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

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

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

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 $X$ is proper over an algebraically closed field $k$ of characteristic$p$, and embeds into a smooth scheme over $k$, then the infinitesimal cohomology of $X$ coincides with etale cohomology with coefficients in $k$ (or more generally $W_n(k)$ if we work with the infinitesimal site of $X$ over $W_n(k)$).

Last revised on March 30, 2011 at 08:47:59.
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