Abstract
In this survey article, we discuss recent work describing the integrally closed rings between a two-dimensional regular local ring D and its quotient field F. A main emphasis is on those rings that can be obtained as an intersection of regular local rings between D and F.
Dedicated to the memory of Paul-Jean Cahen.
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Notes
- 1.
An overring of D is understood to be a subring of F that contains D.
- 2.
Zariski refers to these DVRs as “prime divisors of the second kind,” whereas the essential prime divisors of D are “prime divisors of the first kind.” Abhyankar refers to prime divisors dominating D as “hidden prime divisors” since these are prime divisors that come out on a blowup.
References
S. S. Abhyankar, On the valuations centered in a local domain, Amer. J. Math., 78 (1956) 321-348.
P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials. Amer. Math. Soc. Surveys and Monographs, vol. 48, Providence, 1997.
P.-J. Cahen and J.-L. Chabert, What you should know about integer-valued polynomials, Amer. Math. Monthly, 123 (2016), 311–337.
C. Favre and M. Jonsson, The valuative tree, Springer-Verlag, 2004.
A. Granja, Valuations determined by quadratic transforms of a regular ring, J. Algebra, 280 (2004), no. 2, 699–718.
A. Granja, The valuative tree of a two-dimensional regular local ring, Math. Res. Lett., (2007), no. 1, 17–34.
A. Granja, M. C. Martinez and C. Rodriguez, Valuations dominating regular local rings and proximity relations, J. Pure Appl. Algebra, 209 (2007), no. 2, 371–382.
A. Granja and C. Rodriguez, Proximity relations for real rank-one valuations dominating a local regular ring. Proceedings of the International Conference on Algebraic Geometry and Singularities (Sevilla, 2001). Rev. Mat. Iberoamericana 19 (2003), no. 2, 393–412.
L. Guerrieri, W. Heinzer, B. Olberding and M. Toeniskoetter, Directed Unions of Local Quadratic Transforms of Regular Local Rings and Pullbacks, in Rings, Polynomials, and Modules, Springer, 2017, 257–280.
W. Heinzer, On Krull overrings of a Noetherian domain, Proc. Amer. Math. Soc., 22 (1969), 217–222.
W. Heinzer, B. Johnston, D. Lantz, and K. Shah, Coefficient ideal in and blowups of a commutative Noetherian domain, J. Algebra, 162 (1993), 355–391.
W. Heinzer and K. A. Loper and B. Olberding, The tree of quadratic transforms of a regular local ring of dimension two, J. Algebra, 560 (2020), 383–415.
W. Heinzer, K. A. Loper, B. Olberding, H. Schoutens and M. Toeniskoetter, Ideal theory of infinite directed unions of local quadratic transforms, J. Algebra, 474 (2017), 213–239.
W. Heinzer and B. Olberding, Noetherian intersections of regular local rings of dimension two, J. Algebra, 559 (2020), 320–345.
W. Heinzer and B. Olberding, The ideal theory of intersections of prime divisors dominating a normal Noetherian local domain of dimension two. Rend. Sem. Mat. Univ. Padova, 144 (2020), 145–158.
W. Heinzer, B. Olberding and M. Toeniskoetter, Asymptotic properties of infinite directed unions of local quadratic transforms, J. Algebra, 479 (2017), 216–243.
M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc., 142 (1969), 43–60.
J. Hedstrom and E. Houston, Pseudo-valuation domains, Pacific J. Math., 75 (1978), 137–147.
W. Krull, Beitráge zur Arithmetik kommutativer Integritâtsbereiche II, Math. Zeit., 41 (1936), 665–679.
J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publ. Math. Inst. Haute Etudes Sci., 36 (1969), 195-279.
J. Lipman, Desingularization of two-dimensional schemes, Ann. Math., 107 (1978),151–207.
J. Lipman, On complete ideals in regular local rings, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, (1986) 203-231.
K.A. Loper and F. Tartarone, A classification of the integrally closed rings of polynomials containing \({\mathbb {Z}}[X]\), J. Commut. Algebra, 1 (2009), 91–157.
M. Nagata, On Krull’s conjecture concerning valuation rings, Nagoya Math. J., 4 (1952), 29–33.
M. Nagata, Corrections to my paper “On Krull’s conjecture concerning valuation rings.”, Nagoya Math. J., 9 (1955), 209–212.
B. Olberding, Intersections of valuation overrings of two-dimensional Noetherian domains, in Commutative Algebra: Noetherian and non-Noetherian perspectives, Springer, 2010, 335–362.
B. Olberding, Affine schemes and topological closures in the Zariski-Riemann space of valuation rings, J. Pure Appl. Algebra, 219 (2015), 1720–1741.
B. Olberding and F. Tartarone, Integrally closed rings in birational extensions of two-dimensional regular local rings, Proc. Cambridge Phil. Soc., 155 (2013), 101–127.
P. Ribenboim, Sur une note de Nagata relative à un problème de Krull, Math. Z., 64 (1956), 159–168.
D. Shannon, Monoidal transforms of regular local rings, Amer. J. Math., 95 (1973), 294-320.
O. Zariski and P. Samuel, Commutative Algebra, Vol. 2, Van Nostrand, New York, 1960.
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We thank Dave Lantz for helpful comments on a previous draft of the article.
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Heinzer, W., Loper, K.A., Olberding, B., Toeniskoetter, M. (2023). The Quadratic Tree of a Two-Dimensional Regular Local Ring. In: Chabert, JL., Fontana, M., Frisch, S., Glaz, S., Johnson, K. (eds) Algebraic, Number Theoretic, and Topological Aspects of Ring Theory . Springer, Cham. https://doi.org/10.1007/978-3-031-28847-0_15
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