Cohomological Support and the Geometric Join

Abstract

Let $M,N$ be finitely generated modules over a local complete intersection $R$. Assume that all the modules Tor${}_i^R(M,N)$ are zero for $i>0$. We prove that the cohomological support of $M{\otimes }_{R}N$ (in the sense of Avramov-Buchweitz) is equal to the geometric join of the cohomological supports of $M,N$. This result gives a new connection between two active areas or research, and immediately produces several interesting corollaries. Naturally, it also raises many intriguing new questions about the homological properties of modules over a complete intersection, some of which are investigated in this work.