Three local conditions on a graded ring

Abstract

Let R =ZiGRi be a commutative graded Noetherian ring with unit and let A = -iezAi be 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. Introduction. The following question was posed by Nagata in [9]: if R = kjeZ+Ri is a commutative graded Noetherian ring with unit such that RM is Cohen Macaulay for each maximal graded ideal M of R, then is RP Cohen Macaulay for every prime ideal P of R? Paul Roberts [6], M. Hochster and L. J. Ratliff [4] and the author arrived independently at affirmative answers to the question. An expanded version of the question in the case of a graded ring R, graded by the integers, admitting a finitely generated graded R module A with the property that AM is Cohen Macaulay for each maximal graded ideal M of R is answered in Theorem 1.1 below. In ?2 the same question is answered with R graded by the integers and Cohen Macaulay replaced by regular. If R is graded by the positive integers and is a regular ring, then R admits a structure theorem in the case that R has a finite number of maximal graded ideals. In Proposition 2.3 we show that R is a direct sum of polynomial rings over semilocal regular domains. In ? 3 the question is once again answered when R is graded by the integers, but this time Cohen Macaulay is replaced by Gorenstein. An example is given to show that if R = ZjEZ+Rj is a Gorenstein ring, Ro need not be Gorenstein. In fact in the example RO is not even Cohen Macaulay. Paul Roberts has recently proved these results using methods different from those given below. The basic references for the ideas that follow are [5], [7] and [8].(1) All rings are commutative Noetherian rings with unit element. R = iGeZRi will Received by the editors February 5, 1974. AMS (MOS) subject classifications (1970). Primary 13H05, 13H 10, 13C05, 13E05; Secondary 13C15, 13B25.