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

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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 Tₐ(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.