The multiplicity function of a local ring

Hornell, James

doi:10.1090/S0002-9947-1976-0409491-4

The multiplicity function of a local ring
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by James Hornell

Trans. Amer. Math. Soc. 220 (1976), 321-341

DOI: https://doi.org/10.1090/S0002-9947-1976-0409491-4

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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.

References

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Algèbre locale. Multiplicités

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