Lifting the Cartier transform of Ogus-Vologodsky modulo p n Daxin Xu California Institute of Technology Riemann-Hilbert correspondences 2018, Padova
A theorem of Deligne-Illusie k a perfect field of characteristic p > 0, W ring of Witt vectors, σ : W W Frobenius endomorphism. X/k smooth, X = X k,σ k. X F X/k X F X Spec(k) σ X Spec(k) (i) If smooth W 2 -schemes X, X and F : X X a lifting of F X/k, then: ϕ F : i 0 Ω i X /k [ i] F X/k (Ω X/k, d). (ii) If a smooth lifting X of X sur W 2, then ϕ X : p 1 i=0 Ωi X /k [ i] F X/k (τ <p (Ω X/k, d)),
An analogue of Simpson correspondence for modules with integrable connection in positive characteristic. They provide a nonabelian generalisation of Deligne-Illusie s theorem. p-curvature: an important notion for MIC(X/k). For Der(X/k), p = (p-times) Der(X/k). Given an object (M, ) of MIC(X/k), we consider ψ : Der(X/k) End OX (M), ( ( )) p ( p ). ψ is F X -linear, i.e. ψ(g ) = g p, g O X. We obtain an O X -linear map, called p-curvature of : ψ : M M OX F X (Ω 1 X/k ) satisfying ψ ψ = 0 (can be viewed as a F -Higgs field).
Cartier descent Given an O X -module M, there exists a canonical connection 0 on F X/k (M ) defined by 0 (x m) = d(x) m, x O X, m M. The p-curvature of 0 is zero. Theorem (Cartier) Mod(O X ) { } OX -modules with integrable connection of zero p-curvature M (FX/k (M ), 0 ) M (M, )
Ogus and Vologodsky generalise this correspondence for modules with integrable connection satisfying some nilpotent conditions. For l 0, a Higgs module (M, θ) is nilpotent of level l, if an increasing filtration N of M such that N 0 = 0, N l+1 = M and that Gr N (θ) = 0, i.e. θ(n i ) N i 1 OX Ω 1 X/k. An O X -module with integrable connection is nilpotent of level l, if its p-curvature map is nilpotent of level l. Theorem (Ogus Vologodsky, a special case) Let l < p. A smooth lifting X of X over W 2 induces C 1 X : HIG l (X /k) MIC l (X/k). For (M, θ) HIG l (X /k) and (M, ) = C 1 X (M, θ), X induces an isomorphism in D(O X ) τ <p l (M OX Ω X /k ) F X/k (τ <p l (M OX Ω X/k )).
A local lifting due to Shiho To understand the relation between Cartier transform and p-adic Simpson correspondence, we need to lift C 1 modulo X p n. Shiho constructed a local lifting. X a smooth formal W-scheme with special fiber X. F : X X = X W,σ W a lifting of F X/k. Shiho s functor is defined on the category p-mic(x n / W n ) of O Xn -modules with integrable p-connection. A p-connection on an O Xn -module M is an W n -linear morphism : M Ω 1 X n/ W n OXn M such that (x m) = pd(x) m + x (m), x O Xn, m M. Integrability is defined in the same way for connection.
We have Im(dF X/k ) = 0. df can be divided by p df p : F (Ω 1 X / W ) Ω1 X/ W. Shiho s functor: Φ n : p-mic(x n/ W n ) MIC(X n / W n ) (M, ) (F (M), ) where : F (M) Ω 1 X n/ W n OXn F (M) is defined by (xf (m)) = x ( id df p )( F ( (m)) ) + dx F (m). for x O Xn, m M.
In the following, we will present Oyama s approach for the Cartier transform. Then we use this approach to lift the Cartier transform and globalize Shiho s construction. We define two categories E, E as follows: E objets (U, T, u) U open X, T flat formal W-scheme u : T (= T 1 ) U affine /k E objets (U, T, u) U open X, T flat formal W-scheme u : T U affine /k, T : the closed subscheme of T defined by {x x p = 0} topology: {(U i, T i ) (U, T)} i I of E (resp. E ) is a covering if U = i I U i, and T i TUi, i I, where T Ui = u 1 (U i ), u : T (= T ) U. Oyama topos: Ẽ, Ẽ, topos of sheaves on E, E.
A morphism of topoi realises the Cartier transform. (U, T, u) Ob(E ), F T /k : T T factor through T U F U/k u T F T /k U u T f T /k T T F T /k where u = u σ,k k. E : the first category associated to X. We obtain ρ : E E, (U, T, u) (U, T, u f T /k ). ρ is fully faithful, continuous and cocontinuous. Then it induces a morphism of topoi C X/ W : Ẽ Ẽ such that F Ẽ, C X/ W (F )(U, T) = F (ρ(u, T)).
Structural sheaf: the functor on E (resp. E ) (U, T, u) Γ(T, O Tn ) define a sheaf of rings O E,n (resp. O E,n ). We have C X/W (O E,n) = O E,n. An O E,n -module (resp. O E,n -module) F the data (i) (U, T, u) of E (resp. E ), an u (O Tn )-module F (U,T) of U zar. (ii) f : (U 1, T 1, u 1 ) (U 2, T 2, u 2 ) of E (resp. E ), an u 1 (O T1,n ) linear morphism c f : u 1 (O T1,n ) (u2 (O T2,n )) U1 (F (U2,T 2)) U1 F (U1,T 1), subject to some compatibility condition. (i) F is quasi-cohérent if (U, T), F (U,T) if quasi-coherent. (ii) F is a crystal if f, c f is an isomorphism.
Theorem (Oyama n = 1,X) C X/W, C X/W induces equivalences of categories C qcoh (O E,n) C qcoh (O E,n ). C X/ W : transformée de Cartier (modulo pn ) Suppose a smooth formal W-scheme X with X 1 = X. (X, X) define objects of E and E. We will present an interpretation of crystals in terms of O X -modules with stratification. Based on this interpretation, Oyama show that in the case n = 1, C X/ W coincides with C 1 X (Cartier transform defined by 2 X 2 ).
Interpretation of crystals in terms of modules with stratification We need the envelope of : X X 2 (= X W X) in E, E : We define R X, Q X two adic formal X 2 -schemes which are universal for the following properties: flat formal W-scheme Y, f : Y X 2 / W, if g : Y = (Y 1 ) X (resp. g : Y X) s.t. g Y Y f X X 2 f R X g Y Y f X X 2 (X, R X ) E and (X, Q X ) E q 1, q 2 : (X, R X ) (resp. (X, Q X )) (X, X). f Q X
A local description of O RX. Let t 1,, t d be local coordinates of X and set ξ i = 1 t i t i 1 O X 2. We have q i (O RX ) O X { ξ 1 p,, ξ d } i = 1, 2. p O RX, O QX are O X -bialgebras of X zar. Moreover, they are equipped with a Hopf algebra structure (comutiplication, counit, coinverse). Let A be a Hopf O X -algebra. An A-stratification on an O X -module M is an A-linear isomorphism ε : A OX,d 2 M M OX,d 1 A satisfying some cocycle conditions. { } C (O E,n ) OXn -modules + O RX -stratification F (F (X,X), c q2 c 1 q 1 ) { } C (O E,n ) OXn -modules + O QX -stratification
Let P X be the PD-envelope of : X X 2. We have P X,1 X and then by universal property we obtain an X 2 -morphism P X Q X. An integrable connection is equivalent to a O PX -stratification. The pullback of the above morphism induces a functor µ : C (O E,n ) MIC(X n / W n ). Similarly, there exists a canonical functor ν : C (O E,n ) p-mic(x n / W n ).
Proposition Suppose a lifting F : X X = X W,σ W of F X/k. The diagram C qcoh (O E,n) C X/ W C qcoh (O E,n ) µ p-mic(x n/ W n ) Φn MIC(X n / W) is commutative up to a functorial isomorphism, M C (O E,n): η F : Φ n (µ(m )) ν(c X/ W (M )). In this diagram C X/ W depends only on X, µ, ν depend on X, and Shiho s functor Φ n, the isomorphism η F rely on F : X X. Application of C X/ W : an intrinsic interpretation of Faltings relative Fontaine Laffaille modules and of the computation of their cohomology. ν
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