Spanier–Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories

Abstract

Let $\mathcal {A}$ be an abelian category with enough projective objects, and let $\mathcal {X}$ be a quasi-resolving subcategory of $\mathcal {A}$. In this paper, we investigate the affinity of the Spanier–Whitehead category $\mathsf {SW}(\mathcal {X})$ of the stable category of $\mathcal {X}$ with the singularity category $\mathsf {D}_{\mathsf {sg}}(\mathcal {A})$ of $\mathcal {A}$. We construct a fully faithful triangle functor from $\mathsf {SW}(\mathcal {X})$ to $\mathsf {D}_{\mathsf {sg}}(\mathcal {A})$, and we prove that it is dense if and only if the bounded derived category $\mathsf {D}^{\mathsf {b}}(\mathcal {A})$ of $\mathcal {A}$ is generated by $\mathcal {X}$. Applying this result to commutative rings, we obtain characterizations of the isolated singularities, the Gorenstein rings and the Cohen–Macaulay rings. Moreover, we classify the Spanier–Whitehead categories over complete intersections. Finally, we explore a method to compute the (Rouquier) dimension of the triangulated category $\mathsf {SW}(\mathcal {X})$ in terms of generation in $\mathcal {X}$.