J-multiplicity

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

Definition

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

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

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

References

• Daniel Katz, Javid Validashti, Multiplicities and Rees valuations
• Template:Cite journal

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