Crystalline cohomology of rigid analytic spaces

TL;DR

This paper introduces a new form of infinitesimal cohomology for rigid analytic spaces, extending the theory to non-smooth cases with coefficients in p-adic fields or Fontaine's de Rham ring.

Contribution

It develops a novel cohomology theory for rigid analytic spaces that broadens the scope beyond smooth varieties, incorporating p-adic and Fontaine's period ring coefficients.

Findings

Defines infinitesimal cohomology for non-smooth rigid spaces

Extends cohomological tools to p-adic and Fontaine's rings

Provides foundational results for further research in p-adic geometry

Abstract

In this article, we introduce infinitesimal cohomology for rigid analytic spaces that are not necessarily smooth, with coefficients in a p-adic field or Fontaine's de Rham period ring.