Higher derivations on finitely generated integral domains. II

Abstract:

We prove Theorem. Let $A = k[{x_1}, \ldots ,{x_m}]$ 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 every prime ideal $q \subseteq A$, is equivalent to the following two conditions: (1) no prime of $A$ of height greater than one is differential, and (2) for all $\phi \in {\operatorname {Hom} _k}(A,A),\phi \in \operatorname {Der} _k^n(A)$ if and only if $\Delta \phi \in \Sigma _{i = 1}^{n - 1}\operatorname {Der} _k^i(A) \cup \operatorname {Der} _k^{n - i}(A)(n = 1,2, \ldots )$. Here $\Delta$ denotes the Hochschild coboundary operator, $\cup$ denotes the cup product, and $\operatorname {Der} _k^n(R)$ is the module of higher derivations of rank $n$.