Mod-p Poincare Duality in p-adic Analytic Geometry
Etale cohomology of Fp-local systems does not behave nicely on general smooth p-adic rigid-analytic spaces; e.g., the Fp-cohomology of the 1-dimensional closed unit ball is infinite.
However, it turns out that the situation is much better if one considers only proper rigid-analytic spaces. These spaces have finite Fp cohomology groups and these groups satisfy Poincare Duality if X is smooth and proper.
I will explain how one can prove such results using the concept of almost coherent sheaves that allows to "localize" such questions in an appropriate sense and reduce to some local computations.