Commutative algebra in tensor categories

TL;DR

This paper develops foundational aspects of commutative algebra within symmetric tensor categories, aiming to extend classical algebraic geometry results to this abstract setting, with a focus on observable subgroups and tensor functors.

Contribution

It introduces analogues of classical theorems in commutative algebra for tensor categories and explores the concept of observable subgroups in this context.

Findings

Established analogues of classical algebraic theorems in tensor categories

Provided new insights into observable subgroups of affine group schemes

Linked tensor functors to the structure of tensor categories

Abstract

We develop some foundations of commutative algebra, with a view towards algebraic geometry, in symmetric tensor categories. Most results establish analogues of classical theorems, in tensor categories which admit a tensor functor to some tensor category verifying specific conditions. This is in line with the current program which aims to describe tensor categories by their tensor functors to incompressible categories. We place particular emphasis on the notion of observable subgroups of affine group schemes in tensor categories, which in particular leads to some further insight into observability for classical affine group schemes.