Coefficient fields and scalar extension in positive characteristic (Q1772450)

Let \(k_{(\infty)}\) be the perfect closure of the field \(k(t)\) of rational functions over a perfect field \(k\) of positive characteristic and \(A=k[[X_{1},...,X_{n}]].\) In this paper the authors show that for any maximal ideal \(\mathbf{n}\) of \(A'=k_{(\infty)}\otimes_{k} A,\) the elements in the completion \(\widehat{A'_{\mathbf{n}}}\) which are annihilated by the ``Taylor'' Hasse-Schmidt derivations with respect to the \(X_{i}\) form a coefficient field of \(\widehat{A'_{\mathbf{n}}}.\) For proving the above result the authors give a complete proof of the normalization lemma for power series rings over perfect fields, which is of independent interest as well. This paper extends some results of the second author to the positive characteristic case [cf. \textit{L. Narváez-Macarro}, J. Lond. Math. Soc. (2) 43, No. 1, 12--22 (1991; Zbl 0687.14015)].