Asymptotic Behavior of Tor over Complete Intersections and Applications

Let R be a local complete intersection and M,N are R-modules such that l(Tor i (M,N)) < ∞ for i ≫ 0. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of l(Tor i (M,N)) using Eisenbud operators and show that they have well-behaved growth. We define and study a function η(M,N) which generalizes Serre’s intersection multiplicity χ(M,N) over regular local rings and Hochster’s function θ(M,N) over local hypersurfaces. We use good properties of η(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.