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 formula_1 be a local Noetherian ring of Krull dimension formula_2. Then the j-multiplicity of an ideal "I" is
where formula_4 is the normalized coefficient of the degree "d" − 1 term in the Hilbert polynomial formula_5; formula_6 means the space of sections supported at formula_7.