Generating sequences and semigroups of valuations on 2-dimensional normal local rings
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 619-647.

Dans cet article, nous développons une méthode de calcul de semigroupes d’évaluation pour les évaluations dominant l’anneau d’une singularité de quotient à deux dimensions. Supposons que K est un corps algébriquement clos de caractéristique zéro, K[X,Y] est un anneau polynomial sur K et ν est une évaluation rationnelle non discrète de rang 1 du corps K(X,Y) qui domine K[X,Y] (X,Y) . Étant donné un groupe H abelien fini agissant en diagonale sur K[X,Y] et une suite génératrice de ν dans K[X,Y] dont les membres sont des fonctions propres pour l’action de H, nous calculons le semigroupe S K[X,Y] H (ν) de valeurs d’éléments de l’anneau invariant K[X,Y] H . Nous déterminons en outre quand S K[X,Y] (ν) est un S K[X,Y] H (ν)-module de type fini.

In this paper we develop a method for computing valuation semigroups for valuations dominating the ring of a two dimensional quotient singularity. Suppose that K is an algebraically closed field of characteristic zero, K[X,Y] is a polynomial ring over K and ν is a rational rank 1 non discrete valuation of the field K(X,Y) which dominates K[X,Y] (X,Y) . Given a finite abelian group H acting diagonally on K[X,Y], and a generating sequence of ν in K[X,Y] whose members are eigenfunctions for the action of H, we compute the semigroup S K[X,Y] H (ν) of values of elements of K[X,Y] H . We further determine when S K[X,Y] (ν) is a finitely generated S K[X,Y] H (ν)-module.

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DOI : 10.5802/afst.1642

Arpan Dutta 1

1 Department of Mathematics, IISER Mohali, Knowledge City, Sector 81, Manauli PO, SAS Nagar, Punjab, 140306 (India)
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Arpan Dutta. Generating sequences and semigroups of valuations on 2-dimensional normal local rings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 619-647. doi : 10.5802/afst.1642. https://afst.centre-mersenne.org/articles/10.5802/afst.1642/

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