Torsion in tensor products, and tensor powers, of modules

摘要

For finitely generated modules M and N over a complete intersection R, thevanishing of Tor_i^R(M,N) for all i> 0 gives a tight relationship among depthproperties of M, N and their tensor product. Here we concentrate on theconverse and show, under mild conditions, that the tensor product of M and Nbeing torsion-free (or satisfying higher Serre conditions) forces vanishing ofTor. Special attention is paid to the case of tensor powers of a single module.