A criterion of graded coherentness of tensor algebras and its application to higher dimensional Auslander-Reiten theory

TL;DR

This paper establishes a criterion for graded coherentness of tensor algebras over rings and explores its implications for higher dimensional Auslander-Reiten theory and graded preprojective algebras.

Contribution

It introduces a new criterion for graded coherentness of tensor algebras and links this property to higher Auslander-Reiten theory.

Findings

A[X] is graded coherent with standard grading.

The criterion applies to tensor algebras of certain bimodules.

Relationship established between higher Auslander-Reiten theory and graded coherence.

Abstract

Even if a ring A is coherent, the polynomial ring A[X] in one variable could fail to be coherent. In this note we show that A[X] is graded coherent with the standard grading deg X=1. More generally, we give a criterion of graded coherentness of the tensor algebra T_{A}(M) of a certain class of bi-module M. As an application of the criterion, we show that there is a relationship between higher dimensional Auslander-Reiten theory and graded coherentness of higher preprojective algebras.