Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-29T07:21:54.240Z Has data issue: false hasContentIssue false
  • English
  • Français

Remarks on generalized analytic independence

Published online by Cambridge University Press:  24 October 2008

Giuseppe Valla
Giuseppe Valla
Affiliation:
Istituto Matematico, University of Genoa, Italy
Get access Rights & Permissions [Opens in a new window]

Extract

This paper is concerned with the notion of independence relating sets of elements in a ring A to a proper ideal a of A. A set of elements a1, …, an ∈A is called a-independent if every form in A[X1, …, Xn] vanishing at a1,…, an has all its coefficients in a. This notion leads to many questions (cf. (2) and (12)), which are of some interest in their own right, several of which are considered here. On the other hand, this independence is related to the structure of the graded ring associated to the ideal generated by the set of elements, hence is often relevant to some problems concerning regular sequences and complex.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 2 , March 1979 , pp. 281 - 289
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Atiyah, M. and MacDonald, G.Introduction to commutative algebra (Reading, Mass; Addison–Wesley, 1969).Google Scholar
(2)Barshay, J.Generalized analytic independence. Proc. Amer. Math. Soc. 58 (1976), 32–36.CrossRefGoogle Scholar
(3)Davis, E. D.Ideals of the principal class, R-sequences and a certain monoidal transformation. Pacific J. Math. 20 (1967), 197–205.CrossRefGoogle Scholar
(4)Eisenbud, D., Herrmann, M. and Vogel, W.Remarks on regular sequences. Nagoya Math. J. 67 (1977), 177–180.CrossRefGoogle Scholar
(5)Herrmann, M. and Schmidt, R. Regular sequences and lifting property. (Preprint.)Google Scholar
(6)Kabele, T.Regularity conditions in non-noetherian rings. Trans. Amer. Math. Soc. 155 (1971), 363–374.CrossRefGoogle Scholar
(7)Micali, A., Salmon, P. and Samuel, P. Integrité et factorialité des algèbres symètriques. Atas do IV Coloquio Brasileiro de Matematica, São Paulo, 1965.Google Scholar
(8)Northcott, D. G.Lessons on rings, modules and multiplicities (Cambridge University Press, 1968).CrossRefGoogle Scholar
(9)Rhodes, C. P. L.A multiplicative property of R-sequences and H1-sets. Math. Proc. Cambridge Philos. Soc. 78 (1975), 1–6.CrossRefGoogle Scholar
(10)Sally, J. and Vasconcelos, W.Stable rings and a problem of Bass. Bull. Amer. Math. Soc. 79 (1973), 574–576.CrossRefGoogle Scholar
(11)Valabrega, P. and Valla, G.Form rings and regular sequences. Nagoya Math. J. 72 (1978). (In the press.)CrossRefGoogle Scholar
(12)Valla, G.Elementi independent! rispetto ad un ideale. Rend. Sem. Mat. Univ. Padova 44 (1970), 339–354.Google Scholar
(13)Zariski, O.Foundations of a general theory of birational correspondences. Trans. Amer. Math. Soc. 53 (1943), 497–542.CrossRefGoogle Scholar