Sugli omomorfismi quasi étale e gli anelli eccellenti

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In this paper we define quasi étale ring homomorphisms, as a generalization of étale ones, and we show that if ϕ: A→B is a quasi étale homomorphism of noetherian rings, and A is excellent, then B is excellent. The converse is not always true; however we show that if ϕ is faithfully flat and regular (in particular qnasi étale), if A is universally catenarian and B is excellent, then A is excellent. The above statements can be applied when B is the henselization of A with respect to an ideal; and from this it follows that a Hensel pair is a direct limit of excellent ones.


  1. [1]

    N. Bourbaki,Algèbre Commutative, I, II, Hermann, Paris (1961).

  2. [2]

    —— ——,Algèbre Communtative, cap. III, IV, Hermann, Paris (1961).

  3. [3]

    S. Greco,Algebras over non local hensel rings, J. of Algebia, 8 (1968), pp. 45–59.

  4. [4]

    —— ——,Henselization of a ring with respect to an ideal, Trans. Am. Mat. Soc. 144 (1969), pp. 43–65.

  5. [5]

    —— ——,Sugli omomorfismi piatti e non ramificati, Le Matematiche, 24 (1969), pp. 392–415.

  6. [6]

    A. Grothendieck,Éléments de Géometrie Algébrique, cap. IV parte 1a, Publ. Math. no 20, I. H. E. S. (1960).

  7. [7]

    -- --,Éléments de Géometrie Algébrique, cap. IV parte 2a, Publ. Math. no 24, I.H.E.S. (1965).

  8. [8]

    -- --,Éléments de Géometrie Algébrique, cap. IV parte 3a, Publ. Math. no 28, I.H.E.S. (1965).

  9. [9]

    -- --,Éléments de Géometrie Algébrique, cap. IV parte 4a, Publ. Math. no 32, I.H.E.S. (1967).

  10. [10]

    M. Nagata,Local Rings, Interscience Publ. (1962).

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Lavoro eseguito nell’ambito del Comitato Nazionale per la Matematica del C.N.R.

Entrata in Redazione il 25 Febbraio 1971.

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Greco, S. Sugli omomorfismi quasi étale e gli anelli eccellenti. Annali di Matematica 90, 281–296 (1971).

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