Cohomological support and the geometric join

TL;DR

This paper establishes a new relationship between the cohomological support of tensor products of modules over a complete intersection and the geometric join of their supports, linking homological algebra with geometric concepts.

Contribution

It proves that under certain conditions, the cohomological support of the tensor product equals the geometric join of individual supports, revealing a novel connection between algebraic and geometric properties.

Findings

Cohomological support of tensor product equals geometric join of supports.

New insights into homological properties of modules over complete intersections.

Several surprising corollaries and open questions are presented.

Abstract

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.