Some Remarks on High Order Derivations Ⅱ <Dedicated to Professor Yoshikazu Nakai on his 60th birthday>

概要

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Let A be a commutative ring with 1 and k be a subfield of A. In the previous papers [2] and [3], Y. Nakai and the author showed that the following three statements are equivalent, provided that A is a field. (1゜) D(A/k)＝Homk(A, A), where D(A/k) denotes the derivation algebra of A over k. (2゜) A is purely inseparable over k and [A : k]＜∞. (3゜) IA＝Ker(A⊗kA→A) is a nilpotent ideal of A⊗kA. Subsequently Y. Nakai proved that the statement (1゜) is equivalent in general (i.e. without assumption that A is a field) to the next statement ([4]). (2゜)′　A is a quasi-local ring with the maximal ideal m such that m is nilpotentand the residue field A/m is a purely inseparable and finite extension over k. Moreover his proof had an effect on making it clear that even (1゜) and (3゜) are equivalent in general, although it was not stated explicitly. However it seems to me that not a little part of his proof of (1゜)⇒(2゜)′ is based on our previous theorem. In this paper we shall try to give a simpler proof not making use of our previous theorem, as much as possible, except in the last stage. And we shall give a straightforward proof of the equivalence of (1゜) and (3゜) in general setting. The idea of the proofs of Prop. 5 and Lemma 6 is due to Nakai [4] and [5].