Remark 60.24.12 (Perfectness). Let p be a prime number. Let (A, I, \gamma ) be a divided power ring with p nilpotent in A. Set S = \mathop{\mathrm{Spec}}(A) and S_0 = \mathop{\mathrm{Spec}}(A/I). Let X be a proper smooth scheme over S_0. Let \mathcal{F} be a crystal in finite locally free quasi-coherent \mathcal{O}_{X/S}-modules. Then R\Gamma (\text{Cris}(X/S), \mathcal{F}) is a perfect object of D(A).
Hints: By Remark 60.24.9 we have
By Remark 60.24.11 we have
Using the stupid filtration on the de Rham complex we see that the last displayed complex is perfect in D(A/I) as soon as the complexes
are perfect complexes in D(A/I), see More on Algebra, Lemma 15.74.4. This is true by standard arguments in coherent cohomology using that \mathcal{F}_ X \otimes \Omega ^ q_{X/S_0} is a finite locally free sheaf and X \to S_0 is proper and flat (insert future reference here). Applying More on Algebra, Lemma 15.78.4 we see that
is a perfect object of D(A/I^ n) for all n. This isn't quite enough unless A is Noetherian. Namely, even though I is locally nilpotent by our assumption that p is nilpotent, see Divided Power Algebra, Lemma 23.2.6, we cannot conclude that I^ n = 0 for some n. A counter example is \mathbf{F}_ p\langle x \rangle . To prove it in general when \mathcal{F} = \mathcal{O}_{X/S} the argument of https://math.columbia.edu/~dejong/wordpress/?p=2227 works. When the coefficients \mathcal{F} are non-trivial the argument of [Faltings-very] seems to be as follows. Reduce to the case pA = 0 by More on Algebra, Lemma 15.78.4. In this case the Frobenius map A \to A, a \mapsto a^ p factors as A \to A/I \xrightarrow {\varphi } A (as x^ p = 0 for x \in I). Set X^{(1)} = X \otimes _{A/I, \varphi } A. The absolute Frobenius morphism of X factors through a morphism F_ X : X \to X^{(1)} (a kind of relative Frobenius). Affine locally if X = \mathop{\mathrm{Spec}}(C) then X^{(1)} = \mathop{\mathrm{Spec}}( C \otimes _{A/I, \varphi } A) and F_ X corresponds to C \otimes _{A/I, \varphi } A \to C, c \otimes a \mapsto c^ pa. This defines morphisms of ringed topoi
whose composition is denoted \text{Frob}_ X. One then shows that R\text{Frob}_{X, *}\mathcal{F} is representable by a perfect complex of \mathcal{O}_{X^{(1)}}-modules(!) by a local calculation.
Comments (0)
There are also: