Feuilleter
no. 4
A counterexample to strong local monomialization in a tower of two independent defect Artin–Schreier extensions
Steven Dale Cutkosky1
1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 4, pp. 915-935.

Nous donnons un exemple d’extension d’anneaux locaux réguliers à deux dimensions dans une tour de deux extensions d’Artin–Schreier de défauts indépendants pour lesquelles la monomialisation locale forte ne tient pas.

We give an example of an extension of two dimensional regular local rings in a tower of two independent defect Artin–Schreier extensions for which strong local monomialization does not hold.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1790
Classification : 14B05, 14B25, 13A18
Mots-clés : valuation, positive characteristic, defect, strong monomialization

Steven Dale Cutkosky 1

1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Steven Dale Cutkosky. A counterexample to strong local monomialization in a tower of two independent defect Artin–Schreier extensions. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 4, pp. 915-935. doi : 10.5802/afst.1790. https://afst.centre-mersenne.org/articles/10.5802/afst.1790/

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

[2] Steven Dale Cutkosky Local factorization and monomialization of morphisms, Astérisque, 260, Société Mathématique de France, 1999, 149 pages | Numdam | Zbl

[3] Steven Dale Cutkosky Monomialization of Morphisms from 3 Folds to Surfaces, Lecture Notes in Mathematics, 1786, Springer, 2002, v+235 pages | DOI | MR

[4] Steven Dale Cutkosky Counterexamples to local monomialization in positive characteristic, Math. Ann., Volume 362 (2015) no. 1-2, pp. 321-334 | DOI | MR | Zbl

[5] 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 | DOI | MR | Zbl

[6] Steven Dale Cutkosky Erratic birational behavior of mappings in positive characteristic, Math. Nachr., Volume 296 (2023) no. 11, pp. 5123-5156 | DOI | MR | Zbl

[7] Steven Dale Cutkosky; Hussein Mourtada; Bernard Teissier On the construction of valuations and generating sequences, Algebr. Geom., Volume 8 (2021) no. 6, pp. 705-748 | DOI | MR | Zbl

[8] Steven Dale Cutkosky; Olivier Piltant Ramification of Valuations, Adv. Math., Volume 183 (2004) no. 1, pp. 1-79 | DOI | MR | Zbl

[9] 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 | DOI | MR | Zbl

[10] Samar ElHitti; Laura Ghezzi Dependent Artin-Schreier defect extensions and strong monomialization, J. Pure Appl. Algebra, Volume 220 (2016) no. 4, pp. 1331-1342 | DOI | MR | Zbl

[11] Otto Endler Valuation Theory, Universitext, Springer, 1972, xii+243 pages | DOI | MR | Zbl

[12] Franz-Viktor Kuhlmann Valuation theoretic and model theoretic aspects of local uniformization, Resolution of singularities. A research textbook in tribute to Oscar Zariski (Progress in Mathematics), Volume 181, Birkhäuser, 2000, pp. 381-456 | Zbl

[13] Franz-Viktor Kuhlmann A classification of Artin Schreier defect extensions and characterizations of defectless fields, Ill. J. Math., Volume 54 (2010) no. 2, pp. 397-448 | MR | Zbl

[14] Franz-Viktor Kuhlmann; Olivier Piltant Higher ramification groups for Artin-Schreier defect extensions, 2012 (manuscript)

[15] Franz-Viktor Kuhlmann; Anna Rzepka The valuation theory of deeply ramified fields and its connection with defect extensions, Trans. Am. Math. Soc., Volume 376 (2023) no. 4, pp. 2693-2738 | MR | Zbl

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

[17] Olivier Piltant On the Jung method in positive characteristic, Ann. Inst. Fourier, Volume 53 (2003) no. 4, pp. 1237-1258 (Proceedings of the International Conference in Honor of Frédéric Pham) | DOI | Numdam | MR | Zbl

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

[19] Michel Vaquié Famille admissible de valuations et défaut d’une extension, J. Algebra, Volume 311 (2007) no. 2, pp. 859-876 | DOI | MR | Zbl

[20] Oscar Zariski; Pierre Samuel Commutative algebra. Vol. I, The University Series in Higher Mathematics, D. van Nostrand Company, 1958, xi+329 pages | Zbl

[21] Oscar Zariski; Pierre Samuel Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. van Nostrand Company, 1960, x+414 pages | DOI | MR | Zbl

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