A criterion for graded coherence of tensor algebras and applications to higher dimensional AR-theory

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 grading \(\deg X=1\). More generally, we give a criterion of graded coherence of the tensor algebra Open image in new window of a certain class of bi-module \(\sigma \). As an application of the criterion, we show that there is a relationship between higher dimensional Auslander–Reiten theory and graded coherence of higher preprojective algebras. In the appendix by Srikanth B. Iyengar, we deal with a commutative Noetherian ring R and show that the class of R-R-bi-module to which the criterion is applicable is precisely the class of projective modules.