On the $p$-adic pro-étale cohomology of Drinfeld symmetric spaces

Abstract

Via the relative fundamental exact sequence of $p$-adic Hodge theory, we determine the geometric $p$-adic pro-étale cohomology of the Drinfeld symmetric spaces defined over a $p$-adic field, thus giving an alternative proof of a theorem of Colmez-Dospinescu-Niziol. Along the way, we describe, in terms of differential forms, the geometric pro-étale cohomology of the positive de Rham period sheaf on any connected, paracompact, smooth rigid-analytic variety over a $p$-adic field, and we do it with coefficients. A key new ingredient is the condensed mathematics recently developed by Clausen-Scholze.

83 pages; revised and expanded version

References in corpus (2)

• Espaces de Banach-Colmez et faisceaux cohérents sur la courbe de Fargues-Fontaine
• Overconvergent relative de Rham cohomology over the Fargues-Fontaine curve