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

Abstract

Let A=k[x lt -' , x g ] 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 /^-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 Der {A) = Der| (A) o and inductively define Der(A) 0 as follows: " A) o U DerJ'Wy . The authors show that if e Der (A) 09 then {A) A. Various examples are given which indicate that the above mentioned result is about as good as possible.