Special and general types
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.
Let be a scheme overy a base . The crystalline site of is
the category whose objects are all nilpotent -immersions , where is an open set of and and the ideal on defining this immersion being endowed with a nilpotent divided power structure (…details…).;
the Grothendieck topology on this category is the Zariski topology.
If is of characteristic 0, then coincides with the infinitesimal site of . (…details…).
Related entries: crystal, infinitesimal site, rigid cohomology, Dieudonné module, Monsky-Washnitzer cohomology, Grothendieck connection
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, -Isocrystals and de Rham Cohomology, I, Invent. math. 72, 1983, pp. 159-199
Revised on January 5, 2013 21:03:53
by Urs Schreiber