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 & IHÉS) Schloss Elmau, 7-13 May 2017

Contents X a smooth scheme over a perfect field k of charc. p > 0 . 1- Shiho’s lifting of the “local” Cartier trans- form modulo p n , given a lifting of the relative Frobenius of X / k over W n + 1 ( k ) . 2- Oyama’s interpretation of Ogus-Vologodsky’s Cartier transform modulo p as the pull-back by a morphism of ringed topoi. 3- Xu’s lifting of the Cartier transform mod- ulo p n , using Oyama topoi, given (only) a lifting of X to a smooth formal scheme over W ( k ) .

f ∶ X → S a morphisme of schemes, M an O X - module, λ ∈ Γ ( S, O S ) . A λ -connection on M relatively to S is an O S -linear morphism ∇∶ M → Ω 1 X / S ⊗ O X M such that ∀ t ∈ O X 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 O X -modules with integrable λ -connection, MIC ( X / S ) the category of O X -modules with integrable connection.

� � � � � � � � k a perfect field of characteristic p , W = W ( k ) , S = Spf ( W ) , X a smooth formal S -scheme, X ′ = X × S ,σ S , X n = ( X , O X / p n ) ( ∀ n ≥ 1 ), X = X 1 F X X ′ X X F X / k ◻ Spec ( k ) F k � Spec ( k ) Given an S n + 1 -morphism F n + 1 ∶ X n + 1 → X ′ n + 1 lifting F X / k , d F n + 1 induces an O X n -linear mor- phism n / S n ) → Ω 1 d F n + 1 ∶ F ∗ n ( Ω 1 X ′ X n / S n p that fits into a commutative diagram d F n + 1 F ∗ p � Ω 1 n ( Ω 1 n / S n ) X ′ X n / S n p n + 1 / S n + 1 ) d F n + 1 � F ∗ n + 1 ( Ω 1 Ω 1 X ′ X n + 1 / S n + 1

Shiho defined the functor → MIC ( X n / S n ) Φ n ∶ p - MIC ( X ′ n / S n ) ↦ ( F ∗ ( M ′ , ∇ ′ ) n ( M ′ ) , ∇) , where ∇ is defined ∀ t ∈ O X n and ∀ x ∈ M ′ , by n ( x )) = t ⋅ ( d F n + 1 ∇( t F ∗ ⊗ id )( F ∗ n ( ∇ ′ ( x ))) + dt ⊗ F ∗ n ( x ) , p d F n + 1 ⊗ id � Ω 1 [ F ∗ n ( M ′ ) ] n ( Ω 1 p n M ′ ) X n / S n ⊗ O X n F ∗ n / S n ⊗ O X ′ X ′ For n = 1 , Φ 1 was considered first by Ogus- Vologodsky who proved that it is compatible with their Cartier transform C − 1 . X ′ 2 Proposition 1 (Shiho) Φ n induces an equiv. of cat. n / S n ) ∼ → MIC qn ( X n / S n ) . Φ n ∶ p - MIC qn ( X ′

The quasi-nilpotence is a local condition. If ∃ an étale map X n → Spec ( W n [ T 1 ,...,T d ]) . ( M, ∇ ) an O X -module with ( p -)connection / S n . ∃ O S -linear endomorphisms ∇ ∂ 1 ,..., ∇ ∂ d of M such that ∀ u ∈ M , ∇ ( u ) = dt i ⊗ ∇ ∂ i ( u ) . ∑ 1 ≤ i ≤ d ∇ is integrable ⇔ ∇ ∂ i ○ ∇ ∂ j = ∇ ∂ j ○ ∇ ∂ i ∀ i,j . So we can define the endomorphism ∇ ∂ n = ∏ 1 ≤ i ≤ d ( ∇ ∂ i ) n i of M ∀ n = ( n 1 ,...,n d ) ∈ N d . 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 ∈ N d 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.

� � A Hopf algebra of a (commutative) ringed topos ( T ,A ) is the data of a (commutative) ring B of T and five homomorphisms d 1 � B δ � B ⊗ A B , B π � A , B σ � B , A � B , d 0 where the tensor product is taken on the left (resp. right) for the A -algebra structure de- fined by d 1 (resp. d 0 ), 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 ) = id M and the cocycle condi- tion ε ⊗ B,δ B ⊗ A B � M ⊗ A B ⊗ A B B ⊗ A B ⊗ A M ε ⊗ id B id B ⊗ ε B ⊗ A M ⊗ A B

� � X a smooth formal S -scheme. P X n ( r ) the PD envelop of the diagonal immer- sion X n → X r + 1 compatible with the canonical n PD structure on W n ( n,r ≥ 1 ) . P X ( r ) = ( P X n ( r )) n ≥ 1 is an adic formal S -scheme. P X ∶= P X ( 1 ) has a natural structure of a formal X -groupoid over S ( ⇒ P X = O P X is a formal Hopf O X -algebra of X zar ). X X q 1 , 3 � X 3 � X 2 P X ( 2 ) ∼ ⇒ → P X ( 2 ) → P X . P X × X P X There exists a canonical equiv. of cat. + P X -strat . } ∼ { → MIC qn ( X n / S n ) . O X n -mod .

� � � � U an open of X 2 such that the diagonal X = X 1 → X 2 factors through a closed immersion X → U , I ⊂ O U the associated coherent open ideal, Z the admissible blow-up of I in U , R X the maximal open subscheme of Z where ( I O Z )∣ R X = ( p O Z )∣ R X . ● R X , 1 → X 2 factors through X → X 2 . ● Universal property. Let Y be a flat adic formal S -scheme, f ∶ Y → X 2 an S -morphism that fits into a commutative diagram Y 1 Y f ′ g f � � X 2 X R X Then, there exists a unique map f ′ lifting f . R X has a natural structure of a formal X - groupoid over S ( ⇒ R X = O R X is a formal Hopf O X -algebra of X zar ).

T X n the PD envelop of the closed immer- sion X n → R X ,n lifting the diagonal, compati- ble with the canonical PD-structure on W n . T X ∶= ( T X n ) n ≥ 1 is an adic formal S -scheme. It has a natural structure of a formal X -groupoid over S which lifts that of R X ( ⇒ T X = O T X is a formal Hopf O X -algebra of X zar ). [Shiho] There exists a can. equiv. of cat. + T X -strat . } ∼ { O X n -mod . → p - MIC qn ( X n / S n ) . If ∃ X → Spf ( W { T 1 ,...,T d }) an Example : étale morphism, t i ∈ O X the image of T i and ξ i = 1 ⊗ t i − t i ⊗ 1 . The ideal I is generated by p,ξ 1 ,...,ξ d on an open neighborhood of the diagonal X in X 2 . We have isomorphisms of O X -algebras ∼ O X { ζ 1 ,...,ζ d } → q 1 ∗ ( R X ) , ∼ O X { ζ 1 ,...,ζ d } → q 2 ∗ ( R X ) , that map ζ i to ξ i p .

� � � � � � � � � X a smooth formal S -scheme. Given an S - morphism F ∶ X → X ′ = X × S ,σ S lifting F X / k . There exists a unique morphism g ∶ P X 1 → X ′ which fits into the commutative diagram P X 1 P X φ X 2 g F 2 X ′ ∆ � X ′ 2 R X ′ φ is morphism of groupoids above F . ι P � P X X ϕ φ F � ι R X ′ � R X ′ T X ′ ϕ ∶ P X → T X ′ a morphism of formal groupoids above F . Shiho proved that the functor n ∶ { O X ′ + T X ′ -strat . } � → { O X n -mod . + P X -strat . } n -mod . ϕ ∗ ( M,ε ) � → ( F ∗ n ( M ) ,ϕ ∗ ( ε )) coincides with his functor Φ n

� � � � � � For any k -scheme Y , we denote by Y the schematic image of F Y , i.e. the closed sub- scheme of Y defined by the ideal of O Y con- sisting of sections with vanishing p -th power. F Y / k Y ′ � Y ′ Y f Y / k We construct by dilatation a canonical adic formal X 2 -scheme Q X satisfying the following properties: ● Q X , 1 → Q X , 1 → X 2 factors through X → X 2 . ● Let Y be flat adic formal S -scheme, f ∶ Y → X 2 an S -morphism that fits into a commu- tative diagram Y 1 Y f ′ g f � � X 2 X Q X Then, there exists a unique map f ′ lifting f .

� � Q X has a natural structure of a formal X - groupoid over S ( ⇒ Q X = O Q X is a formal Hopf O X -algebra of X zar ). X → Spf ( W { T 1 ,...,T d }) an étale Example : morphism. We have canonical isomorphisms of O X -algebras ∼ → O X { ζ 1 ,...,ζ d } q 1 ∗ ( Q X ) , ∼ → O X { ζ 1 ,...,ζ d } q 2 ∗ ( Q X ) , that map ζ i to ξ p p . i Q X fits into a commutative diagram of formal X -groupoids over S ϕ � P X T X ′ ̟ φ λ � � R X ′ Q X ψ where λ,̟ are canonical but ψ,ϕ,φ depend on the lifting F ∶ X → X ′ of F X / k .

� � { O X ′ + R X ′ -strat . } { O X n -mod . + Q X -strat . } ψ ∗ n -mod . n ̟ ∗ λ ∗ n � n � { O X n -mod . { O X ′ + T X ′ -strat . } ϕ ∗ + P X -strat . } n -mod . n p - MIC qn ( X ′ n / S n ) Φ n � MIC qn ( X n / S n ) Theorem 2 (Oyama, Xu) There exists an equiv. of cat. that depends only on X X / W ∶ { O X ′ + R X ′ -strat . } → { O X n -mod . + Q X -strat . } n -mod . C ∗ Moreover, given a lifting F ∶ X → X ′ of F X / k , there is a canonical isomorphism n ( M,ε ) ∼ → C ∗ X / W ( M,ε ) . η F ∶ ψ ∗

� � � � � Oyama topoi. X a k -scheme. ● E : the cat. of triples ( U, T ,u ) where U ⊂ X open, T is a flat adic formal S -scheme and u ∶ T ∶= T 1 → U is an affine k -morphism. ● E : the cat. of triples ( U, T ,u ) where U ⊂ X open, T is a flat adic formal S -scheme and u ∶ T → U is an affine k -morphism. ( U, T ,u ) ∈ E u U T T F U / k � F T / k � f T / k F T / k T ′ U ′ � T ′ u ′ ⇒ ( U ′ , T ,u ′ ○ f T / k ) ∈ E ′ ∶= E ( X ′ / S ) , X ′ = X ⊗ k,σ k . ⇒ functor → E ′ ρ ∶ E ↦ ( U ′ , T ,u ′ ○ f T / k ) ( U, T ,u )