 X′/S)

OX×X′X

→ OX×X′X ⊗OX OX×X′X a ⊗ b ↦ a ⊗ 1 ⊗ b

PX/S,(0)

→ PX/S,(0) ⊗OX PX/S,(0) a ⊗ b ↦ a ⊗ 1 ⊗ b

Example ̃ S = Spec(Z/p2Z), ̃ X = Spec(Z/p2Z[t]), ̃ F(t) = tp, (1,t,...,tp−1) basis of OX ⊗OX′ ̂ S(TX′/S) (= Z/pZ[t,∂p]) over ̂ S(TX′/S) (= Z/pZ[tp,∂p]) t → ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⋯ ⋯ tp 1 ⋱ ⋱ ⋱ ⋮ ⋮ ⋱ ⋱ ⋱ ⋯ 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , ∂ → ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 − tp∂p ⋯ 2 − tp∂p ⋱ ⋱ ⋮ ⋮ ⋮ ⋮ ⋱ p − 1 − tp∂p −∂p ⋯ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ .

The Cartier transform of Ogus-Vologodsky Given (only) a smooth ̃ S-lifting ̃ X′ of X′, Ogus and Vologodsky “glue” in a sense the isomorphisms λ ̃

F for

various liftings ̃ F of FX/S. The Morita equivalence of the isomorphism they obtain is the Cartier transform. Γ(TX′/S) the PD-algebra of the OX′-module TX′/S and J = ⊕n≥1Γn(TX′/S) its PD-ideal. ̂ Γ(TX′/S) the completion of Γ(TX′/S) with respect to the PD-filtration (J[n])n≥1.

HIGγ(X′/S) the category of ̂

Γ(TX′/S)-modules [PD-Higgs modules over X′/S].

Dγ

X/S = D(0) X/S ⊗S(TX′/S) ̂

Γ(TX′/S).

MICγ(X/S) the category of left Dγ

X/S-modules.

A left Dγ

X/S-modules ⇔ an OX-module with an inte-

grable connection (E,∇) equipped with a homomor- phism ̂ Γ(TX′/S) → FX/S∗(E ndOX(E,∇)) which extends its p-curvature S(TX′/S) → FX/S∗(E ndOX(E,∇)). Denote by ι the automorphism of ̂ Γ(TX′/S) induced by −idTX′/k

Theorem 12 (Ogus-Vologodsky) Assume given a smooth ̃ S-lifting ̃ X′ of X′. (i) There exists a can. left Dγ

X/k-module B ̃ X′ splitting

the Azumaya algebra FX/S∗(Dγ

X/S) over ̂

Γ(TX′/S). (ii) There exist canonical equiv. of cat. quasi-inverse to each other C ̃

X′∶MICγ(X/S) ∼

→ HIGγ(X′/S), E ↦ ι∗(H omDγ

X/S(B ̃

X′,E)),

C−1

̃ X′∶HIGγ(X′/S) ∼

→ MICγ(X/S), E′ ↦ B ̃

X′ ⊗̂ Γ(TX′/S) ι∗(E′).

Furthermore, they induce equiv. between the subcat.

• f quasi-nilpotent objects.

(iii) Let (E,∇,ψ) ∈ MICγ(X/S) and (E′,θ′) = C ̃

X′(E,∇,ψ).

An ̃ S-lifting ̃ F∶ ̃ X → ̃ X′ of FX/S induces a canonical func- torial isomorphism η̃

F∶(E,ψ) ∼

→ F∗

X/S(E′,−θ′).

(iv) ∀0 ≤ ℓ ≤ p − 1, C−1

̃ X′ induces an equiv. of cat.

C−1

̃ X′∶HIGℓ(X′/S) ∼

→ MICℓ(X/S).

HIGℓ(X′/S) is the cat. of Higgs modules (M′,θ′) (i.e.,

M′ is an OX′-mod., θ′∶M′ → M′⊗OX′Ω1

X′/S is an OX′-linear

map such that θ′ ∧θ′ = 0) which are nilpotent of level ≤ ℓ (i.e., there exists an increasing θ-stable filtration M′

• of

M′ of length ≤ ℓ + 1 such that GrM′

• (θ′) = 0).

MICℓ(X/S) is the cat. of OX-modules with integrable

connection which are nilpotent of level ≤ ℓ (i.e., the associated p-curvature is a nilpotent F-Higgs field of level ≤ ℓ). (v) For (M′,θ′) ∈ HIGℓ(X′/S) and (M,∇) = C−1

̃ X′(M′,θ′), ∃

a canonical isom in Db(OX′) τ<p−ℓ(M′ ⊗ Ω●

X′/S) ∼

→ τ<p−ℓ(FX/S∗(M ⊗ Ω●

X/S)).

Hint for the construction of B ̃

X′ :

̃ F1, ̃ F2 ∶ ̃ X → ̃ X′

1 p!(̃

F∗

2 − ̃

F∗

1) ∶ OX′ → FX/S∗OX is a derivation, i.e. a linear

form u ∶ Ω1

X′/S → FX/S∗OX

σ−1

̃ F2 ○ σ̃ F1 induces an isomorphism

OX×X′X ⊗OX′ S(Ω1

X′/S) ∼

→ OX×X′X ⊗OX′ S(Ω1

X′/S)

(not − ⊗OX′ Γ(Ω1

X′/S) ! : σ̃ Fi(1 ⊗ dt′ i) = τ[p] i

+ ...) σ−1

̃ F2 ○ σ̃ F1 gives the glueing.

Even more explicitly, σ−1

̃ F2 ○ σ̃ F1(1 ⊗ ω′) = 1 ⊗ ω′ − (1 ⊗ u(ω′) − u(ω′) ⊗ 1) ⊗ 1

i.e. a translation on S(OX×X′X ⊗OX′ Ω1

X′/S) whose dual

• n ̂

Γ((OX×X′X ⊗OX′ Ω1

X′/S)∨) is multiplication by exp(u)

(Taylor’s formula). On Uα ∩ Uβ, one glue λα and λβ using conjugation by exp(uαβ).