crystalline site



Topos Theory

topos theory



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Cohomology and homotopy

In higher category theory


under construction



Let C etC_{et} be the etale site of complex schemes of finite type. For XX a scheme, its infinitesimal site Cris(X)Cris(X) is the big site C et/X dRC_{et}/X_{dR} of the de Rham space X dR:C etSetX_{dR} : C_{et} \to Set:

the site whose objects are pairs (SpecA,(SpecA) redX)(Spec A, (Spec A)_{red} \to X) of an affine SpecASpec A and a morphism from its reduced part ((SpecA) red=Spec(A/I)(Spec A)_{red} = Spec (A/I) for II the nilradical of AA) into XX.

More generally, for positive characteristic, the definition is more involved than that.


The abelian sheaf cohomology over Cris(X)Cris(X) is the crystalline cohomology of XX.


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)

A review of some aspects is in

and on page 7 of

In the article

  • Arthur Ogus, Cohomology of the infinitesimal site Annales scientifiques de l’École Normale Supérieure, Sér. 4, 8 no. 3 (1975), p. 295-318 (numdam)

it is shown that if XX is proper over an algebraically closed field kk of characteristic pp, and embeds into a smooth scheme over kk, then the infinitesimal cohomology of XX coincides with etale cohomology with coefficients in kk (or more generally W n(k)W_n(k) if we work with the infinitesimal site of XX over W n(k)W_n(k)).

Last revised on March 30, 2011 at 08:47:59. See the history of this page for a list of all contributions to it.