  Let $M,N$ be finitely generated modules over a local complete intersection
$R$. Assume that for each $i>0$, $\mathrm{Tor}^R_i(M,N)=0$. 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.
