Three local conditions on a graded ring

Abstract

Let R = ^^zRi ^e a comlT>utative graded Noetherian ring with unit and let A = fezAi '3e a finitely generated graded R module. We show that if we assume that AM is a Cohen Macaulay RM module for each maximal graded ideal M of R, then Ap is a Cohen Macaulay Rp module for each prime ideal P of R. With A = R we show that the same is true with Cohen Macaulay replaced by regular and Gorenstein, respectively