Hostname: page-component-7bb8b95d7b-l4ctd Total loading time: 0 Render date: 2024-09-27T01:28:48.237Z Has data issue: false hasContentIssue false
  • English
  • Français

On algebra with universal finite module of differentials

Published online by Cambridge University Press:  22 January 2016

Norio Yamauchi
Norio Yamauchi*
Affiliation:
Shotoku-Gakuen Women’s Junior College
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let k be a field and A a noetherian k-algebra. In this note, we shall study the universal finite module of differentials of A over k, which is denoted by Dk(A). When the characteristic of k is zero, detailed results have been obtained by Scheja and Storch [8]. So we shall treat the positive characteristic case. In § 1, we shall study differential modules of a local ring over subfields. We obtain a criterion of regularity (Theorem (1.14)). In § 2, we shall study the formal fibres and regular locus of A with Dk(A). Our main result is Theorem (2.1) which shows that, if Dk(A) exists, then A is a universally catenary G-ring under a certain assumption. In the local case, this is a generalization of Matsumura’s theorem ([5] Theorem 15), where regularity of A is assumed.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 83 , October 1981 , pp. 107 - 121
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Marot, J., Sur les anneaux universellement japonais, C. R. Acad. Sci. Paris, 277 (1973), 1029-1031.Google Scholar
[2] Matsumura, H., Commutative Algebra, Benjamin, New York, 1970.Google Scholar
[3] Matsumura, H., Formal power series rings over polynomial rings I, in “Number Theory, Algebraic Geometry, and Commutative Algebra” (in honor of Y. Akizuki), Kinokuniya, Tokyo, (1973), 511–520.Google Scholar
[4] Matsumura, H., Quasi-coefficient rings of a local ring, Nagoya Math. J., 68 (1977), 123–130.Google Scholar
[5] Matsumura, H., Noetherian rings with many derivations, in “Contributions to Algebra” (in honor of E. Kolchin), Academic Press, New York, (1977), 279–294.Google Scholar
[6] Nagata, M., Local rings, Wiley, New York, 1962.Google Scholar
[7] Nomura, M., Formal power series rings over polynomial rings II, in “Number Theory, Algebraic Geometry and Commutative Algebra” (in honor of Y. Akizuki), Kinokuniya, Tokyo, (1973), 521–528.Google Scholar
[8] Scheja, G. and Storch, U., Différentielle Eigenschaften der Lokalisierungen analytischer Algebren, Math. Ann., 197 (1972), 137–170.Google Scholar
[9] Valabrega, P., On the excellence property for power series rings over polynomial rings, J. Math, of Kyoto Univ., 15 (1975), 387–395.Google Scholar