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 $\text{ curv}(N)$.
Categories: math.AC