Abstract
Let \(\{R_{n},\mathfrak{m}_{n}\}_{n\geq 0}\) be an infinite sequence of regular local rings with R n+1 birationally dominating R n and \(\mathfrak{m}_{n}R_{n+1}\) a principal ideal of R n+1 for each n. We examine properties of the integrally closed local domain \(S =\bigcup _{n\geq 0}R_{n}\).
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Notes
- 1.
- 2.
If R n = R n+1 for some integer n, then \(R_{n} =\mathcal{ V }\) is a DVR and R n = R m for all m ≥ n.
- 3.
Muhammad Zafrullah has shown us a different proof of Theorem 6.2, an outline of which is as follows. Observe that a Shannon extension S is a Schreier domain, i.e., S is an integrally closed domain such that if r | xy, then r = st where s | x and t | y. A finite conductor Schreier domain is a GCD domain by [32, Theorem 3.6]. By Theorem 2.2, \(\mathfrak{m}_{S}\) is the radical of a principal ideal, and so S is t-local [18, Proposition 1.1(5)]. In particular, \((((x,y)S)^{-1})^{-1} \subseteq \mathfrak{m}_{S}\) for all x, y ∈ S. If S is a GCD domain that is not a valuation domain, then there exist \(x,y \in \mathfrak{m}_{S}\) such that xS ∩ yS = xyS. However, this implies (((x, y)S)−1)−1 = S, a contradiction. Thus if S is a GCD domain, S is a valuation domain. These observations imply the theorem.
References
Abhyankar, S.: On the valuations centered in a local domain. Am. J. Math. 78, 321–348 (1956)
Abhyankar, S.: Desingularization of plane curves. AMS Proc. Symp. Pure Math. (I) 40, 1–45 (1983)
Abhyankar, S.: Resolution of Singularities of Embedded Algebraic Surfaces. Springer, Berlin (1998)
Chase, S.: Direct products of modules. Trans. Am. Math. Soc. 97, 457–473 (1960)
Chevalley, C.: La notion d’anneau de décomposition. Nagoya Math. J. 7, 21–33 (1954)
Cutkosky, D.: Resolution of Singularities. Graduate Studies in Mathematics, vol. 63. American Mathematical Society, Providence, RI (2004)
Fontana, M.: Topologically defined classes of commutative rings. Ann. Mat. Pura Appl. (4) 123, 331–355 (1980)
Fontana, M., Huckaba, J., Papick, I.: Prüfer Domains. Marcel Dekker, New York (1997)
Gabelli, S., Houston, E.: Ideal theory in pullbacks. In: Chapman, S., Glaz, S. (eds.) Non-Noetherian Commutative Ring Theory. Kluwer, Dordrecht (2000)
Gilmer, R.: Multiplicative Ideal Theory. Marcel Dekker, New York (1972)
Gilmer, R., Mott, J., Zafrullah, M.: t-invertibility and comparability. In: Cahen, P.-J., Costa, D., Fontana M., Kabbaj, S.-E. (eds.) Commutative Ring Theory, pp. 141–150. Marcel Dekker, New York (1994)
Glaz, S.: Finite conductor rings. Proc. Am. Math. Soc. 129(10), 2833–2843 (2001)
Granja, A.: Valuations determined by quadratic transforms of a regular ring. J. Algebra 280(2), 699–718 (2004)
Granja, A., Sánchez-Giralda, T.: Valuations, equimultiplicity and normal flatness. J. Pure Appl. Algebra 213(9), 1890–1900 (2009)
Granja, A., Martinez, M.C., Rodriguez, C.: Valuations dominating regular local rings and proximity relations. J. Pure Appl. Algebra 209(2), 371–382 (2007)
Greenberg, B.: Global dimension of cartesian squares. J. Algebra 32, 31–43 (1974)
Hartshorne, R.: Algebraic Geometry. Springer, New York (1977)
Hedstrom, J., Houston, E.: Some remarks on star operations. J. Pure Appl. Algebra 18, 37–44 (1980)
Heinzer, W., Huneke, C., Sally, J.D.: A criterion for spots. J. Math. Kyoto Univ. 26(4), 667–671 (1986)
Heinzer, W., Kim, M.-K., Toeniskoetter, M.: Directed unions of local quadratic transforms of a regular local ring. arXiv:1512.03848
Heinzer, W., Loper, K.A., Olberding, B., Schoutens, H., Toeniskoetter, M.: Ideal theory of infinite directed unions of local quadratic transforms. J. Algebra 474, 213–239 (2017)
Heinzer, W., Olberding, B., Toeniskoetter, M.: Asymptotic properties of infinite directed unions of local quadratic transforms. J. Algebra 479, 216–243 (2017)
Kaplansky, I.: Commutative Rings. Allyn and Bacon, Boston (1970)
Kunz, E.: Introduction to Commutative Algebra and Algebraic Geometry. Birkhäuser, Boston (1985)
Lipman, J.: On complete ideals in regular local rings. In: Algebraic Geometry and Commutative Algebra, vol. I, pp. 203–231. Kinokuniya, Tokyo (1988)
Matsumura, H.: Commutative Ring Theory. Cambridge University Press, Cambridge (1986)
McAdam, S.: Two conductor theorems. J. Algebra 23, 239–240 (1972)
Nagata, M.: Local Rings. Wiley, New York (1962)
Olberding, B., Saydam, S.: Ultraproducts of commutative rings. In: Commutative Ring Theory and Applications (Fez, 2001), pp. 369–386. Lecture Notes in Pure and Appl. Math., vol. 231, Dekker, New York (2003)
Shannon, D.: Monoidal transforms of regular local rings. Am. J. Math. 95, 294–320 (1973)
Vasconcelos, W.: The local rings of global dimension two. Proc. Am. Math. Soc. 35, 381–386 (1972)
Zafrullah, M.: On a property of pre-Schreier domains. Commun. Algebra 15, 1895–1920 (1987)
Zariski, O.: Reductions of the singularities of algebraic three dimensional varieties. Ann. Math. (2) 40, 639–689 (1939)
Zariski, O., Samuel, P.: Commutative Algebra. Vol. II. Reprint of the 1960 edition. Graduate Texts in Mathematics, vol. 29. Springer, New York/Heidelberg (1975)
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Guerrieri, L., Heinzer, W., Olberding, B., Toeniskoetter, M. (2017). Directed Unions of Local Quadratic Transforms of Regular Local Rings and Pullbacks. In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules. Springer, Cham. https://doi.org/10.1007/978-3-319-65874-2_13
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