(m,n)-Quasitilted and (m,n)-Almost Hereditary Algebras

TL;DR

This paper introduces (m,n)-almost hereditary algebras, extending the concept of (m,n)-quasitilted algebras, and explores their module categories, especially focusing on the structure of indecomposable modules and their dimensions.

Contribution

It defines (m,n)-almost hereditary algebras and analyzes their module categories, establishing key properties and relations with (m,n)-quasitilted algebras.

Findings

Indecomposable modules lie in specific parts of the module category.

Characterization of modules with bounded projective or injective dimension.

Special role of (m,1)-almost hereditary algebras in module category structure.

Abstract

Motivated by the study of (m,n)-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension two by a sequence of (co)tiltings involving n-1 tilting modules and m-1 cotilting modules, we introduce (m,n)-almost hereditary algebras. These are the algebras with global dimension m+n and such that any indecomposable module has projective dimension at most m, or else injective dimension at most n. We relate these two classes of algebras, among which (m,1)-almost hereditary ones play a special role. For these, we prove that any indecomposable module lies in the right part of the module category, or else in an m-analog of the left part. This is based on the more general study of algebras the module categories of which admit a torsion-free subcategory such that any indecomposable module lies in that subcategory, or else has injective dimension at most n.