形式的ベキ級数環における高階微分について

Abstract

In [1], the concept of higher differentials in a commutative ring (by means of universal higher derivation) was introduced and it was shown that if a geometric regular local ring R is regular, then the submodule A^n(R) of A(R) generated by elements of degree n over R is R-free. In this paper, we shall consider the case where R is a formal power series ring. When R is a residue class ring k[[X_1,…, X_S]]_q/pk[[X_1,…, X_s]]_q where p, q are prime ideals in k[[X_1,…, Xs]] such that p⊂q, we have the following result under some conditions: The submodule A^n(R) of A(R) generated by elements of degree n over R is R-free if R is regular.