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October 2020 Relative singularity categories II
Huanhuan Li, Zhaoyong Huang
Kodai Math. J. 43(3): 431-453 (October 2020). DOI: 10.2996/kmj/1605063623

Abstract

Let $\mathscr{A}$ be an abelian category with enough projective objects and $\mathscr{C}$ an additive and full subcategory of $\mathscr{A}$, and let $\mathscr{G}(\mathscr{C})$ be the Gorenstein category of $\mathscr{C}$. We study the properties of the $\mathscr{C}$-derived category $D_\mathscr{C}^b(\mathscr{A})$, $\mathscr{C}$-singularity category $D_{\mathscr{C}-sg}(\mathscr{A})$ and $\mathscr{G}(\mathscr{C})$-defect category $D_{\mathscr{G(C)}-defect}(\mathscr{A})$ of $\mathscr{A}$. Let $\mathscr{C}$ be admissible in $\mathscr{A}$. We show that $D_{\mathscr{G(C)}-defect}(\mathscr{A})\simeq D_{\mathscr{C}-sg}(\mathscr{A})$ if and only if $\mathscr{C}=\mathscr{G(C)}$; and $D_{\mathscr{G(C)}-defect}(\mathscr{A})=0$ if and only if the stable category $\underline{\mathscr{G}(\mathscr{C})}$ of $\mathscr{G}(\mathscr{C})$ is triangle-equivalent to $D_{\mathscr{C}-sg}(\mathscr{A})$, and if and only if every object in $\mathscr{A}$ has finite $\mathscr{C}$-proper $\mathscr{G}(\mathscr{C})$-dimension. Then we apply these results to module categories. We prove that under some condition, the Gorenstein derived equivalence of artin algebras induces the Gorenstein singularity equivalence. Finally, for an artin algebra $A$, we establish the stability of Gorenstein defect categories of $A$.

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Huanhuan Li. Zhaoyong Huang. "Relative singularity categories II." Kodai Math. J. 43 (3) 431 - 453, October 2020. https://doi.org/10.2996/kmj/1605063623

Information

Published: October 2020
First available in Project Euclid: 11 November 2020

MathSciNet: MR4173159
Digital Object Identifier: 10.2996/kmj/1605063623

Rights: Copyright © 2020 Tokyo Institute of Technology, Department of Mathematics

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Vol.43 • No. 3 • October 2020
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