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

� � � � � � S a flat scheme over Z / p 2 Z , S = ̃ ̃ S ⊗ Z p F p , X a smooth S -scheme, F X X ′ X F X / S X ◻ � S S F S Theorem 1 (Deligne-Illusie) X ′ and an ̃ (i) Given smooth ̃ S -schemes ̃ X and ̃ S -morphism X ′ lifting F X / S , there exists a canonical quasi- F ∶ ̃ ̃ X → ̃ isomorphism X ′ / S [ − i ] → F X / S ∗ ( Ω ● X / S ) , ⊕ i ≥ 0 Ω i 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 D b ( O X ′ ) X ′ / S [− i ] ∼ ⊕ p − 1 → τ < p F X / S ∗ ( Ω ● X / S ) . i = 0 Ω i X → S a smooth separated morphism, I ⊂ O X × S X the ideal of the diagonal, X / S = O X × S X / I n + 1 ( n ≥ 0 ) ; P n D X / S,n = H om O X ( P n X / S , O X ) P X / S, ( 0 ) the PD-envelop of I , I ⊂ P X / S, ( 0 ) the PD-ideal generated by I , X / S, ( 0 ) = P X / S, ( 0 ) / I [ n + 1 ] ( n ≥ 0 ) . P n

D ( 0 ) X / S,n = H om O X ( P n X / S, ( 0 ) , O X ) [ O X -algebras via the first projection.] The O X -modules ⋃ D X / S D X / S,n = n ≥ 0 D ( 0 ) D ( 0 ) ⋃ X / S X / S,n , = n ≥ 0 are naturally equipped with ring structures. D ( 0 ) X / S is the sheaf of PD-differential operators.

Assume in the following that p O S = 0 and X → S smooth. Proposition 2 The image of the composed homomor- phism D ( 0 ) X / S → D X / S ↪ E nd O S ( O X ) is the ring of O X ′ -linear endomorphisms of O X , and its kernel is the bilateral ideal K of D ( 0 ) X / S locally generated i (for 1 ≤ i ≤ d ∶ = dim ( X / S ) ); the latter by the operators ∂ p are elements of the center Z D ( 0 ) X / S of D ( 0 ) X / S . D ( 0 ) = D ( 0 ) X / S / K , X / S D ( 0 ) ̂ D ( 0 ) X / S / K n . X / S = lim ← � n ≥ 0

⇒ ∃ a canonical isomorphism of O X ′ -algebras F X / S ∗ ( D ( 0 ) X / S ) ∼ → E nd O X ′ ( F X / S ∗ O X ) . 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 nd A ( M ) The functors: ψ ∶ Mod ( B ) → Mod ( A ) , ↦ H om B ( M , E ) , E φ ∶ Mod ( A ) → Mod ( B ) , F ↦ M ⊗ A F , are equivalences of categories quasi-inverse to each other ∼ ∼ ( φ ○ ψ can ψ ○ φ ). ev Id , Id �→ � � →

By Morita equivalence , the functors: Mod ( D ( 0 ) X / S ) → Mod ( O X ′ ) , F X / S ∗ ( H om ( O X , E )) , D ( 0 ) E ↦ X / S → Mod ( D ( 0 ) Mod ( O X ′ ) X / S ) , F ∗ X / S ( F ) = F ⊗ O X ′ O X , F ↦ are equivalences of categories quasi-inverse to each other (Cartier’s Frobenius descent ). ∃ an O X -linear morphism X ( T X / S ) → Z D ( 0 ) D ↦ D p − D ( p ) . c ∶ F ∗ X / S , inducing an isomorphism of O X ′ -algebras S ( T X ′ / S ) ∼ c F X / S ∗ ( Z D ( 0 ) X / S ) . � →

A left D ( 0 ) X / S -module ⇔ an O X -module E with an inte- grable connection ∇∶ E → E ⊗ O X Ω 1 X / S . Then, c induces an O X -linear morphism X / S ( Ω 1 X ′ / S ) , ψ ∶ E → E ⊗ O X F ∗ 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 ̃ ̃ X ′ and an ̃ ̃ S -morphism ̃ F ∶ ̃ X → ̃ X ′ S -schemes X and lifting F X / S , there exists a canonical isomorphism of ̂ S ( T X ′ / S ) -algebras → F X / S ∗ ( D ( 0 ) X / S ) ∼ F ∶ F X / S ∗ ( ̂ D ( 0 ) X / S ) ⊗ O X ′ ̂ S ( T X ′ / S ) λ ̃ compatible with the natural augmentations to F X / S ∗ ( D ( 0 ) X / S ) . We deduce an isomorphism ∼ D ( 0 ) ̂ S ( T X ′ / S ) ( O X ⊗ O X ′ ̂ S ( T X ′ / S )) . → E nd ̂ X / S

̂ The lifting ̃ X / S (̂ D ( 0 ) S ( T X ′ / S )) with a left F equip F ∗ X / S - module structure and induces equiv. of cat. quasi- inverse to each other : Mod ( ̂ D ( 0 ) X / S ) → Mod (̂ S ( T X ′ / S )) , X / S (̂ ( F ∗ S ( T X ′ / S )) , E ) , E ↦ H om ̂ D ( 0 ) X / S S ( T X ′ / S )) → Mod ( ̂ D ( 0 ) Mod (̂ X / S ) , X / S (̂ X / S ( F ) = F ⊗ ̂ S ( T X ′ / S )) . F ↦ F ∗ S ( T X ′ / S ) F ∗ F an ̂ S ( T X ′ / S ) -module ⇒ T X ′ / S ⊗ O X ′ F → F ⇒ θ ∶ F → F ⊗ O X ′ Ω 1 X ′ / S O X ′ -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 F X / S ∗ of the de Rham complex of E is quasi-isomorphic to the Higgs complex of F . Sketch of proof : Koszul resolution of O X ′ : S ( T X ′ / S ) ⊗ O X ′ ⋀ 2 T X ′ / S [ ... S ( T X ′ / S ) ⊗ O X ′ T X ′ / S S ( T X ′ / S )] O X ′ 0 ∼ S ( T X ′ / S ) (̂ S ( T X ′ / S ) ⊗ O X ′ ⋀ ● T X ′ / S , F ) F ⊗ O X ′ Ω ● H om ̂ → X ′ ∼ X / S (̂ ( F ∗ S ( T X ′ / S ) ⊗ O X ′ ⋀ ● T X ′ / S ) , E ) → H om ̂ D ( 0 ) � X / S

� � � � Spencer resolution of O X : D ( 0 ) D ( 0 ) [ ... X / S ⊗ O X ⋀ 2 T X / S X / S ⊗ O X T X / S D ( 0 ) X / S ] O X 0 Reduced to compare two resolutions of O X . � ̂ ̂ D ( 0 ) D ( 0 ) X / S ] X / S ⊗ O X T X / S ⋯ O X P ↦ P. 1 X / S ̂ X / S ̂ S ( T X ′ / S ) ⊗ O X F ∗ S ( T X ′ / S )] � F ∗ � F ∗ X / S T X ′ / S ⋯ � O X commutes with left vertical arrow induced by the dual p ! d ( ̃ F ) ∶ F ∗ of 1 X / S Ω 1 X ′ / S → Ω 1 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 om O X ( P X / S, ( 0 ) , O X ) . Local coordinates t 1 ,...,t d on X , τ i = 1 ⊗ t i − t i ⊗ 1 ∈ I . d ) .τ [ p.k ] is filtered Locally P X / S, ( 0 ) = ⊕ k O X × S X /( τ p 1 ,...,τ p increasingly and exhaustively by d ) .τ [ p.k ] Fil n P X / S, ( 0 ) = ⊕ ∣ k ∣ ≤ n O X × S X /( τ p 1 ,...,τ p on which K n + 1 acts trivially, hence H om O X ( Fil n P X / S, ( 0 ) P X / S, ( 0 ) , O X ) ∼ → D ( 0 ) X / S / K n + 1 .

p ! ( F ∗ × F ∗ ) ∶ I → P X / S, ( 0 ) ; f → f [ p ] Lemma 6 The map 1 composed with the projection P X / S, ( 0 ) → P X / S, ( 0 ) / I . P X / S, ( 0 ) is an F ∗ X -linear map that is zero on I 2 . Linearization gives an O X -linear map : X ′ / S → P X / S, ( 0 ) / I . P X / S, ( 0 ) F ∗ X / S Ω 1 called divided Frobenius . Let Γ ( Ω 1 X ′ / S ) the PD-algebra of the O X ′ -module Ω 1 X ′ / S . Proposition 7 The divided Frobenius map extends uniquely to an isomorphism of PD- O X -algebras : X ′ / S ) ∼ X / S Γ ( Ω 1 → P X / S, ( 0 ) / I . P X / S, ( 0 ) . F ∗ �

Description in local coordinates : t ′ 1 ,....,t ′ d the pull-back of the t i ’s. O X < dt ′ 1 ,....,dt ′ d > → O X < τ i ,....,τ d > /( τ 1 ,...,τ d ) τ [ p ] dt ′ ↦ i i Remark : The composite S ( T X ′ / S ) ∼ c F X / S ∗ ( Z D ( 0 ) X / S ) ↪ F X / S ∗ D ( 0 ) X / S . � → can be obtained by duality from the composite P X / S, ( 0 ) → ∼ � F ∗ P X / S, ( 0 ) / I . P X / S, ( 0 ) X / S Γ ( Ω 1 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 F X / S in the sequel) allows to canonically lift the latter isomorphism X ′ / S ) ∼ X / S Γ ( Ω 1 → P X / S, ( 0 ) / I . P X / S, ( 0 ) F ∗ � to a morphism X / S Γ ( Ω 1 X ′ / S ) → P X / S, ( 0 ) . F ∗

�� Proposition 8 There exists a well-defined map ∼ p ! (̃ F ∗ × ̃ F ∗ ) ∶ ˜ S, ( 0 ) / p P ̃ 1 I ′ → p P ̃ P ̃ X /̃ X /̃ X /̃ S, ( 0 ) S, ( 0 ) ←� .p ! ∼ P X / S, ( 0 ) → � that factors through Ω 1 X ′ / S such that the induced map X ′ / S → P X / S, ( 0 ) is a lifting of the divided Frobenius : Ω 1 � P X / S, ( 0 ) Ω 1 X ′ / S � P X / S, ( 0 ) / I . P X / S, ( 0 ) Ω 1 X ′ / S commutes.

Sketch of proof : want to understand (̃ F × ̃ F ) ∗ ∶ ˜ I ′ → ˜ I . Take x ∈ O X , x ′ ∶ = 1 ⊗ x ∈ O X ′ with lifts ˜ x ′ ∈ O ˜ X , ˜ X ′ , x ∈ O ˜ ̃ F ∗ ( ˜ x ′ ) = ˜ x p + py . ξ ′ ∶ = 1 ⊗ ˜ x ′ − ˜ x ′ ⊗ 1 ξ ∶ = 1 ⊗ ˜ x − ˜ x ⊗ 1 , ˜ ˜ (̃ F × ̃ F ) ∗ ( ˜ ξ ′ ) x p ⊗ 1 + p. ( 1 ⊗ y − y ⊗ 1 ) x p − ˜ 1 ⊗ ˜ = p − 1 ( p i )( ˜ x p − i ⊗ 1 ) ˜ ξ i + p ( 1 ⊗ y − y ⊗ 1 ) ξ p + ˜ ∑ = i = 1 ξ p mod p ˜ ˜ I ≡ Hence (̃ F × ̃ ξ [ p ] + pζ ∈ P ̃ F ) ∗ ( ˜ ξ ′ ) = p ! . ˜ X /̃ S, ( 0 ) with ζ ∈ ˜ I . Then divides by p ! .

Proposition 9 . The divided Frobenius extends canon- ically to a morphism F ∗ X / S Γ ( Ω 1 X ′ / S ) → P X / S, ( 0 ) that, by duality, induces a morphism of O X -modules : D ( 0 ) D ( 0 ) F ∶ ̂ S ( T X ′ / S )) ↪ ̂ X / S (̂ X / S → F ∗ Φ ̃ X / S Warning : Φ ̃ F is not a morphism of rings but its re- striction to the center is one.