Amalgamation over a Graded Ring

Abstract

AbstractLet A and B be commutative rings, let J be an ideal of B, let \(f:A\rightarrow B\) be a ring homomorphism and we note by J(B) the Jacobson radical of B. The purpose of this paper is to study the graduation of the amalgamation \(A\bowtie ^f J\) and \(A\bowtie I\) introduced by D’Anna and Fontana in 2007. We investigate also some homological properties of the amalgamation over a graded ring and we show that if (A, m) is a Noetherian local ring with dimension d, \(f:A\rightarrow B\) a ring homomorphism and \(0\ne J\subseteq J(B)\) an ideal such that J is a finitely generated A-module, then \(H^d_{m\bowtie ^f J}(A\bowtie ^f J)\) is \(FP-gr\)-injective if and only if \(H_m^d(A)\) and \(H_m^d(J)\) are \(FP-gr\)-injective.KeywordsAmalgamationPrimary idealGraded ringGraded module\(FP-gr\)-injective