Categories of modules for elementary abelian p-groups and generalized Beilinson algebras

TL;DR

This paper studies modules over elementary abelian p-groups using a functor from generalized Beilinson algebras, providing a homological framework that distinguishes properties across different ranks and generalizes known modules.

Contribution

It introduces a homological characterization of modules via a functor from generalized Beilinson algebras, extending the understanding of module categories for elementary abelian p-groups.

Findings

Homological methods distinguish module properties for different ranks.

Contrast between r=2 and r>2 cases for equal images modules.

Generalization of W-modules by Carlson, Friedlander, and Suslin.

Abstract

In this paper, we approach the study of modules of constant Jordan type and equal images modules over elementary abelian p-groups E_r of rank r \geq 2 by exploiting a functor from the module category of a generalized Beilinson algebra B(n,r), n \leq p, to mod E_r. We define analogs of the above mentioned properties in mod B(n,r) and give a homological characterization of the resulting subcategories via a P^{r-1}-family of B(n,r)-modules of projective dimension one. This enables us to apply homological methods from Auslander-Reiten theory and thereby arrive at results that, in particular, contrast the findings for equal images modules of Loewy length two over E_2 with the case r > 2. Moreover, we give a generalization of the W-modules defined by Carlson, Friedlander and Suslin.