Homological systems and bocses

TL;DR

This paper generalizes the existence of exact Borel subalgebras to a broader class of finite-dimensional algebras with homological systems, extending previous results beyond quasi-hereditary algebras.

Contribution

It introduces a new approach using differential graded tensor algebras and $A_{ abla}$-algebra structures to describe filtered modules in a more general setting.

Findings

Existence of exact Borel subalgebras up to Morita equivalence for a wider class of algebras.

Use of $A_{ abla}$-algebra and differential graded tensor algebra to analyze module categories.

Generalization of previous theorems from quasi-hereditary to broader homological contexts.

Abstract

We show that, up to Morita equivalence, any finite-dimensional algebra with a suitable homological system, admits an exact Borel subalgebra. This generalizes a theorem by Koenig, K\"ulshammer and Ovsienko, which holds for quasi-hereditary algebras. Our proof follows the same general scheme proposed by these authors, in a more general context: we associate a differential graded tensor algebra with relations, using the structure of $A_{\infty}-$algebra of a suitable Yoneda algebra, and use its category of modules to describe the category of filtered modules associated to the given homological system.