Realizability and the Avrunin-Scott theorem for higher-order support varieties

TL;DR

This paper introduces higher-order support varieties for modules over complete intersection rings and establishes a higher-order Avrunin-Scott theorem linking these varieties with rank varieties in the context of finite elementary abelian groups.

Contribution

It defines higher-order support varieties and provides a complete characterization of their possible forms, extending classical support variety theory.

Findings

Complete description of which higher-order support varieties occur.

Proof of a higher-order Avrunin-Scott theorem linking support and rank varieties.

Extension of support variety theory to higher-order contexts.

Abstract

We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring, and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a finite elementary abelian group, we also prove a higher-order Avrunin-Scott-type theorem, linking higher-order support varieties and higher-order rank varieties for pairs of modules.