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

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

James Hornell, Intersection theory in an equicharacteristic regular local ring and the relative intersection theory, Proc. Amer. Math. Soc. 36 (1972), 8–12. MR 309936, DOI 10.1090/S0002-9939-1972-0309936-6
James Hornell, Divisorial complete intersections, Pacific J. Math. 45 (1973), 217–227. MR 318119
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
D. G. Northcott, Lessons on rings, modules and multiplicities, Cambridge University Press, London, 1968. MR 0231816
D. G. Northcott, The neighbourhoods of a local ring, J. London Math. Soc. 30 (1955), 360–375. MR 71110, DOI 10.1112/jlms/s1-30.3.360

Algèbre locale. Multiplicités

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Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249