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, Multiplicities and Rees valuations
Katz, Daniel; Validashti, Javid (2010). "Multiplicities and Rees valuations". Collectanea Mathematica. 61: 1–24. Zbl 1216.13016. doi:10.1007/BF03191222.

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