An application of the algebra of differentials of infinite rank

Abstract

Let k denote an arbitrary field and let R be an affine local domain over k. Let ( Ω k ( R ) , δ k R ) ({\Omega _k}(R),\delta _k^R) be the universal algebra of k-higher differentials over R. Let K be the quotient field of R and L the residue class field of R. If K is a separable extension of k and L is a separable algebraic extension of k, then it is shown that R is a regular local ring if and only if Ω k ( R ) {\Omega _k}(R) is a free R-algebra. If both K and L are separable extensions of k and R has a separating residue class field, then R is a regular local ring if and only if Ω k ( R ) {\Omega _k}(R) is a free emphR-algebra.