Generating sequences and semigroups of valuations on 2-dimensional normal local rings

Arpan Dutta

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)

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 29 (2020) no. 3, pp. 619-647.

Résumé
Abstract

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.

Reçu le : 2018-05-10
Accepté le : 2018-11-04
Publié le : 2020-11-16

DOI : 10.5802/afst.1642

Affiliations des auteurs :

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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     title = {Generating sequences and semigroups of valuations on 2-dimensional normal local rings},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {619--647},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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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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