Cohomological support and the geometric join

Abstract

Let $M,N$ be finitely generated modules over a local complete intersection $R$. Assume that for each $i>0$, ${\mathrm{Tor}}_i^R(M,N)=0$. 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 $M,N$. Such result gives a new connection between two active areas or research, and immediately produces several surprising corollaries. Naturally, it also raises many intriguing new questions about the homological properties of modules over a complete intersection, some of those are investigated in the second half of this note.

Minor typos are fixed