Asymptotic behavior of Tor over complete intersections and applications

Let $R$ be a local complete intersection and $M,N$ are $R$-modules such that $\ell(\Tor_i^R(M,N))<\infty$ for $i\gg 0$. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of $\ell(\Tor_i^R(M,N))$ 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 $\Tor$ and $\Ext$, vanishing of $\Tor$, depth of tensor products, and dimensions of intersecting modules over local complete intersections.