G-Prime andG-PrimaryG-Ideals onG-Schemes

Abstract

Let G be a flat finite-type group scheme over a scheme S, and X a noetherian S-scheme on which G acts. We define and study G-prime and G-primary G-ideals on X and study their basic properties. In particular, we prove the existence of minimal G-primary decomposition and the well-definedness of G-associated G-prime G-ideals. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for F -regular and F -rational properties.

Uniqueness of Graded Primary Decomposition of Modules Graded Over Finitely Generated Abelian Groups

“…If the scheme is assumed to be Noetherian, [7, Section 5] treats the same problem in more general settings by considering for any abelian group H (finitely generated or not), an H-graded Noetherian ring A, and a finitely generated graded A-module M, set S = Spec G = Spec H (where H is the group algebra of H over ), X = Spec A , and = M. In particular, when we restrict ourselves to the Noetherian rings and finitely generated modules, Corollaries 2.5 and 2.11 of this article can be observed as the consequences of Theorem 5.15 and Proposition 5.17, respectively, of [7].…”

Good filtrations and strong F-regularity of the ring of UP-invariants

“…By Theorem 3.3, G(A) is finitely generated and strongly F -regular, hence is F -rational ([HH1, (3.1)] and [HH3,(4.2)]). By [HM,(7.14)], A is F -rational. 2…”

Equivariant Total Ring of Fractions and Factoriality of Rings Generated by Semi-Invariants

“…For an ideal (quasi-coherent or not) I of O X , the sum of all the F -stable quasi-coherent ideals of I is the largest F -stable quasi-coherent ideal of O X contained in I. We denote this by I * as in [HM,section 4…”

“…As Y is primary, Y * is H-primary, and it does not have an embedded component by [HM,(6.2)]. As Y is primary, Y ′ is primary by [HM,(6.23…”

“…As P ′ is a primary ideal by [HM,(6.23)] and P ′ ⊂ P , it is P -primary or 0-primary. So P ′ = 0 if and only if P ′ is P -primary.…”