On m-adic Higher Differentials in Commutative Rings

Abstract

In [1], it was shown that quotient ring R of affine k-algebra with respect to a separable prime ideal is regular if and only if D^e_N (R, k) is free R-algebra for N≠N. In this paper, we shall define the algebra D^^^∧_N (R, P) of m-adic P-differential of rank N in R. When R is a local ring of equal characteristic, we have the following result under some assumptions : D^^^∧_N (R, k) is m-adic free algebra if and only if R is regular. In this paper, all m-adic ring R are assumed to satisfy the conditions [numerical formula] unless otherwise stated.