# Galois irreducibility implies cohomology freeness for KHT Shimura varieties

## TL;DR

This paper proves that the localized cohomology groups of certain KHT Shimura varieties are free when associated Galois representations are irreducible, under specific field extension conditions.

## Contribution

It extends previous results by establishing cohomology freeness for a broader class of maximal ideals linked to irreducible Galois representations under new field extension hypotheses.

## Key findings

- Cohomology groups are free for irreducible Galois representations.
- The result applies under the condition that the degree of a specific field extension exceeds the variety’s dimension.
- Generalizes prior work to include cases with irreducible Galois representations.

## Abstract

Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{\rho_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2i\pi/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimura variety and $l$ the residual characteristic.