Etale cohomology groups of F_p-local systems do not behave nicely on general p-adic rigid-analytic spaces. For example, F_p cohomology groups of an affinoid space are usually infinite. This happens even for nice spaces such as the one-dimensional closed unit ball.

However, it turns out the recent theory of perfectoid spaces, developed by P. Scholze, is very useful to understand F_p cohomology groups for p-adic rigid-analytic spaces. For example, Scholze showed that proper rigid-analytic varieties do have finite cohomology groups for any F_p-local system.

I am going to introduce the concept of almost coherent sheaves, and show how it can be used (together with the theory of perfectoid spaces) to give a new proof of the finiteness theorem. It can be also used to prove Poincare Duality for p-torsion coefficients on smooth and proper p-adic rigid-analytic spaces.

This is work in progress. Speaker(s): Bogdan Zavyalov (Stanford University / U(M))