Remarks on m-adic Higher Differential Theoretic Characterization of Regular Local Rings

Abstract

This is a suite of the previous paper[1]. In that paper, we showed under some assumptions that when R is a local ring of equal characteristic with maximal ideal m, then the algebra D^^^^_N (R, P) of m-adic P-differentials of rank N (≐̸N) in R is free if and only if R is regular. In this paper, we shall show under some assumptions that when R is a local ring of unequal characteristic with maximal ideal m, then D^^^^_N (R, P) is free if and only if R is regular and m^2 does not contain a prime element u of P.