logo AFST

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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1642
@article{AFST_2020_6_29_3_619_0,
     author = {Arpan Dutta},
     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},
     volume = {Ser. 6, 29},
     number = {3},
     year = {2020},
     doi = {10.5802/afst.1642},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1642/}
}
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/

[1] Shreeram Abhyankar On the valuations centered in a local domain, Am. J. Math., Volume 78 (1956), pp. 321-348 | Article | MR 82477 | Zbl 0074.26301

[2] David J. Benson Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, Volume 190, Cambridge University Press, 1993, x+118 pages | MR 1249931 | Zbl 0864.13001

[3] Steven Dale Cutkosky Ramification of valuations and local rings in positive characteristic, Commun. Algebra, Volume 44 (2016) no. 7, pp. 2828-2866 | Article | MR 3507154 | Zbl 1345.13005

[4] Steven Dale Cutkosky The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation, Algebra Number Theory, Volume 11 (2017) no. 6, pp. 1461-1488 | Article | MR 3687103 | Zbl 1411.14007

[5] Steven Dale Cutkosky Finite generation of extensions of associated graded rings along a valuation, J. Lond. Math. Soc., Volume 98 (2018) no. 1, pp. 177-203 | Article | MR 3847237 | Zbl 1401.14019

[6] Steven Dale Cutkosky; Pham An Vinh Valuation semigroups of two-dimensional local rings, Proc. Lond. Math. Soc., Volume 108 (2014) no. 2, pp. 350-384 | Article | MR 3166356 | Zbl 1294.13004

[7] Olga Kashcheyeva Constructing examples of semigroups of valuations, J. Pure Appl. Algebra, Volume 220 (2016) no. 12, pp. 3826-3860 | Article | MR 3517559 | Zbl 1346.13006

[8] Franz-Viktor Kuhlmann Valuation theoretic and model theoretic aspects of local uniformization, Resolution of singularities (Obergurgl, 1997) (Progress in Mathematics) Volume 181, Birkhäuser, 2000, pp. 381-456 | Article | MR 1748629 | Zbl 1046.14001

[9] Serge Lang Algebra, Graduate Texts in Mathematics, Volume 211, Springer, 2002, xvi+914 pages | Zbl 0984.00001

[10] Mohammad Moghaddam A construction for a class of valuations of the field k(X 1 ,,X d ,Y) with large value group, J. Algebra, Volume 319 (2008) no. 7, pp. 2803-2829 | Article | MR 2397409 | Zbl 1229.12007

[11] Josnei Novacoski; Mark Spivakovsky Key polynomials and pseudo-convergent sequences, J. Algebra, Volume 495 (2018), pp. 199-219 | Article | MR 3726108 | Zbl 1391.13009

[12] Jean-Pierre Serre A course in arithmetic, Graduate Texts in Mathematics, Volume 7, Springer, 1973, viii+115 pages | MR 344216 | Zbl 0256.12001

[13] Mark Spivakovsky Valuations in function fields of surfaces, Am. J. Math., Volume 112 (1990) no. 1, pp. 107-156 | Article | MR 1037606 | Zbl 0716.13003

[14] Bernard Teissier Valuations, deformations, and toric geometry, Valuation theory and its applications, Vol. II (Saskatoon, SK, 1999) (Fields Institute Communications) Volume 33, American Mathematical Society, 2003, pp. 361-459 | MR 2018565 | Zbl 1061.14016

[15] Bernard Teissier Overweight deformations of affine toric varieties and local uniformization, Valuation theory in interaction (EMS Series of Congress Reports), European Mathematical Society, 2014, pp. 474-565 | Zbl 1312.14126

[16] Oscar Zariski; Pierre Samuel Commutative algebra. Vol. II, The University Series in Higher Mathematics, Princeton University Press, 1960, x+414 pages | Zbl 0121.27801