Asymptotic behavior of Tor over complete intersections and applications

TL;DR

This paper studies the asymptotic growth of Tor modules over complete intersection rings, introducing a new invariant that generalizes classical intersection multiplicities and applying it to various homological properties.

Contribution

It introduces the invariant η^R(M,N) to analyze asymptotic Tor behavior over complete intersections, extending classical multiplicity concepts.

Findings

Tor modules have well-behaved asymptotic growth

η^R(M,N) generalizes Serre's and Hochster's multiplicities

Results on Tor vanishing, depth, and dimension over complete intersections

Abstract

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.