[Paper Review] Relative Singularity Categories and Gorenstein-Projective Modules

Xiao‐Wu Chen|arXiv (Cornell University)|Sep 12, 2007

Algebraic structures and combinatorial models15 references4 citations

TL;DR

This paper introduces the relative singularity category with respect to a self-orthogonal subcategory 𝜔 in an abelian category, establishing a triangle-equivalence between the stable category of 𝜔-Cohen-Macaulay objects and the relative singularity category under suitable conditions. The key contribution extends Buchweitz-Happel duality to Gorenstein rings, showing that the stable category of Gorenstein-projective modules is compactly generated, with compact objects precisely the finitely-generated Gorenstein-projective modules up to direct summands.

ABSTRACT

We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory $ω$ of an abelian category. We introduce the Frobenius category of $ω$-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of $ω$-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.

Motivation & Objective

• To define and study the relative singularity category $ D_{\omega}(\mathcal{A}) $ with respect to a self-orthogonal subcategory $ \omega \subseteq \mathcal{A} $, generalizing the classical singularity category.
• To introduce the Frobenius category of $ \omega $-Cohen-Macaulay objects and establish its stable category as a triangulated quotient.
• To relate the stable category of Gorenstein-projective modules over a Gorenstein ring to singularity categories via relative singularity categories.
• To prove that the stable category of Gorenstein-projective modules over a Gorenstein ring is compactly generated, with compact objects corresponding to finitely-generated Gorenstein-projective modules up to direct summands.
• To generalize Buchweitz-Happel duality to the case of possibly infinitely-generated Gorenstein-projective modules.

Proposed method

• Define the relative singularity category $ D_{\omega}(\mathcal{A}) $ as the Verdier quotient $ D^b(\mathcal{A}) / K^b(\omega) $, where $ \omega $ is self-orthogonal.
• Introduce the subcategories $ \widehat{\omega} $, $ {}_\omega\mathcal{X} $, and $ \omega^\perp $, and show that under self-orthogonality, $ \omega \subseteq \widehat{\omega} \subseteq {}_\omega\mathcal{X} \subseteq \omega^\perp $.
• Construct a full exact embedding from the stable category of $ \omega $-Cohen-Macaulay objects into $ D_{\omega}(\mathcal{A}) $, and prove it is an equivalence under reasonable conditions.
• Use the dimension-shift technique in homological algebra to show that exact complexes of projectives have cocycles in $ {}^\perp R\text{-Proj} $, hence in $ R\text{-Gproj} $.
• Apply the Thomason-Trobaugh-Yao-Neeman theorem on compact generation of quotient triangulated categories to show that $ R\text{-\underline{GProj}} $ is compactly generated.
• Use the functor $ Z^0 $, taking zeroth cocycles, and the functor $ \mathbf{a} $, which assigns to a complex its associated complete resolution, to relate $ K(R\text{-Proj}) $ and $ R\text{-\underline{GProj}} $.

Experimental results

Research questions

• RQ1 Under what conditions is the stable category of $ \omega $-Cohen-Macaulay objects triangle-equivalent to the relative singularity category $ D_{\omega}(\mathcal{A}) $?
• RQ2 How does the relative singularity category relate to the classical singularity category and to tilting subcategories?
• RQ3 Is the stable category of Gorenstein-projective modules over a Gorenstein ring compactly generated?
• RQ4 Do the compact objects in the stable category of Gorenstein-projective modules coincide with the finitely-generated Gorenstein-projective modules up to direct summands?
• RQ5 Can Buchweitz-Happel duality be extended to the case of possibly infinitely-generated Gorenstein-projective modules?

Key findings

• The stable category of $ \omega $-Cohen-Macaulay objects embeds fully and faithfully into the relative singularity category $ D_{\omega}(\mathcal{A}) $, and this embedding becomes an equivalence under suitable conditions.
• For a Gorenstein ring $ R $, the stable category $ R\text{-\underline{GProj}} $ of Gorenstein-projective modules is compactly generated as a triangulated category.
• The subcategory of compact objects in $ R\text{-\underline{GProj}} $ consists precisely of the direct summands of finitely-generated Gorenstein-projective $ R $-modules.
• The functor $ Z^0: K^{\rm ex}(R\text{-Proj}) \to R\text{-\underline{GProj}} $ is a triangle-equivalence, linking exact complexes of projectives to Gorenstein-projective modules.
• The relative singularity category $ D_{\omega}(\mathcal{A}) $ generalizes the classical singularity category $ D_{\rm sg}(\mathcal{A}) $, and recovers it via tilting subcategories.
• The result extends Buchweitz-Happel duality beyond the finitely-generated case, showing that $ D_{\rm sg}(R) \simeq R\text{-\underline{GProj}} $ holds even when Gorenstein-projective modules are not necessarily finitely generated.