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Topos Theory

topos theory

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Toposes

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Definition

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.

Properties

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

Logical characterization

Let kk be a ring. Let kRk \to R be a finitely presented kk-algebra. Then the big infinitesimal topos of the Spec(k)Spec(k)-scheme Spec(R)Spec(R) classifies the theory of commutative squares of ring homomorphisms

A B k R \array{A & \rightarrow & B \\ \uparrow &&\uparrow \\ k & \rightarrow& R }

where the rings AA and BB are local, the top arrow ABA \to B is surjective and has nilpotent kernel (i.e. every element of the kernel is nilpotent) This result is due to (Hutzler 2018). By the conditions on the top morphism, it is enough to require that AA or BB is local.

Routine arguments, to be made explicit in a further revision of this entry, allow to generalize this description to the non-affine case. Let f:XSf \colon X \to S be a scheme over SS. Assume that XX is locally of finite presentation over SS. Then the big infinitesimal topos of the SS-scheme XX classifies, as a Sh(X)Sh(X)-topos, the Sh(X)Sh(X)-theory of commutative squares of ring homomorphisms

A B f 1𝒪 S 𝒪 X \array{A & \rightarrow & B \\ \uparrow &&\uparrow \\ f^{-1}\mathcal{O}_S & \rightarrow& \mathcal{O}_X }

where the rings AA and BB are local, the top arrow ABA \to B is surjective and has nilpotent kernel (i.e. every element of the kernel is nilpotent), and both vertical arrows are local (i.e. reflect invertibility).

References

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

A review of some aspects is in

and on page 7 of

In the article

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)).

The result about the geometric theory classified by the big infinitesimal topos appears in