Reduction techniques of singular equivalences

Abstract

It is shown that a singular equivalence induced by tensoring with a suitable complex of bimodules defines a singular equivalence of Morita type with level, in the sense of Wang. This result is applied to homological ideals and idempotents to produce new reduction techniques for testing the properties of syzygy-finite and injectives generation of finite dimensional algebras over a field.

Characterization of eventually periodic modules in the singularity categories

“…Moreover, Λ is non-simple and self-injective if and only if it is Morita equivalent to the bound quiver algebra kZ e /R N for some e ≥ 1 and N ≥ 2, where R denotes the arrow ideal of the path algebra kZ e . Now, we show the following key lemma, which is a consequence of results due to Qin [24] and Shen [29,27]. Lemma 3.9.…”

“…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.…”

“…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.…”