Higher derivations on finitely generated integral domains

Abstract

In this paper, we prove the following theorem: Let A = k [ x 1 , ⋯ , x t ] 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 {A_q} is regular for all primes q ⊆ A q \subseteq A , is equivalent to the following two conditions: (1) No nonminimal prime of A is differential, and (2) der n ⁡ ( A / k ) = D e r n ( A / k ) \operatorname {der}^n (A/k) = \mathrm {Der}^n (A/k) for all n. Here Der n ⁡ ( A / k ) \operatorname {Der}^n (A/k) denotes the A-module of all nth order derivations of A into A which are zero or k, and der n ⁡ ( A / k ) \operatorname {der}^n(A/k) denotes the A-submodule of Der n ⁡ ( A / k ) \operatorname {Der}^n(A/k) generated by composites δ 1 ∘ ⋯ ∘ δ j ( 1 ≦ j ≦ n ) {\delta _1} \circ \cdots \circ {\delta _j}(1 \leqq j \leqq n) of first order derivations δ i {\delta _i} .