On support varieties for modules over complete intersections

TL;DR

This paper demonstrates that over local complete intersections, any variety can be realized as a module's support variety, and the support variety of certain modules is connected, revealing deep geometric properties of modules.

Contribution

It establishes that all varieties can be realized as support varieties over complete intersections and proves connectedness of support varieties for indecomposable maximal Cohen-Macaulay modules.

Findings

Any variety can be realized as a support variety over a local complete intersection.

Support varieties of indecomposable maximal Cohen-Macaulay modules are connected.

Provides geometric insights into module theory over complete intersections.

Abstract

We show that, over a local complete intersection, every possible variety is realized as the cohomological support variety of some module. Moreover, we show that the projective variety of a complete indecomposable maximal Cohen-Macaulay module is connected.