Singular equivalences of functor categories via Auslander-Buchweitz approximations

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 $(\mathrm{mod}A)/[T]$ are triangle equivalent. In particular, the canonical module $\omega$ over a commutative Noetherian ring induces a singular equivalence between $(\mathsf{CM}R)/[\omega ]$ and $(\mathrm{mod}R)/[\omega ]$, 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 ${\mathsf{D}}_{\mathsf{sg}}(\mathcal{A})\simeq {\mathsf{D}}_{\mathsf{sg}}(\mathcal{X})$, which is a functor category version of Xiao-Wu Chen's theorem.

The proof of Theorem B is corrected. Some results are added. Title is changed and references are modified, accordingly

References in corpus (2)

• Classifications of exact structures and Cohen-Macaulay-finite algebras
• Tilting Cohen-Macaulay representations