Retractions and Gorenstein Homological Properties

• DOI: 10.1007/s10468-013-9415-1
• Publication Date: 2013-03-23T04:46:38Z
• Authors: Xiao-Wu Chen; Yu Ye
• Keywords: Rings and Algebras (math.RA); FOS: Mathematics; Mathematics - Rings and Algebras; Representation Theory (math.RT); 0101 mathematics; 01 natural sciences; Mathematics - Representation Theory

We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free). We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra $A$, there exists a connected self-injective Nakayama algebra $A'$ such that there is a sequence of left retractions linking $A$ to $A'$; in particular, the singularity category of $A$ is triangle equivalent to the stable category of $A'$. We classify connected Nakayama algebras with at most three simple modules according to Gorenstein homological properties.