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

Let A be an abelian category with enough projective objects, and let X be a quasi-resolving subcategory of A. In this paper, we investigate the affinity of the Spanier-Whitehead category SW(X) of the stable category of X with the singularity category D-sg(A) of A. We construct a fully faithful triangle functor from SW(X) to D-sg(A), and we prove that it is dense if and only if the bounded derived category D-b(A) of A is generated by 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 SW(X) in terms of generation in X.