J-multiplicity explained

In algebra, a j-multiplicity is a generalization of a Hilbert–Samuel multiplicity. For m-primary ideals, the two notions coincide.

Definition

Let

(R, ak{m}) be a local Noetherian ring of Krull dimension d > 0 . Then the j-multiplicity of an ideal I isj(I) = j(\operatorname{gr} _{I} R) where j(\operatorname{gr} _{I} R) is the normalized coefficient of the degree d - 1 term in the Hilbert polynomial \Gamma _{ak{m}(\operatorname{gr}} _{I} R) ; \Gamma _{ak{m}} means the space of sections supported at ak{m} .

References

• Daniel Katz, Javid Validashti, Multiplicities and Rees valuations
• Katz . Daniel . Validashti . Javid . Multiplicities and Rees valuations . 1216.13016 . Collectanea Mathematica . 61 . 1–24 . 2010 . 10.1007/BF03191222 . 10.1.1.509.99 . 2014-05-18 . https://web.archive.org/web/20120621164601/http://www.collectanea.ub.edu/index.php/Collectanea/article/viewArticle/5243 . 2012-06-21 . dead .