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Retractions and Gorenstein Homological Properties

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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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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Xiao-Wu Chen & Yu Ye

  2. Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei, 230026, Anhui, People’s Republic of China

    Xiao-Wu Chen & Yu Ye

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  1. Xiao-Wu Chen
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  2. Yu Ye
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Correspondence to Xiao-Wu Chen.

Additional information

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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