Featured Speaker:  James Myer  

Title:  Introduction to prismatic cohomology

Abstract:  The study of the cohomology of algebraic varieties is depicted by Peter Scholze as a “plane worth” of pairs of primes (p,ℓ), each indexing a cohomology theory for varieties over 𝔽_p with coefficients in 𝔽_ℓ. The singular cohomology occupies a vertical line over ∞; the étale cohomology dances around, avoiding the pairs (p,p); the analytic de Rham cohomology occupies the top right corner, intersecting the singular cohomology @ (∞,∞), symbolizing the classical de Rham comparison theorem, while the diagonal is picked off by the algebraic de Rham cohomology. Zooming in on a point on the diagonal, we begin to wonder whether there is a cohomology theory interpolating between the étale to the crystalline (and de Rham). In fact, the depiction of the plane of pairs of primes is striated by lines from each of the various cohomology theories, but no cohomology theory seems to “wash over” any 2-dimensional part of the picture and “phase in and out” between any one or the other. The prismatic cohomology theory is this “2-dimensional” theory interpolating between the étale and crystalline (and de Rham) theories.

The classical de Rham comparison theorem between the (dual of the) analytic de Rham cohomology and the singular homology offers a geometric interpretation of a (co)homology class as an obstruction to (globally) integrating a differential form. This geometric interpretation loses steam when faced with torsion classes because the integral over a torsion class is always zero. It is also worthwhile to note the relative ease with which we may calculate the de Rham cohomology of a variety (this can be done by machine, e.g. Macaulay2) as opposed to the singular cohomology of a variety. So, how do we detect these torsion cycles algebraically? We will see via a calculation applying the universal coefficients theorem that the hypothesis of equality of dimensions of the analytic and algebraic de Rham cohomology groups of a variety implies lack of torsion in singular cohomology. Somewhat conversely, we’ll see that the presence of torsion in the singular cohomology of the analytic space associated to a variety forces the algebraic de Rham cohomology group to be larger than expected. This interplay between the various cohomology theories for varieties, e.g. singular, étale, analytic de Rham, algebraic de Rham, crystalline, is facilitated by a (specialization of a sequence of) remarkable theorem(s) whose proof depends on the existence of, and motivates the construction of, the prismatic cohomology theory.

Following this introduction, we will venture into some detail, set up some notation for the next speaker, and elaborate a bit more on the story to come.

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