The homogeneous spectrum of a graded commutative ring

Abstract

Suppose Γ \Gamma is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a Γ \Gamma -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose A A is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring A [ Γ ] A[\Gamma ] have Noetherian spectrum. If rank ⁡ Γ ≤ 2 \operatorname {rank} \Gamma \le 2 , we show that A [ Γ ] A[\Gamma ] has Noetherian spectrum, while for each n ≥ 3 n \ge 3 we establish existence of an example where the homogeneous spectrum of A [ Γ ] A[\Gamma ] is not Noetherian.