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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The multiplicity function of a local ring


Author: James Hornell
Journal: Trans. Amer. Math. Soc. 220 (1976), 321-341
MSC: Primary 14M10; Secondary 13H15
MathSciNet review: 0409491
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Abstract: Let A be a local ring with maximal ideal m. Let $ f \in A$, and define $ {\mu _A}(f)$ to be the multiplicity of the A-module $ A/Af$ with respect to m. Under suitable conditions $ {\mu _A}(fg) = {\mu _A}(f) + {\mu _A}(g)$. The relationship of $ {\mu _A}$ to reduction of A, normalization of A and a quadratic transform of A is studied. It is then shown that there are positive integers $ {n_1}, \ldots ,{n_s}$ and rank one discrete valuations $ {v_1}, \ldots ,{v_s}$ of A centered at m such that $ {\mu _A}(f) = {n_1}{v_1}(f) + \cdots + {n_s}{v_s}(f)$ for all regular elements f of A.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0409491-4
PII: S 0002-9947(1976)0409491-4
Article copyright: © Copyright 1976 American Mathematical Society