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

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