On Hodge--Witt cohomology of Drinfeld's upper half space over a finite field

TL;DR

This paper investigates the Hodge-Witt cohomology of Drinfeld's upper half space over finite fields, introducing a differential operator theory over Witt vectors to analyze the structure of associated cohomology modules.

Contribution

It develops a theory of differential operators over Witt vectors for smooth schemes and applies it to study the module structure of local cohomology groups in Hodge-Witt cohomology.

Findings

Local cohomology groups are finitely generated modules over differential operator sheaves.

The action of Galois groups on cohomology is explicitly described.

New tools for analyzing cohomology of p-adic and finite field varieties.

Abstract

In this dissertation we study the Hodge-Witt cohomology of the $d$-dimensional Drinfeld's upper half space $\mathcal{X} \subset \mathbb{P}_k^d$ over a finite field $k$. We consider the natural action of the $k$-rational points $G$ of the linear group $\mathrm{GL}_{d+1}$ on $H^0(\mathcal{X},\mathrm{W}_n\Omega_{\mathbb{P}_k^d}^i)$, making them natural $\mathrm{W}_n(k)[G]$-modules. To study these representations, we introduce a theory of differential operators over the Witt vectors for smooth $k$-schemes $X$, through a quasi-coherent sheaf of $\mathrm{W}_n(k)$-algebras $\mathcal{D}_{\mathrm{W}_n(X)}$. We apply this theory to equip suitable local cohomology groups arising from $H^0(\mathcal{X},\mathrm{W}_n\mathcal{O}_{\mathbb{P}_k^d})$ with a $\Gamma(\mathbb{P}_k^d,\mathcal{D}_{\mathrm{W}_n(\mathbb{P}_k^d)})$-module structure. Those local cohomology groups are naturally modules over some parabolic subgroup of $\mathrm{GL}_{d+1}(k)$, and we prove that they are finitely generated $\Gamma(\mathbb{P}_k^d,\mathcal{D}_{\mathrm{W}_n(\mathbb{P}_k^d)})$-modules.