Cohomology of Non-Smooth Rigid Analytic Spaces

Abstract

This thesis considers the cohomology theories of rigid analytic spaces, with a focus on spaces that might be singular. Analogous to the complex algebraic geometry, we generalize derived de Rham cohomology, infinitesimal cohomology, and Deligne–Du Bois cohomology to their rigid analytic counterparts, and study their relations. We also consider the p-adic etale cohomology of rigid analytic spaces, extending the Hodge–Tate decomposition theorem of Faltings and Scholze to non-smooth rigid analytic spaces. The strategy to the latter is the simplicial method and the resolution of singularities. Furthermore, joint with Shizhang Li, we reproduce the period sheaves and the p-adic Poincaré sequence in p-adic Hodge theory, using the derived de Rham complex.