Espaces de Banach-Colmez et faisceaux coh\'erents sur la courbe de Fargues-Fontaine

TL;DR

This paper introduces a new, simplified definition of Banach-Colmez spaces and clarifies their relationship with coherent sheaves on the Fargues-Fontaine curve, also describing related cohomology computations.

Contribution

It provides an equivalent, simpler definition of Banach-Colmez spaces and links them to coherent sheaves via t-structure changes, enriching the understanding of their interplay.

Findings

New simplified definition of Banach-Colmez spaces

Explicit relationship with coherent sheaves on the Fargues-Fontaine curve

Description of pro-étale cohomology of open disk and affine space

Abstract

We give a new definition, simpler but equivalent, of the abelian category of Banach-Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues-Fontaine curve. One goes from one category to the other by changing the t-structure on the derived category. Along the way, we obtain a description of the pro-\'etale cohomology of the open disk and the affine space, of independent interest.