Invariant rings through categories

TL;DR

This paper introduces a categorical framework for invariant theory, showing that a condition called adequacy ensures finite generation of invariant rings across various algebraic categories.

Contribution

It defines a new categorical notion of geometric reductivity (adequacy) and proves it guarantees finite generation of invariants in diverse settings.

Findings

Adequacy implies finite generation of invariant rings.

Applicable to modules and comodules over bialgebras.

Extends to quasi-coherent sheaves on algebraic stacks.

Abstract

We formulate a notion of "geometric reductivity" in an abstract categorical setting which we refer to as adequacy. The main theorem states that the adequacy condition implies that the ring of invariants is finitely generated. This result applies to the category of modules over a bialgebra, the category of comodules over a bialgebra, and the category of quasi-coherent sheaves on a finite type algebraic stack over an affine base.