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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Colength of derivation ideals

Author: Kenneth Kramer
Journal: Proc. Amer. Math. Soc. 40 (1973), 346-350
MSC: Primary 13B10
MathSciNet review: 0318122
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Abstract: In this paper, $ D$ is a derivation acting on the formal power series ring $ K[[{t_1}, \cdots ,{t_r}]]$ over a field $ K$ of characteristic $ p \ne 0$. We conjecture that the colengths of the ideals ( $ ({D^{{p^n}}}{t_1}, \cdots ,{D^{{p^n}}}{t_r})$ and $ ({D^{{p^{n - 1}}}}{t_1}, \cdots ,{D^{{p^{n - 1}}}}{t_r})$ are congruent modulo $ {p^n}$, provided they are finite. We give a proof for the case $ r = 1$ and any $ n \geqq 1$, and for the case $ n = 1$ and any $ r \geqq 1$.

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Article copyright: © Copyright 1973 American Mathematical Society