Proposition 60.17.4. The functor
of Lemma 60.17.3 is an equivalence of categories.
Proposition 60.17.4. The functor
of Lemma 60.17.3 is an equivalence of categories.
Proof. Let (M, \nabla ) be given. We are going to construct a crystal in quasi-coherent modules \mathcal{F}. Write \nabla (m) = \sum \theta _ i(m)\text{d}x_ i. Then \theta _ i \circ \theta _ j = \theta _ j \circ \theta _ i and we can set \theta _ K(m) = (\prod \theta _ i^{k_ i})(m) for any multi-index K = (k_ i) with k_ i \geq 0 and \sum k_ i < \infty .
Let (U, T, \delta ) be any object of \text{Cris}(X/S) with T affine. Say T = \mathop{\mathrm{Spec}}(B) and the ideal of U \to T is J_ B \subset B. By Lemma 60.5.6 there exists an integer e and a morphism
where T_ e = \mathop{\mathrm{Spec}}(D_ e) as in the proof of Lemma 60.17.3. Choose such an e and f; denote f : D \to B also the corresponding divided power A-algebra map. We will set \mathcal{F}_ T equal to the quasi-coherent sheaf of \mathcal{O}_ T-modules associated to the B-module
However, we have to show that this is independent of the choice of f. Suppose that g : D \to B is a second such morphism. Since f and g are morphisms in \text{Cris}(X/S) we see that the image of f - g : D \to B is contained in the divided power ideal J_ B. Write \xi _ i = f(x_ i) - g(x_ i) \in J_ B. By analogy with the proof of Lemma 60.17.3 we define an isomorphism
by the formula
which makes sense by our remarks above and the fact that \nabla is topologically quasi-nilpotent (so the sum is finite!). A computation shows that
if given a third morphism h : (U, T, \delta ) \longrightarrow (X, T_ e, \bar\gamma ). It is also true that c_{f, f} = 1. Hence these maps are all isomorphisms and we see that the module \mathcal{F}_ T is independent of the choice of f.
If a : (U', T', \delta ') \to (U, T, \delta ) is a morphism of affine objects of \text{Cris}(X/S), then choosing f' = f \circ a it is clear that there exists a canonical isomorphism a^*\mathcal{F}_ T \to \mathcal{F}_{T'}. We omit the verification that this map is independent of the choice of f. Using these maps as the restriction maps it is clear that we obtain a crystal in quasi-coherent modules on the full subcategory of \text{Cris}(X/S) consisting of affine objects. We omit the proof that this extends to a crystal on all of \text{Cris}(X/S). We also omit the proof that this procedure is a functor and that it is quasi-inverse to the functor constructed in Lemma 60.17.3. \square
Comments (0)
There are also: