Relative support varieties

TL;DR

This paper introduces relative support varieties for modules over finite dimensional algebras, demonstrating their properties and utility in understanding module homology and classifying algebra representation types.

Contribution

It defines and studies relative support varieties, establishing their properties and linking them to homological data and algebra classification.

Findings

Relative support varieties share key properties with classical support varieties.

Finite generation conditions allow these varieties to encode homological information.

A new criterion for determining wild representation type of selfinjective algebras is provided.

Abstract

We define relative support varieties with respect to some fixed module over a finite dimensional algebra. These varieties share many of the standard properties of classical support varieties. Moreover, when introducing finite generation conditions on cohomology, we show that relative support varieties contain homological information on the modules involved. As an application, we provide a new criterion for a selfinjective algebra to be of wild representation type.