Jennetta George: Representation Theory of Noetherian Categories

Abstract: In this paper, we examine the main results of Sam and Snowden in their paper titled "A Gröbner Approach to Combinatorial Categories”. The work concerns the category of functors from a small category C to the category of modules over a ring. The functor category from C to an arbitrary category D is the category whose objects are all functors from C to D and whose morphisms are all natural transformations between such functors. When we let D be the category M of modules over k , the functor category from C to M is called the representation category of C, Rep(C).

One category of particular interest to us is FI, the category of finite sets and injections. Representations of FI come up in the study of cohomology of configuration spaces in the work of Benson Farb, Thomas Church, Jordan S. Ellenberg, and Rohit Nagpal to encode sequences of representations of symmetric groups.

In the study of categories of representations, a basic question one may ask is whether or not a certain category is Noetherian. The purpose of this paper, in coherence with the work of Sam and Snowden, is to develop general criteria for proving a a category is Noetherian. We will use the approach of Gröbner bases of representations of categories to prove our results.