Algebras Determined by Their Supports

TL;DR

This paper introduces ada algebras, a class of artin algebras characterized by their modules lying in specific parts of the module category, and explores their structure and properties, including conditions for simple connectivity.

Contribution

It defines ada algebras and analyzes their Auslander-Reiten components, showing their representation theory is determined by their tilted support algebras and linking simple connectivity to Hochschild cohomology.

Findings

Representation theory is contained in left and right supports.

Auslander-Reiten components are described explicitly.

Simple connectivity is characterized by Hochschild cohomology vanishing.

Abstract

In this paper, we introduce and study a class of algebras which we call ada algebras. An artin algebra is ada if every indecomposable projective and every indecomposable injective module lies in the union of the left and the right parts of the module category. We describe the Auslander-Reiten components of an ada algebra, showing in particular that its representation theory is entirely contained in that of its left and right supports, which are both tilted algebras. Also, we prove that an ada algebra over an algebraically closed field is simply connected if and only if its first Hochschild cohomology group vanishes.