Multiple Tor modules: rigidity and Mayer-Vietoris spectral sequences

TL;DR

This paper develops a homological theory for spectral sequences from multiple complexes, extending properties of Tor modules to multiple ideals and analyzing their support regions in multigraded settings.

Contribution

It introduces new complexes related to sums and products of ideals, establishing their homologies and extending rigidity and Tor-independence properties.

Findings

Extended rigidity property to multiple ideals.

Developed a homological theory for spectral sequences from multiple complexes.

Described support regions of Tor modules in multigraded settings.

Abstract

We extend some properties of a pair of ideals described in terms of Tor modules to any number of ideals, including the well-known rigidity property. Those extensions require the development of a homological theory for spectral sequences arising from multiple complexes. Out of this theory, two new complexes associated with quotients by sums and quotients by products of the given ideals emerge, and their homologies are related via the Tor-independence property. In the multigraded setting, we describe the support regions of Tor modules for quotients by sums and products of ideals generated by variables in terms of each other.