Abstract
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 algebras). 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.
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The authors are supported by National Natural Science Foundation of China (No.s 10971206 and 11201446), and NECT-12-0507.
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Chen, XW., Ye, Y. Retractions and Gorenstein Homological Properties. Algebr Represent Theor 17, 713–733 (2014). https://doi.org/10.1007/s10468-013-9415-1
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DOI: https://doi.org/10.1007/s10468-013-9415-1