Two homological proofs of the Noetherianity of $FI_G$

TL;DR

This paper presents two homological proofs demonstrating the Noetherian property of the category $FI$, a key concept in algebraic and representation theory, originally established by Church, Ellenberg, Farb, and Nagpal.

Contribution

It introduces two novel homological methods to prove the Noetherianity of the $FI$ category, providing alternative approaches to a foundational algebraic result.

Findings

Two distinct homological proofs of $FI$'s Noetherianity

Enhanced understanding of $FI$ category structure

Potential applications to related algebraic categories

Abstract

We give two homological proofs of the Noetherianity of the category $FI$, a fundamental result discovered by Church, Ellenberg, Farb, and Nagpal.