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

${\displaystyle j(I)=j(\operatorname {gr} _{I}R)}$

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

References

• Daniel Katz, Javid Validashti, Multiplicities and Rees valuations
• Katz, Daniel; Validashti, Javid (2010). "Multiplicities and Rees valuations". Collectanea Mathematica. 61: 1–24. CiteSeerX 10.1.1.509.99. doi:10.1007/BF03191222. Zbl 1216.13016. Archived from the original on 2012-06-21. Retrieved 2014-05-18.