j-multiplicity
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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 ) {\displaystyle (R,{\mathfrak {m}})} be a local Noetherian ring of Krull dimension d > 0 {\displaystyle d>0}. Then the j-multiplicity of an ideal I is
j ( I ) = j ( gr I R ) {\displaystyle j(I)=j(\operatorname {gr} _{I}R)}
where j ( gr I R ) {\displaystyle j(\operatorname {gr} _{I}R)} is the normalized coefficient of the degree d − 1 term in the Hilbert polynomial Γ m ( gr I R ) {\displaystyle \Gamma _{\mathfrak {m}}(\operatorname {gr} _{I}R)}; Γ m {\displaystyle \Gamma _{\mathfrak {m}}} means the space of sections supported at m {\displaystyle {\mathfrak {m}}}.