Remark 60.23.2. The proof of Proposition 60.23.1 shows that the conclusion
for i > 0 is true for any \mathcal{O}_{X/S}-module \mathcal{F} which satisfies conditions (1) and (2) of Proposition 60.21.1. This applies to the following non-crystals: \Omega ^ i_{X/S} for all i, and any sheaf of the form \underline{\mathcal{F}}, where \mathcal{F} is a quasi-coherent \mathcal{O}_ X-module. In particular, it applies to the sheaf \underline{\mathcal{O}_ X} = \underline{\mathbf{G}_ a}. But note that we need something like Lemma 60.15.1 to produce a de Rham complex which requires \mathcal{F} to be a crystal. Hence (currently) the collection of sheaves of modules for which the full statement of Proposition 60.23.1 holds is exactly the category of crystals in quasi-coherent modules.
Comments (0)