Relative Singularity Categories and Gorenstein-Projective Modules

Details Relative Singularity Categories and Gorenstein-Projective Modules Relative Singularity Categories and Gorenstein-Projective Modules

2007-09-12

arxiv.org/abs/0709.1762v1

Mathematische Nachrichten, 284 (No. 2-3) (2011), 199-212

Mathematics

Rings and Algebras

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Scientific paper

We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory $\omega$ of an abelian category. We introduce the Frobenius category of $\omega$-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of $\omega$-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable category of (unnecessarily finitely-generated) Gorenstein-projective modules with singularity categories of rings. 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 summands.

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