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Triangulated Equivalences Involving Gorenstein Projective Modules

  Published:2017-09-04
 Printed: Dec 2017
  • Yuefei Zheng,
    Department of Applied Mathematics, College of Science, Northwest A&F University, Yangling 712100, Shaanxi Province, China
  • Zhaoyong Huang,
    Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China
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Abstract

For any ring $R$, we show that, in the bounded derived category $D^{b}(\operatorname{Mod} R)$ of left $R$-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ (resp. $\overline{\mathbf{GI}}(\operatorname{Mod} R)$) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if $R$ is a left and right noetherian ring admitting a dualizing complex, then $\underline{\mathbf{GP}}(\operatorname{Mod} R)$ and $\overline{\mathbf{GI}}(\operatorname{Mod} R)$ are equivalent.
Keywords: triangulated equivalence, Gorenstein projective module, stable category, derived category, homotopy category triangulated equivalence, Gorenstein projective module, stable category, derived category, homotopy category
MSC Classifications: 18G25, 16E35 show english descriptions Relative homological algebra, projective classes
Derived categories
18G25 - Relative homological algebra, projective classes
16E35 - Derived categories
 

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