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 &minus; 1 term in the Hilbert polynomial $$\Gamma_\mathfrak{m}(\operatorname{gr}_I R)$$; $$\Gamma_\mathfrak{m}$$ means the space of sections supported at $$\mathfrak{m}$$.