The infinitesimal site and algebraic de Rham cohomology

Abstract: The de Rham cohomology of the analytification of a smooth projective variety over $\mathbb{C}$ can be computed via an algebraic de Rham complex. Unfortunately, the algebraic de Rham complex is somewhat poorly behaved in positive characteristic. To solve this problem, Grothendieck showed first how to reinterpret de Rham cohomology in characteristic 0 as cohomology on a site (the infinitesimal site), and second how to modify the infinitesimal site to obtain a site that works well also in characteristic p (the crystalline site).

In this talk, we will explain algebraic de Rham cohomology and define the infinitesimal and stratifying sites. We also will define the notion of a classical Weil cohomology theory, which de Rham cohomology (char 0) and crystalline cohomology give examples of.

algebraic geometrynumber theory

Audience: advanced learners

( slides )

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2025 topic: Weil conjectures.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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