Singular equivalences induced by bimodules and quadratic monomial algebras

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.