cohomology

# Contents

## Idea

Crystalline cohomology is the abelian sheaf cohomology with respect to the crystalline site of a scheme. Hence, put more generally, it is the cohomology of de Rham spaces/coreduced objects.

Crystalline cohomology serves to refine the notion of de Rham cohomology for schemes.

Crystalline cohomology is in particular a Weil cohomology? and is generalized by the notion of rigid cohomology.

## Definition

Let $X$ be a scheme overy a base $S$. The crystalline site $\mathrm{Cris}\left(X/S\right)$ of $X$ is

• the category whose objects are all nilpotent $S$-immersions $U↪T$, where $U$ is an open set of $X$ and and the ideal on $T$ defining this immersion being endowed with a nilpotent divided power structure (…details…).;

• the Grothendieck topology on this category is the Zariski topology.

If $S$ is of characteristic 0, then $\mathrm{Cris}\left(X/S\right)$ coincides with the infinitesimal site of $X$. (…details…).

## References

An original account of the definition of the crystalline topos is section 7, page 299 of

• Alexander Grothendieck, Crystals and de Rham cohomology of schemes , chapter IX in Dix Exposes sur la cohomologie des schema (pdf)

• Jacob Lurie, Notes on Crystals and algebraic D-modules (pdf)

• wikipedia crystalline cohomology

• P. Berthelot, A. Ogus, Notes on crystalline cohomology, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9

• P. Berthelot, A. Ogus, $F$-Isocrystals and de Rham Cohomology, I, Invent. math. 72, 1983, pp. 159-199

Revised on January 5, 2013 21:03:53 by Urs Schreiber (89.204.138.93)