Some Topological Properties of the Derivation Algebra D(K/k) for a Field Extension K/k

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  • Some Topological Properties of the Deri

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Let K be a field extension of a field k, and D(K/k) be its derivation algebra. In the previous paper [1], we gave some topological considerations on D(K/k). And we showed, for example, that D(K/k) is dense in Homk(K,K) with respect to the finite topology if and only if the center Z(D)(K/k)) of D(K/k) coincides with k. On the other hand, we know already that D(K/k)＝D(K/ks) and Z(D(K/ks))⊃ks (where ks denotes the separable algebraic closure of k in K) and that if [K:k] is finite then D(K/k) is discrete. Consequently D(K/k) is discrete if [K:ks〕 is finite. In this paper first we show that the converses of these are false. That is, we show the existence of an example K/k such that Z(D(/k))≠ks and D(K/k) is discrete nevertheless [K:ks] is not finite. Next we shall prove that, under the assumption Z(D(K/k)) ＝k, D(K/k) is discrete if and only if [K:k] is finite. Finally we shall notice that, under a natural isomorphismΦ:Homk(K ？ K, K)CSH→Homk (K,K) such thatΦ(f)(x) ＝f(1 ？ x), Φ-1(D(K/k)) is the totality of the K-linear maps f:K ？ K→K which are continuous with respect to the Ik/k-adic topology of k ？ K and the discrete topology of K. Notation and terminology are the sarrle as [1], [2] and [3]. The finite topology of Homk(K, K) or D(K/k) defined in [1] will be called the finite k-topology in this paper. ？ means always ？ k.

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