Localization in a graded ring

Abstract

If one wants to investigate the properties and relations of homogeneous ideals in a commutative graded ring, one has as a model on the one hand the well known case of a polynomial ring and on the other hand the general commutative ideal theory. The case of a polynomial ring has been studied for the sake of algebraic geometry, and one of the methods was traditionally the passage to nonhomogeneous coordinates by choosinig a suitable hyperplane of infinity [5, pp. 750-755; 7, pp. 491-496]. On the other hand it appears that the homogeneous ideals of a graded ring form a system that is closed under the usual ideal operations, as is the system of all ideals of a commutative ring. Thus one may try to copy the whole ideal theory, but now for homogeneous ideals (and homogeneous elements) only. Samuel [4], Northcott [3] and Yoshida [6] have proved the elementary properties of homogen-eous ideals for a graded ring. In this paper we investigate how far the process of localization can be carried over to graded rings. The degrees in our graded ring are the integers; the case of a bigraded ring is each time treated as a corollary. In ?1 we summarize the elementary properties of homogeneous ideals. Having formulated and proved Lemma 1, all proofs become straightforward. In ?2 we study the localization (i.e., passage to a ring of fractions) with respect to a prime ideal or a finite set of prime ideals. Here we introduce the concept of a relevant prime ideal, as did Yoshida. In ?3 we discuss the transition to a nonhomogeneous ring by choosing a hyperplane of infinity. This may be called localization in the sense of the Zariski topology. By a hyperplane of infinity we mean simply a homogeneous element I of degree one, which is not nilpotent. The corresponding nonhomogeneous ring can be obtained in two ways, namely as R/(l-1)R, but also as follows: Let [1] be the multiplicatively closed subset of R, consisting of all powers of 1. Then the ring of fractions R[z] is again a graded ring. The zero-degree subring R[z]o of R[l] is our nonhomogeneous ring, i.e., R/(l-1)R-R[z]o. As elements of degree one may happen to be