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Integrally closed rings in birational extensions of two-dimensional regular local rings

Published online by Cambridge University Press:  12 February 2013

BRUCE OLBERDING  and
FRANCESCA TARTARONE
BRUCE OLBERDING
Affiliation:
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, U.S.A. e-mail: olberdin@nmsu.edu
FRANCESCA TARTARONE
Affiliation:
Dipartimento di Matematica, Università degli Studi “Roma Tre”Largo San Leonardo Murialdo 1, Roma, 00146, Italy. e-mail: tfrance@mat.uniroma3.it
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Abstract

Let D be an integrally closed local Noetherian domain of Krull dimension 2, and let f be a nonzero element of D such that fD has prime radical. We consider when an integrally closed ring H between D and Df is determined locally by finitely many valuation overrings of D. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of D and, when D is analytically normal, this property holds for D if and only if it holds for the completion of D. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where D is a regular local ring and f is a regular parameter of D, then H is determined locally by a single valuation. As a consequence, we show that if H is also the integral closure of a finitely generated D-algebra, then the exceptional prime ideals of the extension H/D are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 155 , Issue 1 , July 2013 , pp. 101 - 127
Copyright
Copyright © Cambridge Philosophical Society 2013 

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References

REFERENCES

[1]Abhyankar, S.On the valuations centered in a local domain. Amer. J. Math. 78 (1956), 321–348.CrossRefGoogle Scholar
[2]Abhyankar, S.Local uniformization on algebraic surfaces over ground fields of characteristic p ≠ 0. Ann. of Math. (2) 63 (1956), 491–526.CrossRefGoogle Scholar
[3]Chang, C. C. and Keisler, H. J.Model Theory. Third edition. Studies in Logic and the Foundations of Mathematics, 73 (North-Holland Publishing Co., Amsterdam, 1990).Google Scholar
[4]Cohen, I. S.On the structure and ideal theory of complete local rings. Trans. Amer. Math. Soc. 59, (1946). 54–106.CrossRefGoogle Scholar
[5]Engler, A. and Prestel, A.Valued Fields. Springer Monographs in Mathematics (Springer–Verlag, Berlin, 2005).Google Scholar
[6]Evans, E. G.A generalization of Zariski's main theorem. Proc. Amer. Math. Soc. 26 (1970), 45–48.Google Scholar
[7]Gilmer, R.Multiplicative Ideal Theory (Marcel Dekker, New York, 1972); rep. Queen's Papers in Pure and Applied Mathematics, vol. 90 (Queen's University, Kingston, 1992).Google Scholar
[8]Göhner, H.Semifactoriality and Muhly's condition (N) in two dimensional local rings. J. Algebra 34 (1975), 403–429.CrossRefGoogle Scholar
[9]Grothendieck, A. and Dieudonné, J.Eléments de Géométrie Algébrique IV. Étude locale des schḿas et des morphismes de schémas, première partie. Publ. Math. Inst. Hautes Etudes Sci. 20 (1964), 101–346.CrossRefGoogle Scholar
[10]Grothendieck, A. and Dieudonné, J.Eléments de Géométrie Algébrique III. Étude cohomologique des faisceaux cohérents, seconde partie. Publ. Math. Inst. Hautes Etudes Sci. 17 (1963), 5–91.Google Scholar
[11]Hartshorne, R.Algebraic Geometry. Graduate Texts in Mathematics, No. 52 (Springer-Verlag, New York-Heidelberg, 1977).CrossRefGoogle Scholar
[12]Heinzer, W.On Krull overrings of a Noetherian domain. Proc. Amer. Math. Soc. 22 (1969), 217–222.CrossRefGoogle Scholar
[13]Heinzer, W., Huneke, C. and Sally, J.A criterion for spots. J. Math. Kyoto Univ. 26 (1986), no. 4, 667–671.Google Scholar
[14]Heinzer, W. and Lantz, D.Exceptional prime divisors of two-dimensional local domains. Commutative Algebra (Berkeley, CA, 1987, 279–304. Math. Sci. Res. Inst. Publ. 15 (Springer, New York, 1989).Google Scholar
[15]Heinzer, W. and Ohm, J.Noetherian intersections of integral domains. Trans. Amer. Math. Soc. 167 (1972), 291–308.CrossRefGoogle Scholar
[16]Katz, D. and Validashti, J.Multiplicities and Rees valuations. Collect. Math. 61 (2010), 1–24.CrossRefGoogle Scholar
[17]Lipman, J.Desingularization of two-dimensional schemes. Ann. Math. (2) 107 (1978), no. 1, 151–207.CrossRefGoogle Scholar
[18]Loper, K.A. and Tartarone, F.A classification of the integrally closed rings of polynomials containing ℤ[X]. J. Comm. Algebra 1 (2009), 91–157.Google Scholar
[19]MacLane, S.A construction for absolute values in polynomial rings. Trans. Amer. Math. Soc. 40, no. 3 (1936), 363–395.CrossRefGoogle Scholar
[20]Matsumura, H.Commutative Ring Theory (Cambridge University Press, 1986).Google Scholar
[21]Nagata, M.Local rings (Interscience, New York, 1962).Google Scholar
[22]Olberding, B.Irredundant intersections of valuation overrings of two-dimensional Noetherian domains. J. Algebra 318 (2007), 834–855.CrossRefGoogle Scholar
[23]Olberding, B.Intersections of valuation overrings of two-dimensional Noetherian domains. In Commutative algebra: Noetherian and non-Noetherian perspectives, 331–357 (Springer, 2011).Google Scholar
[24]Olberding, B. and Saydam, S.Ultraproducts of commutative rings. Commutative ring theory and applications (Fez, 2001), 369–386. Lecture Notes in Pure and Appl. Math. 231, (Dekker, New York, 2003).Google Scholar
[25]Sally, J. D.One-fibered ideals. Commutative Algebra (Berkeley, CA, 1987), 437–442. Math. Sci. Res. Inst. Publ. 15 (Springer, New York, 1989).Google Scholar
[26]Swanson, I.Rees valuations. In Commutative algebra: Noetherian and non-Noetherian perspectives, 417–436 (Springer, 2011).Google Scholar
[27]Swanson, I. and Huneke, C.Integral closure of ideals, rings, and modules. London Math. Soc. Lecture Note Series, 336 (Cambridge University Press, Cambridge, 2006).Google Scholar