Smooth representations and sheaves

TL;DR

This paper explores the geometric interpretation of smooth representations of automorphism groups of universal domains, analyzing their properties and computing cohomology groups for various classes of these representations.

Contribution

It introduces a geometrization approach for smooth automorphism group representations and provides explicit cohomology calculations for certain classes.

Findings

Computed cohomology groups for multiple classes of smooth representations

Established properties of geometric representations of automorphism groups

Enhanced understanding of the structure of smooth automorphism group representations

Abstract

The paper is concerned with `geometrization' of smooth (i.e. with open stabilizers) representations of the automorphism group of universal domains, and with the properties of `geometric' representations of such groups. As an application, we calculate the cohomology groups of several classes of smooth representations of the automorphism group of an algebraically closed extension of infinite transcendence degree of an algebraically closed field.