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. doi:10.1007/BF03191222. Zbl 1216.13016.