Criteria for vanishing of Tor over complete intersections

Olgur Celikbas, Srikanth B. Iyengar, Greg Piepmeyer and Roger Wiegand

Vol. 276 (2015), No. 1, 93–115

DOI: 10.2140/pjm.2015.276.93

Abstract

We exploit properties of Dao’s $\eta$-pairing (see Trans. Amer. Math. Soc. 365:6 (2013), 2803–2821), as well as techniques of Huneke, Jorgensen, and Wiegand (J. Algebra 238:2 (2001), 684–702), to study the vanishing of ${Tor}_i\left(M,N\right)$ for finitely generated modules $M$, $N$ over complete intersections. We prove vanishing of ${Tor}_i\left(M,N\right)$ for all $i\ge 1$ under depth conditions on $M$, $N$, and $M\otimes N$. Our arguments improve a result of Dao and establish a new connection between the vanishing of $Tor$ and the depth of tensor products.

Keywords

complete intersection, tensor product, torsion, vanishing of Tor

Mathematical Subject Classification 2010

Primary: 13C40, 13D07

Milestones

Received: 27 October 2014

Revised: 17 December 2014

Accepted: 17 December 2014

Published: 1 July 2015