Algebra & Number Theory

The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation

Steven Cutkosky

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Suppose that R is a 2-dimensional excellent local domain with quotient field K, K is a finite separable extension of K and S is a 2-dimensional local domain with quotient field K such that S dominates R. Suppose that ν is a valuation of K such that ν dominates S. Let ν be the restriction of ν to K. The associated graded ring grν(R) was introduced by Bernard Teissier. It plays an important role in local uniformization. We show that the extension (K,ν) (K,ν) of valued fields is without defect if and only if there exist regular local rings R1 and S1 such that R1 is a local ring of a blowup of R, S1 is a local ring of a blowup of S, ν dominates S1, S1 dominates R1 and the associated graded ring grν(S1) is a finitely generated grν(R1)-algebra.

We also investigate the role of splitting of the valuation ν in K in finite generation of the extensions of associated graded rings along the valuation. We say that ν does not split in S if ν is the unique extension of ν to K which dominates S. We show that if R and S are regular local rings, ν has rational rank  1 and is not discrete and grν(S) is a finitely generated grν(R)-algebra, then S is a localization of the integral closure of R in K, the extension (K,ν) (K,ν) is without defect and ν does not split in S. We give examples showing that such a strong statement is not true when ν does not satisfy these assumptions. As a consequence, we deduce that if ν has rational rank 1 and is not discrete and if R R is a nontrivial sequence of quadratic transforms along ν, then grν(R) is not a finitely generated grν(R)-algebra.

Article information

Source
Algebra Number Theory, Volume 11, Number 6 (2017), 1461-1488.

Dates
Received: 4 January 2017
Revised: 15 March 2017
Accepted: 17 April 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842807

Digital Object Identifier
doi:10.2140/ant.2017.11.1461

Mathematical Reviews number (MathSciNet)
MR3687103

Zentralblatt MATH identifier
06763332

Subjects
Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
Secondary: 11S15: Ramification and extension theory 13B10: Morphisms 14E22: Ramification problems [See also 11S15]

Keywords
valuation local uniformization

Citation

Cutkosky, Steven. The role of defect and splitting in finite generation of extensions of associated graded rings along a valuation. Algebra Number Theory 11 (2017), no. 6, 1461--1488. doi:10.2140/ant.2017.11.1461. https://projecteuclid.org/euclid.ant/1510842807


Export citation

References

  • S. Abhyankar, “On the valuations centered in a local domain”, Amer. J. Math. 78 (1956), 321–348.
  • S. Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Math. Studies 43, Princeton University Press, 1959.
  • S. D. Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260, Société Mathématique de France, Paris, 1999.
  • S. D. Cutkosky, “Counterexamples to local monomialization in positive characteristic”, Math. Ann. 362:1-2 (2015), 321–334.
  • S. D. Cutkosky, “A generalisation of the Abhyankar Jung theorem to associated graded rings of valuations”, Math. Proc. Cambridge Philos. Soc. 160:2 (2016), 233–255.
  • S. D. Cutkosky, “Ramification of valuations and local rings in positive characteristic”, Comm. Algebra 44:7 (2016), 2828–2866.
  • S. D. Cutkosky and O. Piltant, “Ramification of valuations”, Adv. Math. 183:1 (2004), 1–79.
  • S. D. Cutkosky and P. A. Vinh, “Valuation semigroups of two-dimensional local rings”, Proc. Lond. Math. Soc. $(3)$ 108:2 (2014), 350–384.
  • A. Grothendieck, “Eléments de géométrie algébrique, II: Étude globale élémentaire de quelques classes de morphismes”, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 5–222.
  • A. Grothendieck, “Eléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I”, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 5–167.
  • O. Endler, Valuation theory, Springer, 1972.
  • L. Ghezzi and O. Kashcheyeva, “Toroidalization of generating sequences in dimension two function fields of positive characteristic”, J. Pure Appl. Algebra 209:3 (2007), 631–649.
  • L. Ghezzi, H. T. Hà, and O. Kashcheyeva, “Toroidalization of generating sequences in dimension two function fields”, J. Algebra 301:2 (2006), 838–866.
  • R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977.
  • F. J. Herrera Govantes, M. A. Olalla Acosta, M. Spivakovsky, and B. Teissier, “Extending valuations to formal completions”, pp. 252–265 in Valuation theory in interaction, edited by A. Campillo et al., European Mathematical Society, Zürich, 2014.
  • F.-V. Kuhlmann, “Valuation theoretic and model theoretic aspects of local uniformization”, pp. 381–456 in Resolution of singularities (Obergurgl, 1997), edited by H. Hauser et al., Progr. Math. 181, Birkhäuser, Basel, 2000.
  • F.-V. Kuhlmann, “A classification of Artin–Schreier defect extensions and characterizations of defectless fields”, Illinois J. Math. 54:2 (2010), 397–448.
  • S. Mac Lane, “A construction for absolute values in polynomial rings”, Trans. Amer. Math. Soc. 40:3 (1936), 363–395.
  • J. S. Milne, Étale cohomology, Princeton Mathematical Series 33, Princeton University Press, 1980.
  • M. Raynaud, Anneaux locaux henséliens, Lecture Notes in Math. 169, Springer, 1970.
  • M. Spivakovsky, “Valuations in function fields of surfaces”, Amer. J. Math. 112:1 (1990), 107–156.
  • B. Teissier, “Valuations, deformations, and toric geometry”, pp. 361–459 in Valuation theory and its applications (Saskatoon, SK, 1999), vol. 2, edited by F.-V. Kuhlmann et al., Fields Inst. Commun. 33, American Mathematical Society, Providence, RI, 2003.
  • B. Teissier, “Overweight deformations of affine toric varieties and local uniformization”, pp. 474–565 in Valuation theory in interaction, edited by A. Campillo et al., European Mathematical Society, Zürich, 2014.
  • M. Vaquié, “Famille admissible de valuations et défaut d'une extension”, J. Algebra 311:2 (2007), 859–876.
  • O. Zariski and P. Samuel, Commutative algebra, vol. 2, Van Nostrand, Princeton, 1960.