[Paper Review] Relative singularity category of a non-commutative resolution of singularities

Igor Burban, Martin Kalck|Kölner Universitäts PublikationsServer (Universität zu Köln)|Mar 21, 2011

Algebraic structures and combinatorial models25 references3 citations

TL;DR

This paper introduces and fully describes the relative singularity category Δ_Y(𝕐) for non-commutative resolutions of nodal curves, showing it decomposes into p blocks, each equivalent to a category Δ_nd defined via a gentle algebra. The key result is a complete classification of indecomposable objects, computation of the Auslander–Reiten quiver, and determination of the Grothendieck group K₀(Δ_nd) ≅ ℤ².

ABSTRACT

In this article we study the triangulated category of singularities associated with a non-commutative resolution of singularities. In particular, we give a complete description of this category in the case of a curve with nodal singularities, classifying its indecomposable objects and computing its Auslander-Reiten quiver and K-group.

Motivation & Objective

• To define and study the relative singularity category Δ_X(𝕏) as a triangulated category measuring the difference between Perf(X) and D^b(Coh(𝕏)) for a non-commutative resolution 𝕏 of a singular variety X.
• To generalize Buchweitz's equivalence for Gorenstein singularities to non-commutative resolutions via a relative singularity category.
• To provide a complete description of Δ_Y(𝕐) when Y is a nodal curve, including classification of indecomposable objects and computation of K₀ and Auslander–Reiten quiver.
• To establish a link between the relative singularity category and the stable category of maximal Cohen–Macaulay modules over the completed local ring at a node.
• To show that Δ_Y(𝕐) splits into p blocks, each equivalent to Δ_nd, where p is the number of singular points of Y.

Proposed method

• Define the relative singularity category as the idempotent completion of the Verdier quotient D^b(Coh(𝕏))/Perf(X), where 𝕏 = (X, End_X(𝒪⊕ℱ′)) is a non-commutative resolution.
• Use the condition that ℱ′ is locally free on U = X∖Z to establish a localization equivalence analogous to Buchweitz’s theorem.
• Employ negative K-theory of Schlichting to compute the Grothendieck group K₀(Δ_X(𝕏)).
• For nodal curves, construct 𝕐 = (Y, End_Y(𝒪⊕ℐ_Z)) and show Δ_Y(𝕐) decomposes into p copies of Δ_nd via a splitting result.
• Realize Δ_nd as a quotient of bounded homotopy categories of pro- and add-categories over the completed path algebra A_nd of a quiver with relations.
• Use the functor Hom_{A_nd}(P_*, −) to induce an equivalence between Δ_nd / Tria(S_+, S_-) and the stable MCM category over O_nd = k[[u,v]]/(uv).

Experimental results

Research questions

• RQ1 How can the relative singularity category Δ_X(𝕏) be defined and characterized for non-commutative resolutions of singularities?
• RQ2 Does the localization equivalence of Buchweitz extend to non-commutative resolutions, particularly for nodal curves?
• RQ3 What is the structure of the relative singularity category Δ_Y(𝕐) when Y is a nodal curve with p singular points?
• RQ4 Can the indecomposable objects and morphisms in Δ_Y(𝕐) be explicitly classified?
• RQ5 What is the Grothendieck group K₀ of the relative singularity category Δ_nd for a nodal curve?

Key findings

• The relative singularity category Δ_Y(𝕐) for a nodal curve Y with p singular points decomposes as a direct sum of p copies of Δ_nd, i.e., Δ_Y(𝕐) ≅ ⋁_{i=1}^p Δ_i with each Δ_i ≅ Δ_nd.
• The category Δ_nd is Hom-finite and representation discrete, with all indecomposable objects and morphism spaces explicitly classified.
• The Auslander–Reiten quiver of Δ_nd is computed and shown to have a specific structure related to the quiver with relations for A_nd.
• The Grothendieck group of Δ_nd is K₀(Δ_nd) ≅ ℤ², computed via negative K-theory and the structure of the derived category.
• There is a quotient equivalence Δ_nd / Tria(S_+, S_-) ≅ ⌜MCM(O_nd), showing a direct link to the stable category of MCM modules over the node.
• An alternative quiver description of Δ_nd is given in terms of representations of a gentle algebra Λ, providing a combinatorial model.