Weak normality and seminormality

Marie A. Vitulli

doi:10.1007/978-1-4419-6990-3_17

Commutative Algebra pp 441-480

Weak normality and seminormality

Authors
Authors and affiliations

Marie A. VitulliEmail author

Chapter

First Online:: 27 August 2010

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.

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)MATHCrossRefGoogle 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)MATHMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
6.
Bass, H.: Torsion free and projective modules. Trans. Am. Math. Soc. 102, 319–327 (1962)MATHMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHMathSciNetGoogle 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)MATHMathSciNetGoogle Scholar
15.
Davis, E.D.: On the geometric interpretation of seminormality. Proc. Am. Math. Soc. 68(1), 1–5 (1978)MATHGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
20.
Greco, S., Traverso, C.: On seminormal schemes. Compositio Math. 40(3), 325–365 (1980)MATHMathSciNetGoogle Scholar
21.
Greither, C.: Seminormality, projective algebras, and invertible algebras. J. Algebra 70(2), 316–338 (1981)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
23.
Hamann, E.: On the R-invariance of R[X]. J. Algebra 35, 1–17 (1975)MATHCrossRefMathSciNetGoogle Scholar
24.
Heitmann, R.C.: Lifting seminormality. Michigan Math. J. 57, 439–445 (2008)MATHCrossRefMathSciNetGoogle Scholar
25.
Itoh, S.: On weak normality and symmetric algebras. J. Algebra 85(1), 40–50 (1983)MATHCrossRefMathSciNetGoogle 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)MATHMathSciNetGoogle 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)MATHMathSciNetGoogle 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 folderMATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
38.
Reid, L., Roberts, L.G., Singh, B.: On weak subintegrality. J. Pure Appl. Algebra 114, 93–109 (1996)MATHCrossRefMathSciNetGoogle Scholar
39.
Reid, L., Vitulli, M.A.: The weak subintegral closure of a monomial ideal. Comm. Alg. 27(11), 5649–5667 (1999)MATHCrossRefMathSciNetGoogle 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)MATHMathSciNetGoogle Scholar
42.
Roberts, L.G., Singh, B.: Invertible modules and generic subintegrality. J. Pure Appl. Algebra 95, 331–351 (1994)MATHCrossRefMathSciNetGoogle Scholar
43.
Rush, D.E.: Seminormality. J. Algebra 67, 377–387 (1980)MATHCrossRefMathSciNetGoogle 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)MATHMathSciNetGoogle 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 213MATHGoogle Scholar
47.
Swan, R.G.: On seminormality. J. Algebra 67, 210–229 (1980)MATHCrossRefMathSciNetGoogle 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)MATHMathSciNetGoogle Scholar
51.
Vitulli, M.A.: The hyperplane sections of weakly normal varieties. Am. J. Math. 105(6), 1357–1368 (1983)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
55.
Yanagihara, H.: On glueings of prime ideals. Hiroshima Math. J. 10(2), 351–363 (1980)MATHMathSciNetGoogle Scholar
56.
Yanagihara, H.: Some results on weakly normal ring extensions. J. Math. Soc. Jpn. 35(4), 649–661 (1983)MATHCrossRefMathSciNetGoogle Scholar
57.
Yanagihara, H.: On an intrinisic definition of weakly normal rings. Kobe J. Math. 2, 89–98 (1985)MATHMathSciNetGoogle Scholar

Copyright information

Authors and Affiliations

Marie A. Vitulli
- 1
Email author

1.Department of Mathematics1222 University of OregonEugeneUSA

About this chapter

Cite this chapter as:: Vitulli M.A. (2011) Weak normality and seminormality. In: Fontana M., Kabbaj SE., Olberding B., Swanson I. (eds) Commutative Algebra. Springer, New York, NY

DOI https://doi.org/10.1007/978-1-4419-6990-3_17
Publisher Name Springer, New York, NY
Print ISBN 978-1-4419-6989-7
Online ISBN 978-1-4419-6990-3
eBook Packages Mathematics and Statistics

Personalised recommendations

Actions

Buy eBook

USD 159.00

Instant download
Readable on all devices
Own it forever
Local sales tax included if applicable

Weak normality and seminormality

Abstract

Preview

Copyright information

Authors and Affiliations

Personalised recommendations

Export citation

Cookies