Tensor products of higher almost split sequences

TL;DR

This paper explores the relationship between higher almost split sequences over tensor products of algebras and those over individual factors, providing explicit descriptions in specific finite cases.

Contribution

It offers a complete description of higher almost split sequences over tensor products of certain finite algebras using chain map techniques.

Findings

Characterization of higher almost split sequences over tensor products.

Explicit description when tensor product is (n+m)-representation finite.

Use of chain map mapping cones to describe sequences.

Abstract

We investigate how the higher almost split sequences over a tensor product of algebras are related to those over each factor. Herschend and Iyama gave a precise criterion for when the tensor product of an $n$-representation finite algebra and an $m$-representation finite algebra is $(n+m)$-representation finite. In this case we give a complete description of the higher almost split sequences over the tensor product by expressing every higher almost split sequence as the mapping cone of a suitable chain map and using a natural notion of tensor product for chain maps.