Obstructions to curvature of modules over Cohen-Macaulay rings

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring with residue field $k$. If $M$ is a finitely generated $A$-module then set $\text{curv}(M)={\mathrm{lim\hspace{0.17em}sup}}_n\sqrt[n]{{\beta }_n^A(M)}$. We show that under mild hypotheses the existence of a single module $M$ with $1\le \text{curv}(M)<\text{curv}(k)$ imposes obstructions to both $\text{curv}(k)$ and $\text{curv}(M)$. Similarly we show that the condition ${\text{Tor}}_n^A(M,N)=0$ for $n\gg 0$ imposes constraints on both $\text{curv}(M)$ and $\text{curv}(N)$.