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 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 generall ...
. Then the j-multiplicity of an
ideal ''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
{{commutative-algebra-stub