under construction
Let be the etale site of complex schemes of finite type. For a scheme, its infinitesimal site is the big site of the de Rham space :
the site whose objects are pairs of an affine and a morphism from its reduced part ( for the nilradical of ) into .
More generally, for positive characteristic, the definition is more involved than that.
The abelian sheaf cohomology over is the crystalline cohomology of .
Let be a ring. Let be a finitely presented -algebra. Then the big infinitesimal topos of the -scheme classifies the theory of commutative squares of ring homomorphisms
where the rings and are local, the top arrow 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 or 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 be a scheme over . Assume that is locally of finite presentation over . Then the big infinitesimal topos of the -scheme classifies, as a -topos, the -theory of commutative squares of ring homomorphisms
where the rings and are local, the top arrow 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).
An original account of the definition of the crystalline topos is section 7, page 299 of
A review of some aspects:
and on page 7 of
In the article
it is shown that if is proper over an algebraically closed field of characteristic , and embeds into a smooth scheme over , then the infinitesimal cohomology of coincides with etale cohomology with coefficients in (or more generally if we work with the infinitesimal site of over ).
The result about the geometric theory classified by the big infinitesimal topos appears in
Matthias Hutzler, Internal language and classified theories of
toposes in algebraic geometry_, Master’s thesis at the University of Augsburg, 2018, GitLab, pdf download
Matthias Hutzler, Syntactic presentations for glued toposes and for crystalline toposes, Phd. diss. Universität Augsburg 2021. (arXiv:2206.11244)
Last revised on May 4, 2023 at 12:07:07. See the history of this page for a list of all contributions to it.