A note on projective dimension over twisted commutative algebras

TL;DR

The paper studies the asymptotics of homological invariants for GL-equivariant modules over twisted commutative algebras. It proves that for finitely generated $A$-modules $M$, the projective dimension and depth of $M(\mathbb{C}^n)$ as $A(\mathbb{C}^n)$-modules grow linearly with $n$ (slope at most $d$), by establishing that the key invariant $\gamma_M(n)$ is eventually linear and relating homological data to a GL-representation-theoretic framework via $K(A)$ and the formal character $\Theta_M$. In the rank-$\le r$ case, explicit formulas $\mathrm{pdim}_M(n)=(d-r)n-(d-r)r$ and $\mathrm{depth}_M(n)=rn+r(d-r)$ reproduce the Auslander–Buchsbaum relation and illuminate the general structure through the linear strands $F_k(M)$. The Krull dimension of quotient tca's $B$ is also shown to be eventually linear in $n$, with precise dependence on irreducible components of $\mathrm{Spec}(B)$ and a Grassmannian-fibration argument. These results confirm the Le–Nagel–Nguyen–Römer conjecture in the GL case and enhance the understanding of asymptotic invariants in equivariant commutative algebra.

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 Römer for a special class of modules.

Theorems & Definitions (13)

• Theorem 3.1
• Example 3.2
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['thm:gamma']}
• Remark 3.4
• Remark 3.5
• Theorem 4.1
• Example 4.2
• Lemma 4.3