Differential graded enhancements of singularity categories
Xiao-Wu Chen, Zhengfang Wang
TL;DR
The paper develops two explicit differential graded (dg) enhancements of the singularity category $\mathbf{D}_{\rm sg}(R)$ for rings: the Vogel dg category and the singular Yoneda dg category. It defines the Vogel dg category $\mathcal{V}(\mathfrak{a})$ via almost-cochain maps and bounded morphisms, proves a fundamental orthogonal decomposition of $H^0(\mathcal{V})$ into minus and plus parts, and relates this to standard Verdier-quotient constructions; it also establishes a quasi-equivalence with the dg singularity category. Separately, it constructs the singular Yoneda dg category $\mathcal{SY}_{\Lambda/E}$ using bar resolutions and noncommutative differential forms, showing $\mathbf{S}_{\rm dg}(\Lambda) \simeq \mathcal{SY}^{f}_{\Lambda/E}$ and $\mathbf{D}_{\rm sg}(\Lambda) \simeq H^0(\mathcal{SY}^{f}_{\Lambda/E})$, thereby providing an explicit dg model for $\mathbf{D}_{\rm sg}(\Lambda)$. The work also analyzes long exact sequences and Buchweitz-type results relating $\mathbf{D}_{\rm sg}(R)$ to Tor/Ext groups, discusses the Singular Presilting Conjecture and its consequences for the Auslander–Reiten Conjecture, and situates these two dg enhancements within a broader landscape of noncommutative geometry and homological algebra. Overall, the paper advances concrete dg realizations of singularity categories and demonstrates their deep structural connections via semi-orthogonal decompositions and bar/Yoneda formalisms, with implications for singularity theory and representation theory.
Abstract
The singularity category of a ring detects the homological singularity of the given ring, and appears in many different contexts. We describe two different dg enhancements of the singularity category, that is, the Vogel dg category and the singular Yoneda dg category. These two dg enhancements turn out to be quasi-equivalent. We report some progress on the Singular Presilting Conjecture.