# Rigid Cohomology of Drinfeld's Upper Half Space over a Finite Field

## TL;DR

This paper computes the rigid cohomology of Drinfeld's upper half space over a finite field using two methods, confirming a known cohomology formula that aligns with previous l-adic cohomology results.

## Contribution

It provides a detailed computation of the rigid cohomology of Drinfeld's upper half space via two distinct approaches, enhancing understanding of its cohomological properties.

## Key findings

- Cohomology formula matches previous l-adic cohomology results.
- Two methods yield consistent cohomology computations.
- Confirms known cohomology structure for Drinfeld's upper half space.

## Abstract

In this paper the rigid cohomology of Drinfeld's upper half space over a finite field is computed in two ways. The first method proceeds by computation of the rigid cohomology of the complement of Drinfeld's upper half space in the ambient projective space and then use of the associated long exact sequence for rigid cohomology with proper supports. The second method proceeds by direct computation of rigid cohomology as a direct limit of de Rham cohomologies of a family of strict open neighborhoods of the tube of Drinfeld's upper half space in the ambient rigid-analytic projective space. The resulting cohomology formula has been known since 2007, when Grosse-Kloenne proved that it is the same as the one obtained from l-adic cohomology.