Abstract
In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past 15 years. We develop the theories for general commutative rings, but specialize to reduced Noetherian rings when necessary. We hope to acquaint the reader with many of the consequences of the theories.
Keywords
Prime Ideal Local Ring Commutative Ring Algebraic Variety Projective Module
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
- 1.Anderson, D.F.: Seminormal graded rings. II. J. Pure Appl. Algebra 23(3), 221–226 (1982)zbMATHCrossRefGoogle Scholar
- 2.Andreotti, A., Bombieri, E.: Sugli omeomorfismi delle varietà algebriche. Ann. Scuola Norm. Sup. Pisa 23, 430–450 (1969)MathSciNetGoogle Scholar
- 3.Andreotti, A., Norguet, F.: La convexité holomorphe dans l’espace analytique de cycles d’une variété algébrique. Ann. Scuola Norm. Sup. Pisa 21, 31–82 (1967)zbMATHMathSciNetGoogle Scholar
- 4.Asanuma, T.: D-algebras which are D-stably equivalent to D[Z]. In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto University, Kyoto, 1977), pp. 447–476. Kinokuniya Book Store, Tokyo (1978)Google Scholar
- 5.Barhoumi, S., Lombardi, H.: An algorithm for the traverso-swan theorem on seminormal rings. J. Algebra 320, 1531–1542 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
- 6.Bass, H.: Torsion free and projective modules. Trans. Am. Math. Soc. 102, 319–327 (1962)zbMATHMathSciNetGoogle Scholar
- 7.Bombieri, E.: Seminormalità e singolarità ordinarie. In: Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Roma, Novembre 1971), pp. 205–210. Academic, London (1973)Google Scholar
- 8.Brenner, H.: Test rings for weak subintegral closure. Preprint pp. 1–12 (2006)Google Scholar
- 9.Brewer, J.W., Costa, D.L.:Seminormality and projective modules. J. Algebra 58, 208–216 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
- 10.Cartan, H.: Quotients of complex analytic spaces. In: Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960), pp. 1–15. Tata Institute of Fundamental Research, Bombay (1960)Google Scholar
- 11.Coquand, T.: On seminormality. J. Algebra 305(1), 577–584 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
- 12.Costa, D.L.: Seminormality and projective modules. In: Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 34th Year (Paris, 1981), Lecture Notes in Mathematics, vol. 924, pp. 400–412. Springer, Berlin (1982)Google Scholar
- 13.Cumino, C., Greco, S., Manaresi, M.: Bertini theorems for weak normality. Compositio Math. 48(3), 351–362 (1983)zbMATHMathSciNetGoogle Scholar
- 14.Cumino, C., Greco, S., Manaresi, M.: Hyperplane sections of weakly normal varieties in positive characteristic. Proc. Am. Math. Soc. 106(1), 37–42 (1989)zbMATHMathSciNetGoogle Scholar
- 15.Davis, E.D.: On the geometric interpretation of seminormality. Proc. Am. Math. Soc. 68(1), 1–5 (1978)zbMATHGoogle Scholar
- 16.Dayton, B.H.: Seminormality implies the Chinese remainder theorem. In: Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Mathematics, vol. 854, pp. 124–126. Springer, Berlin (1981)Google Scholar
- 17.Dayton, B.H., Roberts, L.G.: Seminormality of unions of planes. In: Algebraic K-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980), Lecture Notes in Mathematics, vol. 854, pp. 93–123. Springer, Berlin (1981)Google Scholar
- 18.Gaffney, T., Vitulli, M.A.: Weak subintegral closure of ideals. ArXiV 0708.3105v2, 1–40 (2008)Google Scholar
- 19.Gilmer, R., Heitmann, R.C.: On Pic(R[X]) for R seminormal. J. Pure Appl. Algebra 16(3), 251–257 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
- 20.Greco, S., Traverso, C.: On seminormal schemes. Compositio Math. 40(3), 325–365 (1980)zbMATHMathSciNetGoogle Scholar
- 21.Greither, C.: Seminormality, projective algebras, and invertible algebras. J. Algebra 70(2), 316–338 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
- 22.Gubeladze, J.: Anderson’s conjecture and the maximal monoid class over which projective modules are free. Math. USSR Sbornik 63, 165–180 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
- 23.Hamann, E.: On the R-invariance of R[X]. J. Algebra 35, 1–17 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
- 24.Heitmann, R.C.: Lifting seminormality. Michigan Math. J. 57, 439–445 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
- 25.Itoh, S.: On weak normality and symmetric algebras. J. Algebra 85(1), 40–50 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
- 26.Kleiman, S.L.: Equisingularity, multiplicity, and dependence. In: Commutative algebra and algebraic geometry (Ferrara), Lecture Notes in Pure and Applied Mathematics, vol. 206, pp. 211–225. Dekker, New York (1999)Google Scholar
- 27.Lam, T.Y.: Serre’s problem on projective modules. Springer Monographs in Mathematics. Springer, Berlin (2006)CrossRefGoogle Scholar
- 28.Leahy, J.V., Vitulli, M.A.: Seminormal rings and weakly normal varieties. Nagoya Math. J. 82, 27–56 (1981)zbMATHMathSciNetGoogle Scholar
- 29.Leahy, J.V., Vitulli, M.A.: Weakly normal varieties: the multicross singularity and some vanishing theorems on local cohomology. Nagoya Math. J. 83, 137–152 (1981)zbMATHMathSciNetGoogle Scholar
- 30.Lejeune-Jalabert, M., Teissier, B.: Clôture integrale des ideaux et equisingularité. In: Seminaire Lejeune-Teissier. Université Scientifique et Médicale de Grenoble, St-Martin-d’Heres (1974)Google Scholar
- 31.McAdam, S.: Asymptotic Prime Divisors, Lecture Notes in Mathematics, vol. 1023. Springer, Berlin (1983)Google Scholar
- 32.Mumford, D.: The Red Book of Varieties and Schemes, Lecture Notes in Mathematics, vol. 1358, expanded edn. Springer, Berlin (1999). Includes the Michigan lectures (1974) on curves and their Jacobians, With contributions by Enrico ArbarelloGoogle Scholar
- 33.Ratliff, L.J., Rush, D.E.: Two notes on reductions of ideals. Indiana Univ. Math. J. 27(6), 929–934 (1978). Ratliff folderzbMATHCrossRefMathSciNetGoogle Scholar
- 34.Reid, L., Roberts, L., Singh, B.: Finiteness of subintegrality. In: P. Goerss, J. Jardine (eds.) Algebraic K-Theory and Algebraic Topology, NATO ASI Series, Series C, vol. 407, pp. 223–227. Kluwer, Dordrecht (1993)Google Scholar
- 35.Reid, L., Roberts, L.G.: Generic weak subintegrality (1995)Google Scholar
- 36.Reid, L., Roberts, L.G.: A new criterion for weak subintegrality. Comm. Algebra 24(10), 3335–3342 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
- 37.Reid, L., Roberts, L.G., Singh, B.: The structure of generic subintegrality. Proc. Indian Acad. Sci. (Math. Sci.) 105(1), 1–22 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
- 38.Reid, L., Roberts, L.G., Singh, B.: On weak subintegrality. J. Pure Appl. Algebra 114, 93–109 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
- 39.Reid, L., Vitulli, M.A.: The weak subintegral closure of a monomial ideal. Comm. Alg. 27(11), 5649–5667 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
- 40.Roberts, L.G.: Integral dependence and weak subintegrality. In: Singularities in algebraic and analytic geometry (San Antonio, TX, 1999), Contemp. Math., vol. 266, pp. 23–28. Am. Math. Soc., Providence, RI (2000)Google Scholar
- 41.Roberts, L.G., Singh, B.: Subintegrality, invertible modules and the Picard group. Compositio Math. 85, 249–279 (1993)zbMATHMathSciNetGoogle Scholar
- 42.Roberts, L.G., Singh, B.: Invertible modules and generic subintegrality. J. Pure Appl. Algebra 95, 331–351 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
- 43.Rush, D.E.: Seminormality. J. Algebra 67, 377–387 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
- 44.Salmon, P.: Singolarità e gruppo di Picard. In: Symposia Mathematica, Vol. II (INDAM, Rome, 1968), pp. 341–345. Academic, London (1969)Google Scholar
- 45.Seidenberg, A.: The hyperplane sections of normal varieties. Trans. Am. Math. Soc. 69, 357–386 (1950)zbMATHMathSciNetGoogle Scholar
- 46.Shafarevich, I.R.: Basic Algebraic Geometry. Springer, New York (1974). Translated from the Russian by K. A. Hirsch, Die Grundlehren der Mathematischen Wissenschaften, Band 213zbMATHGoogle Scholar
- 47.Swan, R.G.: On seminormality. J. Algebra 67, 210–229 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
- 48.Swan, R.G.: Gubeladze’s proof of Anderson’s conjecture. In: Azumaya algebras, actions, and modules (Bloomington, IN, 1990), Contemp. Math., vol. 124, pp. 215–250. Am. Math. Soc., Providence, RI (1992)Google Scholar
- 49.Swanson, I., Huneke, C.: Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006)Google Scholar
- 50.Traverso, C.: Seminormality and Picard group. Ann. Scuola Norm. Sup. Pisa 24, 585–595 (1970)zbMATHMathSciNetGoogle Scholar
- 51.Vitulli, M.A.: The hyperplane sections of weakly normal varieties. Am. J. Math. 105(6), 1357–1368 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
- 52.Vitulli, M.A.: Corrections to: “Seminormal rings and weakly normal varieties” [Nagoya Math. J. 82 (1981), 27–56; MR 83a:14015] by J. V. Leahy and M.A. Vitulli. Nagoya Math. J. 107, 147–157 (1987)Google Scholar
- 53.Vitulli, M.A.: Weak normalization and weak subintegral closure. In: C. Grant-Melles, R. Michler (eds.) Singularities in algebraic and analytic geometry (San Antonio, TX, 1999), Contemp. Math., vol. 266, pp. 11–21. Am. Math. Soc., Providence, RI (2000)Google Scholar
- 54.Vitulli, M.A., Leahy, J.V.: The weak subintegral closure of an ideal. J. Pure Appl. Algebra 141(2), 185–200 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
- 55.Yanagihara, H.: On glueings of prime ideals. Hiroshima Math. J. 10(2), 351–363 (1980)zbMATHMathSciNetGoogle Scholar
- 56.Yanagihara, H.: Some results on weakly normal ring extensions. J. Math. Soc. Jpn. 35(4), 649–661 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
- 57.Yanagihara, H.: On an intrinisic definition of weakly normal rings. Kobe J. Math. 2, 89–98 (1985)zbMATHMathSciNetGoogle Scholar
Copyright information
© Springer Science+Business Media, LLC 2011