Singular equivalences of functor categories via Auslander-Buchweitz approximations

TL;DR

This paper constructs singular equivalences between functor categories using Auslander-Buchweitz approximations, generalizing known results and providing new conditions for such equivalences in additive categories.

Contribution

It introduces a sufficient condition for inducing singular equivalences between categories and their subcategories, extending previous theorems to functor categories.

Findings

Established a singular equivalence from cotilting modules.

Generalized Matsui-Takahashi's theorem for canonical modules.

Provided a functor category version of Chen's theorem.

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 $\omega$ over a commutative Noetherian ring induces a singular equivalence between $(\mathsf{CM}R)/[\omega]$ and $(\mod R)/[\omega]$, 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.