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 [math]\displaystyle{ (R, \mathfrak{m}) }[/math] be a local Noetherian ring of Krull dimension [math]\displaystyle{ d \gt 0 }[/math]. Then the j-multiplicity of an ideal I is

[math]\displaystyle{ j(I) = j(\operatorname{gr}_I R) }[/math]

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

References

• Daniel Katz, Javid Validashti, Multiplicities and Rees valuations
• Katz, Daniel; Validashti, Javid (2010). "Multiplicities and Rees valuations". Collectanea Mathematica 61: 1–24. doi:10.1007/BF03191222. http://www.collectanea.ub.edu/index.php/Collectanea/article/viewArticle/5243. Retrieved 2014-05-18.