Asymptotic behavior of Tor over complete intersections and applications

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.

Cited by in corpus (8)

• The radius of a subcategory of modules
• Vanishing of Tor, and why we care about it
• Bounds on depth of tensor products of modules
• Decent intersection and Tor-rigidity for modules over local hypersurfaces
• Torsion in tensor products, and tensor powers, of modules
• On the vanishing of theta invariant and a conjecture of Huneke and Wiegand
• The vanishing of a higher codimension analog of Hochster's theta invariant
• On the ideal case of a conjecture of Auslander and Reiten