For an abelian category A we investigate when the stable categories GProj(A) and GInj(A) are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions on A that ensure the existence of a Quillen equivalence between the model categories in question. We also study when such a Quillen equivalence transfers from A to categories naturally associated to A, such as Ch(A), the category of chain complexes in A, or Rep(Q, A), the category A-valued representations of a quiver Q. de Ciencia y Tecnología de la Región de Murcia in the framework of III PCTRM 2011-2014 . 1 In the important special case where A is a quasi-Frobenius ring, for example, if A = kG is the group algebra of a finite group G with coefficients in a field k, the category MCM(A) is just the category mod(A) of all finitely generated A-modules.2 The singularity category D sg (A) is defined to be the Verdier quotient D b (A)/D b perf (A) of the bounded derived category D b (A), whose objets are complexes of A-modules with bounded and finitely generated homology, by the subcategory D b perf (A), whose objects are isomorphic (in D b (A)) to a perfect complex, that is, to a bounded complex of finitely generated projective A-modules. The name singularity category and the symbol D sg (A) seem to be the popular choices nowadays, however, in the work of Buchweitz [8, Def. 1.2.2], this category is called the stabilized derived category and denoted by D b (A), and in the work of Orlov [31], it is called the triangulated category of singularities and denoted by D sg (A).

Rickard proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules, induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self-injective case.

Rickard proved that for certain self‐injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite‐dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self‐injective case.

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