Higher derivations on finitely generated integral domains

Abstract:

In this paper, we prove the following theorem: Let $A = k[{x_1}, \cdots ,{x_t}]$ be a finitely generated integral domain over a field k of characteristic zero. Then A regular, i.e. the local ring ${A_q}$ is regular for all primes $q \subseteq A$, is equivalent to the following two conditions: (1) No nonminimal prime of A is differential, and (2) $\operatorname {der}^n (A/k) = \mathrm {Der}^n (A/k)$ for all n. Here $\operatorname {Der}^n (A/k)$ denotes the A-module of all nth order derivations of A into A which are zero or k, and $\operatorname {der}^n(A/k)$ denotes the A-submodule of $\operatorname {Der}^n(A/k)$ generated by composites ${\delta _1} \circ \cdots \circ {\delta _j}(1 \leqq j \leqq n)$ of first order derivations ${\delta _i}$.