A note on projective dimension over twisted commutative algebras

TL;DR

The paper proves that the projective dimension of certain modules over twisted commutative algebras grows linearly with dimension, confirming a specific conjecture for this class.

Contribution

It establishes the linear growth of projective dimension for modules over free twisted commutative algebras generated in degree one.

Findings

Projective dimension grows linearly with n.

Confirms a conjecture for modules over twisted commutative algebras.

Applicable to modules finitely generated in degree one.

Abstract

Let $M$ be a finitely generated module over a free twisted commutative algebra $A$ that is finitely generated in degree one. We show that the projective dimension of $M({\bf C}^n)$ as an $A({\bf C}^n)$-module is eventually linear as a function of $n$. This confirms a conjecture of Le, Nagel, Nguyen, and R\"omer for a special class of modules.