Singular equivalences induced by bimodules and quadratic monomial algebras

Abstract

We investigate the problem when the tensor functor by a bimodule yields a singular equivalence. It turns out that this problem is equivalent to the one when the Hom functor given by the same bimodule induces a triangle equivalence between the homotopy categories of acyclic complexes of injective modules. We give conditions on when a bimodule appears in a pair of bimodules, that defines a singular equivalence with level. We construct an explicit bimodule, which yields a singular equivalence between a quadratic monomial algebra and its associated algebra with radical square zero. Under certain conditions which include the Gorenstein cases, the bimodule does appear in a pair of bimodules defining a singular equivalence with level.

Singular equivalences and Auslander-Reiten conjecture

“…Typical examples are singular equivalences of Morita type (with level) when X • is a module ([38, Theorem 3.1],[33, Definition 2.1]). See [8,Proposition 4.8] for recent progress in this direction. In the general case, it was proved recently by Dalezios [12,Theorem 3.6] that F induces a singular equivalence if…”

Reduction techniques of singular equivalences

“…In [11], Chen-Liu-Wang gave a sufficient condition on when a tensor functor with a bimodule defines a singular equivalence Morita type with level, and in [16], Dalezios proved that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a complex of bimodules always induces a singular equivalence of Morita type with level. Our first theorem is a complex version of Chen-Liu-Wang's work, and it generalizes the result of Dalezios to arbitrary algebra (not limited to Gorenstein algebra).…”

Characterization of eventually periodic modules in the singularity categories

“…The notion of singular equivalences of Morita type with level was introduced by Wang [35], and it is known that such equivalences induce triangle equivalences between singularity categories (see Remark 3.2). Recently, singular equivalences of Morita type with level have been intensively studied (see [11,13,24,31,35]). In particular, Skartsaeterhagen [31], Qin [24] and Wang [35,37] have discovered invariants under singular equivalence of Morita type with level.…”