Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Efficient generation of maximal ideals in polynomial rings


Authors: E. D. Davis and A. V. Geramita
Journal: Trans. Amer. Math. Soc. 231 (1977), 497-505
MSC: Primary 13F20; Secondary 13E05
DOI: https://doi.org/10.1090/S0002-9947-1977-0472800-5
MathSciNet review: 0472800
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The cardinality of a minimal basis of an ideal I is denoted $ \nu (I)$. Let A be a polynomial ring in $ n > 0$ variables with coefficients in a noetherian (commutative with $ 1 \ne 0$) ring R, and let M be a maximal ideal of A. In general $ \nu (M{A_M}) + 1 \geqslant \nu (M) \geqslant \nu (M{A_M})$. This paper is concerned with the attaining of equality with the lower bound. It is shown that equality is attained in each of the following cases: (1) $ {A_M}$ is not regular (valid even if A is not a polynomial ring), (2) $ M \cap R$ is maximal in R and (3) $ n > 1$. Equality may fail for $ n = 1$, even for R of dimension 1 (but not regular), and it is an open question whether equality holds for R regular of dimension $ > 1$. In case $ n = 1$ and $ \dim (R) = 2$ the attaining of equality is related to questions in the K-theory of projective modules. Corollary to (1) and (2) is the confirmation, for the case of maximal ideals, of one of the Eisenbud-Evans conjectures; namely, $ \nu (M) \leqslant \max \{ \nu (M{A_M}),\dim (A)\} $. Corollary to (3) is that for R regular and $ n > 1$, every maximal ideal of A is generated by a regular sequence--a result well known (for all $ n \geqslant 1$) if R is a field (and somewhat less well known for R a Dedekind domain).


References [Enhancements On Off] (What's this?)

  • [A-T] E. Artin and J. T. Tate, A note on finite ring extensions, J. Math. Soc. Japan 3 (1951), 74-77. MR 13, 427. MR 0044509 (13:427c)
  • [B] H. Bass, Libération des modules projectifs sur certain anneaux de polynômes, Séminaire Bourbaki 1973/1974, Exposé no. 448, Lecture Notes in Math., vol. 431, Springer-Verlag, Berlin and New York, 1975, pp. 228-254. MR 51 # 18. MR 0472826 (57:12516)
  • [B-M] H. Bass and M. P. Murthy, Grothendieck groups and Picard groups of abelian group rings, Ann. of Math. (2) 86 (1967), 16-73. MR 36 #2671. MR 0219592 (36:2671)
  • [Bo] E. Bombieri, Seminormalità e singolarità ordinarie, Symposia Mathematica, Vol. XI (INDAM, Rome, 1971), Academic Press, New York and London, 1973, pp. 205-210. MR 49 #11361. MR 0429874 (55:2884)
  • [D-1] E. D. Davis, Ideals of the principal class, R-sequences and a certain monoidal transformation, Pacific J. Math. 20 (1967), 197-205. MR 34 #5860. MR 0206035 (34:5860)
  • [D-2] -, Regular sequences and minimal bases, Pacific J. Math. 26 (1971), 323-326. MR 43 #1968. MR 0276220 (43:1968)
  • [D-3] -, On the geometric interpretation of seminormality (in preparation).
  • [D-G] E. D. Davis and A. V. Geramita, Maximal ideals in polynomial rings, Conf. on Commutative Algebra, Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin and New York, 1973, pp. 57-60. MR 49 #2697. MR 0337928 (49:2697)
  • [E] S. Endô, Projective modules over polynomial rings, J. Math. Soc. Japan 15 (1963), 339-352. MR 27 #5808. MR 0155875 (27:5808)
  • [Fd] D. Ferrand, Courbes gauches et fibres de rang 2, C. R. Acad. Sci. Paris Ser. A 281 (1975), A345-A347. MR 0379517 (52:422)
  • [F] O. Forster, Über die Anzahl der Erzeugenden eines Ideals in einem Noetherschen Ring, Math. Z. 84 (1964), 80-87; erratum, ibid. 86 (1964), 190. MR 29 # 1231. MR 0163932 (29:1231)
  • [E-E] D. Eisenbud and G. Evans, Three conjectures about modules over polynomial rings, Conf. on Commutative Algebra, Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin and New York, 1973, pp. 78-89. MR 48 #8466. MR 0432627 (55:5614)
  • [G] A. V. Geramita, Maximal ideals in polynomial rings, Proc. Amer. Math. Soc. 41 (1973), 34-36. MR 47 #6672. MR 0318123 (47:6672)
  • [Mo] T. T. Moh, On the unboundedness of generators of prime ideals in power series rings of three variables, J. Math. Soc. Japan 26 (1974), 722-734. MR 50 #7158. MR 0354680 (50:7158)
  • [M] M. P. Murthy, Projective $ A[X]$-modules, J. London Math. Soc. 41 (1966), 453-456. MR 34 #188. MR 0200289 (34:188)
  • [N] M. Nagata, Local rings, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)
  • [Q] D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167-171. MR 0427303 (55:337)
  • [S] P. Salmon, Singolarità e gruppo di Picard, Symposia Mathematica, Vol. II, (INDAM, Rome, 1968), Academic Press, New York and London, 1969, pp. 341-345. MR 40 #4263. MR 0251032 (40:4263)
  • [Sg] A. Seidenberg, On the dimension theory of rings. II, Pacific J. Math. 4 (1954), 603-614. MR 16,441. MR 0065540 (16:441g)
  • [Sr] J.-P. Serre, Sur les modules projectif, Séminaire Dubreil-Pisot (1960/1961), No. 2, Sécretariat Mathématique, Paris, 1963. MR 28 #3911.
  • [Sw] R. Swan, Serre's problem, Conf. on Commutative Algebra-1975 (A. V. Geramita, Editor), Queen's Papers in Pure and Applied Math., no. 42, Queen's Univ., Kingston, Ontario, Canada. MR 0396531 (53:394)
  • [T] C. Traverso, Seminormality and Picard group, Ann. Scuola Norm. Sup. Pisa (3) 24 (1970), 585-595. MR 43 #3275. MR 0277542 (43:3275)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 13F20, 13E05

Retrieve articles in all journals with MSC: 13F20, 13E05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0472800-5
Keywords: Polynomial ring, maximal ideal, number of generators, Hilbert rings, regular sequences, projective modules, $ {K_0}$, seminormal ring
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society