Ind-Banach approach to Grothendieck duality in Rigid-analytic geometry

Abstract

We prove a duality theorem for quasi-compactly supported cohomology of quasi-coherent sheaves on rigid-analytic spaces, with respect to a smooth and Kiehl partially-proper morphism. This includes an identification of the dualizing object with volume forms. The functional analysis underlying our theory does not use condensed mathematics, but rather Ind-Banach spaces, following Ben-Bassat--Kelly--Kremnizer. Nevertheless, our overall strategy is inspired by that of Clausen--Scholze in the complex-analytic setting.