In algebra, a j-multiplicity is a generalization of a
Hilbert–Samuel multiplicity. For ''m''-primary ideals, the two notions coincide.
Definition
Let
be a
local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
of
Krull dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
. Then the j-multiplicity of an
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
''I'' is
:
where
is the normalized coefficient of the degree ''d'' − 1 term in the Hilbert polynomial
;
means the space of sections supported at
.
References
*Daniel Katz, Javid Validashti
Multiplicities and Rees valuations*
Commutative algebra
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