Vol. 63, No. 2, 1976

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On extending higher derivations generated by cup products to the integral closure

Joseph Becker and William C. Brown

Vol. 63 (1976), No. 2, 325–334

Let A = k[x1,,xg] be a finitely generated integral domain over a field k of characteristic zero. Let A denote the integral closure of A in its quotient field. A well known result due to A. Seidenberg says that any first order k-derivation of A can be extended to A. This result is known to be false for higher order derivations. In this paper, the authors investigate what types of higher derivations on A can be extended to A. The main results are for higher derivations which are cup products. Set Derk1(A) = Derk1(A)0 and inductively define Derkn(A)0 as follows:

   n              n        n∑− 1   i        n− i
Derk(A)0 = {φ ∈ Derk(A)|Δ φ ∈   Derk(A )0 ∪Derk  (A)0}.

The authors show that if φ Derkn(A)0, then φ(A) A. Various examples are given which indicate that the above mentioned result is about as good as possible.

Mathematical Subject Classification 2000
Primary: 13B10
Secondary: 14F10
Received: 25 March 1975
Published: 1 April 1976
Joseph Becker
William C. Brown