Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-11T12:31:47.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Two theorems on excellent rings1)

Published online by Cambridge University Press:  22 January 2016

Silvio Greco
Silvio Greco*
Affiliation:
Istituto Matematico del Politecnico di Torino
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 f: A → B be a homomorphism of commutative noetherian rings. The main results of this paper are:

(a) Assume f is finite and induces a surjective map on the spectra. Then if B is quasi-excellent A is quasi-excellent and is excellent if it is universally catenarian (Th. 3.1); and

(b) If f is absolutely flat and A is excellent then B is excellent (Th. 5.3). In particular the strict henselization of an excellent local ring is excellent (Cor. 5.6.).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 60 , February 1976 , pp. 139 - 149
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

Footnotes

1)

This research was supported by CNR.

References

[1] André, M., Localisation de la lissité formelle, Man. Math., 1974.Google Scholar
[2] Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Addison-Wesley, Reading (Mass.), 1969.Google Scholar
[3] Ferrand, D., Monomorphismes et morphismes absolument plats, Bull. Soc. Math. France, 100 (1972), 97–128.Google Scholar
[4] Greco, S., Sugli omomorfismi piatti e non ramificati, Le Matematiche (Catania), 24 (1969), 392–415.Google Scholar
[5] Greco, S., Sugli omomorfismi quasi étale e gli anelli eccellenti, Ann. Mat. Pura ed Appl., 90 (1971), 281–296.Google Scholar
[6] Greco, S. and Salmon, P., Topics in m-adic topologies, Erg. der Math. b. 58, Springer Verlag, 1971.Google Scholar
[7] Grothendieck, A. and Dieudonné, J., Eléments de Géométrie Algébrique I, Grund. der Math. b. 166, Springer Verlag, 1971.Google Scholar
[8] Grothendieck, A. and Dieudonné, J., Eléments de Géométrie Algébrique IV, Publ. Math. 20 . . ., IHES.Google Scholar
[9] Matsumura, H., Commutative Algebra, W.A. Benjamin Inc., 1970.Google Scholar
[10] Matsumura, H., Formal power series rings over polynomial rings I, Number Theory, Algebraic Geometry, and Commutative Algebra in honor of Y. Akizuki, Tokio, 1973, 511–520.Google Scholar
[11] Nomura, M., Formal power series rings over polynomial rings II, Number Theory, Algebraic Geometry and Commutative Algebra in honor of Y. Akizuki, Tokio, 1973, 521–528.Google Scholar
[12] Olivier, J. P., Montée des propriétés par morphismes absolument plats, C. R. des Journées d’Algèbre Pure et Appliquée, Montpellier, 1971, 86–109.Google Scholar
[13] Pedrini, C., Incollamenti di ideali primi e gruppi di Picard, Rend. Sem. Mat. Univ. Padova, 48 (1973), 39–66.Google Scholar
[14] Ratliff, L. J. Jr., Catenary rings and the altitude formula, Amer. J. Math., 94 (1972), 458–466.Google Scholar
[15] Seydi, H., Anneaux henséliens et conditions des chaines, Bull. Soc. Math. France, 98 (1970), 9–31.CrossRefGoogle Scholar
[16] Seydi, H., Sur la théorie des anneaux de Weierstrass I, Bull. Sc. Math., 96 (1971), 227–235.Google Scholar
[17] Seydi, H., Sur la théorie des anneaux excellents en caractéristique p, I, Bull. Sc. Math., 96 (1972), 193–198.Google Scholar
[18] Valabrega, P., On two-dimensional regular local rings and a lifting problem, Ann. Sc. Norm. Sup. Pisa, 27 (1973), 1–21.Google Scholar
[19] Valabrega, P., On the excellent property for rings of restricted power series, Boll. UMI, 9 (1974), 486–494.Google Scholar
[20] Valabrega, P., On the excellent property for power series rings over polynomial rings, Kyoto J. Math., Vol. 15, No. 2, 1975.Google Scholar