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The Stacks project

Remark 60.24.14 (Complete comparison). Let p be a prime number. Let (A, I, \gamma ) be a divided power ring with A a Noetherian p-adically complete ring and p nilpotent in A/I. Set S = \mathop{\mathrm{Spec}}(A) and S_0 = \mathop{\mathrm{Spec}}(A/I). Let Y be a proper smooth scheme over S and set X = Y \times _ S S_0. Let \mathcal{F} be a finite type crystal in quasi-coherent \mathcal{O}_{X/S}-modules. Then

  1. there exists a coherent \mathcal{O}_ Y-module \mathcal{F}_ Y endowed with integrable connection

    \nabla : \mathcal{F}_ Y \longrightarrow \mathcal{F}_ Y \otimes _{\mathcal{O}_ Y} \Omega _{Y/S}

    such that \mathcal{F}_ Y/p^ e\mathcal{F}_ Y is the module with connection over A/p^ eA found in Remark 60.24.11, and

  2. we have

    R\Gamma (\text{Cris}(X/S), \mathcal{F}) = R\Gamma (Y, \mathcal{F}_ Y \otimes _{\mathcal{O}_ Y} \Omega ^\bullet _{Y/S})

    in D(A).

Hints: The existence of \mathcal{F}_ Y is Grothendieck's existence theorem (insert future reference here). The isomorphism of cohomologies follows as both sides are computed as R\mathop{\mathrm{lim}}\nolimits of the versions modulo p^ e (see Remark 60.24.10 for the left hand side; use the theorem on formal functions, see Cohomology of Schemes, Theorem 30.20.5 for the right hand side). Each of the versions modulo p^ e are isomorphic by Remark 60.24.11.


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