Obstructions to curvature of modules over Cohen-Macaulay rings
Tony J. Puthenpurakal
TL;DR
The paper investigates obstructions to curvature growth for modules over Cohen–Macaulay local rings that are not complete intersections. By relating the curvature $\operatorname{curv}(M)$, Betti numbers $\beta_n^A(M)$, and multiplicity $e(A)$ through base-change, superficial elements, and syzygy methods, it derives bounds linking $\operatorname{curv}(M)$ and $\operatorname{curv}(k)$. Key results show that the existence of a module with $1\le\operatorname{curv}(M)<\operatorname{curv}(k)$ forces obstructions to both curvatures via inequalities like $e(A)\ge\left(1+\frac{\beta_1(M)}{\beta_0(M)}\right)(1+\operatorname{curv}(k))$, yielding concrete upper bounds $\operatorname{curv}(k)\le\frac{e(A)}{2}-1$ and $\operatorname{curv}(M)<\sqrt{e(A)}-1$. The study also connects Tor-vanishing (and Ext/injective-vanishing) to constraints on the pair of curvatures, showing, for instance, that $\operatorname{Tor}$-vanishing forces $\min\{\operatorname{curv}(M),\operatorname{curv}(N)\}\le\sqrt{e(A)}-1$ and, under infinite projective dimensions, $\max\{\operatorname{curv}(M),\operatorname{curv}(N)\}\le e(A)/2-1$. These results contribute to understanding how curvature behaves under homological vanishing and provide obstructions tied to multiplicity and dimension reduction.
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 $\text{curv}(M) = \limsup_n\sqrt[n]{β_n^A(M)}$. 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 $\text{curv}(N)$.