Twisted support varieties

TL;DR

This paper introduces twisted support varieties for modules over Artin algebras, exploring their properties under automorphisms, and relates them to module periodicity and Dade's Lemma, with applications to Frobenius algebras.

Contribution

It defines twisted support varieties induced by automorphisms and establishes their key properties, linking them to module periodicity and extending classical results.

Findings

Twisted support varieties satisfy Dade's Lemma.

The variety is one-dimensional if and only if the module is automorphism-periodic.

Results apply to DTr-periodic modules over Frobenius algebras.

Abstract

We define and study twisted support varieties for modules over an Artin algebra, where the twist is induced by an automorphism of the algebra. Under a certain finite generation hypothesis, we show that the twisted variety of a module satisfies Dade's Lemma and is one dimensional precisely when the module is periodic with respect to the twisting automorphism. As a special case we obtain results on DTr-periodic modules over Frobenius algebras.