Vanishing of Tor, and why we care about it

Abstract

Given finitely generated modules $M$ and $N$ over a local ring $R$, the tensor product $M{\otimes }_{R}N$ typically has nonzero torsion. Indeed, the assumption that the tensor product is torsion-free influences the structure and vanishing of the modules $\Tor^R_i(M,N)$ for all $i\ge 1$. In turn, the vanishing of $\Tor^R_i(M,N)$ imposes restrictions on the depth properties of the modules $M$ and $N$. These connections made their first appearance in Auslander's 1961 paper "Modules over unramified regular local rings". We will survey the literature on these topics, with emphasis on progress during the past twenty years.

21 pages, slightly revised version

References in corpus (11)

• Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations
• Auslander-Reiten conjecture and Auslander-Reiten duality
• Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension
• Asymptotic behavior of Tor over complete intersections and applications
• Vanishing of Tor Over Complete Intersections
• Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free
• Syzygies and tensor product of modules
• Vanishing of Tate homology and depth formulas over local rings
• Torsion in tensor products, and tensor powers, of modules
• A note on the depth formula and vanishing of cohomology
• Asymmetric complete resolutions and vanishing of Ext over Gorenstein rings

Cited by in corpus (6)

• On the ideal case of a conjecture of Huneke and Wiegand
• Bounds on depth of tensor products of modules
• Factorization Properties of Leamer Monoids
• Maximal Cohen-Macaulay tensor products
• On the vanishing of theta invariant and a conjecture of Huneke and Wiegand
• On the vanishing of (co)homology for modules admitting certain filtrations