Pure Descent for the Module of Zariski Differentials
نویسنده
چکیده
It will be shown that for any given pure extension A -» B of noetherian k -algebras, with k being a field of characteristic zero, and for any prime ideal b Q A the Zariski-Lipman conjecture for Av is solvable, if £ is a locally factorial domain for which the finite differential module is reflexive. We will also discuss an embedding property with respect to the module of Zariski differentials of A9. 1. Preliminaries. A homomorphism S is equivalent to saying that E -*• E <8>R S is injective, where E denotes the injective hull of the Ä-module R/mR, see (6.11) in [3]. Let x G E generate the one-dimensional socle of the Ä-module E and c := Anns(x ® 1) with mRS Q c G S. Then Sq is pure, too. Secondly, we will discuss a class of pure extensions in characteristic zero. Let R ®t K is faithfully flat, and the extended homomorphism R ®kK-+S is nondegenerate. Therefore we may even assume k = S/ms. Now, let r := dim S/mRS and *,, . . ., tr be elements of ms whose residue classes modulo mRS form a system of parameters in S/mRS. The extended homomorphism R' := R[TU ..., T¡¡-*S with <p'\R = <¡p and Received by the editors January 30, 1980 and, in revised form, March 24, 1980. 1980 Mathematics Subject Classification. Primary 13B02; Secondary 14B15. © 1981 American Mathematical Society 0002-9939/81/0000-0201/$02.50 7 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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