Finite Generation of Noetherian Graded Rings

Gotō, Shirō; Yamagishi, Kikumichi

doi:10.2307/2045060

Cited by 9 publications

(14 citation statements)

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“…In the case of a finite group action (not necessarily commutative), an analogous claim was recently proved by Gabber; see [ILO14, Proposition IV.2.2.3]. It seems that the work [GY83] is not widely known in algebraic geometry; at least, we had reproved the theorem (in a more complicated way!) before finding the reference.…”

Section: Noetherian L-graded Ringsmentioning

confidence: 67%

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Luna's fundamental lemma for diagonalizable groups

Temkin¹

2018

Alg. Geom.

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We study relatively affine actions of a diagonalizable group G on locally noetherian schemes. In particular, we generalize Luna's fundamental lemma when applied to a diagonalizable group: we obtain criteria for a G-equivariant morphism f : X → X to be strongly equivariant, namely the base change of the morphism f G of quotient schemes, and establish descent criteria for f G to be an open embedding,étale, smooth, regular, syntomic, or lci.

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Section: Noetherian L-graded Ringsmentioning

confidence: 67%

“…A theorem of Goto and Yamagishi states that any noetherian L-graded ring is finitely generated over the subring of invariants; see [GY83]. In the case of a finite group action (not necessarily commutative), an analogous claim was recently proved by Gabber; see [ILO14, Proposition IV.2.2.3].…”

Section: Noetherian L-graded Ringsmentioning

confidence: 91%

Luna's fundamental lemma for diagonalizable groups

Temkin¹

2018

Alg. Geom.

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“…As in the title, we may refer to a tensor triangulated category K satisfying hypotheses (a) and (b) as being affine and weakly regular, respectively. Note that R being noetherian implies that R 0 = End K (1) is a noetherian ring and that R is a finitely generated R 0 -algebra, by [Goto and Yamagishi 1983].…”

Section: Introduction and Resultsmentioning

confidence: 99%

Affine weakly regular tensor triangulated categories

Dell’Ambrogio

Stanley

2016

Pacific J. Math.

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We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.Dell'Ambrogio was partially supported by the Labex CEMPI (ANR-11-LABX-0007-01) .

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“…Therefore the graded ring RT, where T = h(R -P*) is h-noetherian by the "homogeneous" Cohen theorem (see Theorems 6,7,8 in [4] with suitable easy modifications). It follows that RT is noetherian (see [3], Theorem 1 .I .) and the assertion follows since Rp* is a localization of RT.…”

Section: Proposition (47)mentioning

confidence: 95%

Rees algebras and gröbner bases

Portelli

Spangher

1990

Communications in Algebra

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I-34100-Trieste (Italy). This paper deals with the following problem. Robbiano showed in [9] that standard bases, Grobner bases, Macaulay bases are all instances of the same general situation. In this paper, we develop this philosophy from the point of view of the Rees algebra R of a ring A w.r.t. a filtration F given on A. The ring R plays a fine job between A and the graded ring G associated to (A, F). The use of R and the properties of termorderings and their relate Grobner bases led naturally to the definition of Grobner filtrations in general commutative rings.

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Finite Generation of Noetherian Graded Rings

Abstract: Abstract. Let H be an additive abelian group. Then a commutative ring A is said to be //-graded if there is given a family [Ah}h<=H of subgroups of A such that A = ®heHAh and AhAg C Ah+g for all h,

Cited by 9 publications

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Luna's fundamental lemma for diagonalizable groups

Luna's fundamental lemma for diagonalizable groups

Affine weakly regular tensor triangulated categories

Rees algebras and gröbner bases

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