Relative singularity categories and Gorenstein‐projective modules

By Xiao-Wu Chen. 2007

79 citations recorded by the source corpus.

We introduce the notion of relative singularity category with respect to a self‐orthogonal subcategory ω of an abelian category. We introduce the Frobenius category of ω‐Cohen‐Macaulay objects, and under certain conditions, we show that the stable category of ω‐Cohen‐Macaulay objects is triangle‐equivalent to the relative singularity category. As applications, we rediscover theorems by Buchweitz, Happel and Beligiannis, which relate the stable categories of (unnecessarily finitely‐generated) Gorenstein‐projective modules to the (big) singularity categories of rings. For the case where ω is the additive closure of a self‐orthogonal object, we relate the category of ω‐Cohen‐Macaulay objects to the category of Gorenstein‐projective modules over the opposite endomorphism ring of the self‐orthogonal object. We prove that for a Gorenstein ring, the stable category of Gorenstein‐projective modules is compactly generated and its compact objects coincide with finitely‐generated Gorenstein‐projective modules up to direct summand. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5c3854b0f2593ed0e1d77f3eaa66c9eb35ee19d1

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In Algebra and Number Theory

1. Oyvind Solberg
2. Oystein Skartsaeterhagen
3. Chrysostomos Psaroudakis

In Geometry and Topology

1. Oyvind Solberg
2. Oystein Skartsaeterhagen
3. Chrysostomos Psaroudakis

In Mathematical Physics

1. Oyvind Solberg
2. Oystein Skartsaeterhagen
3. Chrysostomos Psaroudakis

Citations in the corpus

• Gorenstein categories, singular equivalences and finite generation of cohomology rings in recollements (2014, background): “See [18] for a discussion of more general quotients of D(B) by K(X), where X is a selforthogonal subcategory of B.”
• On singular equivalences of Morita type (2013, unknown): “We mention in particular Xiao-Wu Chen’s work here [7, 8, 9, 10], but also Bernhard Keller, Daniel Murfet and Michel Van den Bergh [16] as well as Osamu Iyama, Kiriko Kato and Jun-Ichi Miyachi [14].”
• Totally acyclic complexes over noetherian schemes (2009, unknown): “ete resolutions. This triangulated category has been studied by several authors, including Buchweitz [Buc87], Beligiannis [Bel00], Jørgensen [Jør01, Jør07], Iyengar and Krause [IK06] and Xiao-Wu Chen [Che07]. There is one feature that we need to mention here: the complete projective resolution of an”