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2007

Relative Singularity Categories and Gorenstein-Projective Modules

Chen, Xiao-Wu

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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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