Obstructions to curvature of modules over Cohen-Macaulay rings

TL;DR

This paper investigates how the existence of modules with certain curvature properties over Cohen-Macaulay rings imposes restrictions on the curvature of the residue field and related modules, revealing obstructions to their possible values.

Contribution

It establishes new constraints on module curvature over Cohen-Macaulay rings based on the existence of modules with specific curvature bounds and vanishing Tor conditions.

Findings

Existence of modules with curvature between 1 and curvature of the residue field imposes obstructions.

Vanishing Tor conditions impose constraints on the curvature of modules involved.

Results reveal obstructions to the possible values of module and residue field curvatures.

Abstract

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with residue field $k$. If $M$ is a finitely generated $A$-module then set $\text{curv}(M) = \limsup_n\sqrt[n]{\beta_n^A(M)}$. We show that under mild hypotheses the existence of a single module $M$ with $1 \leq \text{curv}(M) < \text{curv}(k)$ imposes obstructions to both $\text{curv}(k)$ and $\text{curv}(M)$. Similarly we show that the condition $\text{Tor}^A_n(M, N) = 0$ for $n \gg 0$ imposes constraints on both $\text{curv}(M)$ and $\text{curv}(N)$.