Representation dimension and finitely generated cohomology

TL;DR

This paper investigates the relationship between representation dimension and module complexity in selfinjective Artin algebras with finitely generated cohomology, providing bounds and new examples of high-dimensional algebras.

Contribution

It establishes that the representation dimension exceeds the maximal module complexity in such algebras, offering a unified method to estimate lower bounds across various algebra classes.

Findings

Representation dimension is greater than the maximal module complexity.

Provides lower bounds for the representation dimension of group and exterior algebras.

Constructs new examples of algebras with arbitrarily large representation dimension.

Abstract

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior algebras and Artin complete intersections. We also obtain new examples of classes of algebras with arbitrarily large representation dimension.