Reduction techniques of singular equivalences

TL;DR

This paper introduces new reduction techniques for analyzing properties of finite dimensional algebras using singular equivalences induced by tensoring with complexes of bimodules, extending Morita theory.

Contribution

It demonstrates that singular equivalences induced by tensoring define a singular equivalence of Morita type with level, providing novel tools for algebra property testing.

Findings

Established a link between tensor-induced singular equivalences and Morita type equivalences with level.

Applied these results to homological ideals and idempotents for algebra property analysis.

Provided new methods for testing syzygy-finiteness and injective generation in finite dimensional algebras.

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.