Singular equivalences of functor categories via Auslander-Buchweitz approximations
Yasuaki Ogawa
TL;DR
This work extends singularity theory to functor categories by constructing a functor-category analogue of Chen's theorem and an Auslander-Buchweitz approximation framework that yields triangle equivalences between singularity categories of quotient categories $A/[\omega]$ and $X/[\omega]$. It provides concrete mechanisms to obtain cotilting-based singularities, including a generalized Matsui-Takahashi result and an alternative MT proof, and connects these equivalences to properties like Iwanaga-Gorensteinness and AR dualities. The approach unifies several known results under a common AB-approximation umbrella and yields new both structural and computational insights, illustrated by algebro-geometric and representation-theoretic examples. Overall, the paper broadens the applicability of singular equivalences to functor categories and cotilting contexts, enabling broader comparisons between module categories, costable quotients, and their stable/homological invariants.
Abstract
The aim of this paper is to construct singular equivalences between functor categories. As a special case, we show that there exists a singular equivalence arising from a cotilting module $T$, namely, the singularity category of $(^\perp T)/[T]$ and that of $(\mod A)/[T]$ are triangle equivalent. In particular, the canonical module $ω$ over a commutative Noetherian ring induces a singular equivalence between $(\mathsf{CM}R)/[ω]$ and $(\mod R)/[ω]$, which generalizes Matsui-Takahashi's theorem. Our result is based on a sufficient condition for an additive category $\mathcal{A}$ and its subcategory $\mathcal{X}$ so that the canonical inclusion $\mathcal{X}\hookrightarrow\mathcal{A}$ induces a singular equivalence $\mathsf{D_{sg}}(\mathcal{A})\simeq \mathsf{D_{sg}}(\mathcal{X})$, which is a functor category version of Xiao-Wu Chen's theorem.