Injective Presentations of Induced Modules over Cluster-Tilted Algebras

TL;DR

This paper provides an explicit construction linking injective presentations of modules over a tilted algebra to those over its cluster-tilted extension, offering new insights into their homological properties.

Contribution

It introduces a method to explicitly construct injective presentations of induced modules over cluster-tilted algebras, connecting them to the underlying tilted algebra modules.

Findings

Constructs injective presentations for induced modules over cluster-tilted algebras.

Provides a module-theoretic proof of the 1-Gorenstein property of cluster-tilted algebras.

Extends understanding of homological relations between tilted and cluster-tilted algebras.

Abstract

Every cluster-tilted algebra $B$ is the relation extension $C\ltimes \text{Ext}^2_C(DC,C)$ of a tilted algebra $C$. A $B$-module is called induced if it is of the form $M\otimes_C B$ for some $C$-module $M$. We study the relation between the injective presentations of a $C$-module and the injective presentations of the induced $B$-module. Our main result is an explicit construction of the modules and morphisms in an injective presentation of any induced $B$-module. In the case where the $C$-module, and hence the $B$-module, is projective, our construction yields an injective resolution. In particular, it gives a module theoretic proof of the well-known 1-Gorenstein property of cluster-tilted algebras.