j-multiplicity

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

Definition

Let $(R, \mathfrak{m})$ be a local Noetherian ring of Krull dimension $d > 0$ . Then the j-multiplicity of an ideal I is

: $j(I) = j(\operatorname{gr}_I R)$

where $j(\operatorname{gr}_I R)$ is the normalized coefficient of the degree d − 1 term in the Hilbert polynomial $\Gamma_\mathfrak{m}(\operatorname{gr}_I R)$ ; $\Gamma_\mathfrak{m}$ means the space of sections supported at $\mathfrak{m}$ .

References

Daniel Katz, Javid Validashti, [https://web.archive.org/web/20160305005558/http://www.math.ku.edu/~dlk/dkjv_final.pdf Multiplicities and Rees valuations]
{{cite journal | last1=Katz | first1=Daniel | last2=Validashti | first2=Javid | title=Multiplicities and Rees valuations | zbl=1216.13016 | journal=Collectanea Mathematica | volume=61 | pages=1–24 | year=2010 | doi=10.1007/BF03191222 | url=http://www.collectanea.ub.edu/index.php/Collectanea/article/viewArticle/5243 | citeseerx=10.1.1.509.99 | access-date=2014-05-18 | archive-url=https://web.archive.org/web/20120621164601/http://www.collectanea.ub.edu/index.php/Collectanea/article/viewArticle/5243 | archive-date=2012-06-21 | url-status=dead }}

Category:Commutative algebra