Asymptotic behavior of Tor over complete intersections and applications

Let $R$ be a local complete intersection and $M,N$ are $R$-modules such that $\mathrm{\ell }({\text{Tor}}_i^R(M,N))<\mathrm{\infty }$ for $i\gg 0$. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of $\mathrm{\ell }({\text{Tor}}_i^R(M,N))$ using Eisenbud operators and show that they have well-behaved growth. We define and study a function ${\eta }^R(M,N)$ which generalizes Serre’s intersection multiplicity ${\chi }^R(M,N)$ over regular local rings and Hochster’s function ${\theta }^R(M,N)$ over local hypersurfaces. We use good properties of ${\eta }^R(M,N)$ to obtain various results on complexities of $\text{Tor}$ and $\text{Ext}$, vanishing of $\text{Tor}$, depth of tensor products, and dimensions of intersecting modules over local complete intersections.