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Symbolic Powers of Regular Primes

Published online by Cambridge University Press:  20 November 2018

Yasunori Ishibashi
Yasunori Ishibashi*
Affiliation:
Hiroshima University, Hiroshima, Japan
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In a recent paper [6], P. Seibt has obtained the following result: Let k be a field of characteristic 0, k[T1, … , Tr] the polynomial ring in r indeterminates over k, and let P be a prime ideal of k[T1, … , Tr]. Then a polynomial F belongs to the n-th symbolic power P(n) of P if and only if all higher derivatives of F from the 0-th up to the (n – l)-st order belong to P.

In this work we shall naturally generalize this result so as to be valid for primes of the polynomial ring over a perfect field k. Actually, we shall get a generalization as a corollary to a theorem which asserts: For regular primes P in a k-algebra R of finite type, a certain differential filtration of R associated with P coincides with the symbolic power filtration (P(n))n≧0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 6 , 01 December 1981 , pp. 1331 - 1337
Copyright
Copyright © Canadian Mathematical Society 1981

References

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6. Seibt, P., Differential filiations and symbolic powers of regular primes, Math. Z. 166 (1979), 159–164.Google Scholar