Members’ Seminar

It is well known that a finitely generated torsion abelian group A is finite, and thus carries no `rational' information. However, if the torsion group A is not finitely generated, then there exist meaningful ways to extract rational information out of A. For example, the Tate module Q/Z (which, by definition, is the inverse limit of its n-torsion as n varies) is the profinite completion of Z, and thus has rank one in a suitable sense. The goal of this talk is to describe an example of a similar phenomenon recently uncovered in p-adic geometry. More precisely, I will explain an algebro-geometric construction of a chain complex H associated to an algebraic variety X of dimension d over a p-adic field with the following two features: H[1/p] calculates the coherent cohomology of X (and thus has roughly d non-zero cohomology groups) while the p-adic completion of H gives the topological cohomology of X (and thus has roughly 2d non-zero cohomology groups). This two-fold increase in size is explained entirely by the presence of large p-primary torsion in the cohomology of H. The complex H was originally constructed by Scholze using perfectoid spaces, and plays a fundamental role in his work on p-adic Hodge theory.