In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,[1] of a nonzero finitely generated module
over a commutative Noetherian local ring
and a primary ideal
It is related to the Hilbert function of the associated graded module
by the identity For sufficiently large
, it coincides with a polynomial function of degree equal to
, often called the Hilbert-Samuel polynomial (or Hilbert polynomial).
[2] For the ring of formal power series in two variables
taken as a module over itself and the ideal
generated by the monomials x2 and y3 we have Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence.
However, it is still reasonably close to being additive, as a consequence of the Artin–Rees lemma.
the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.
be a Noetherian local ring and I an m-primary ideal.
If is an exact sequence of finitely generated R-modules and if
has finite length,[3] then we have:[4] where F is a polynomial of degree strictly less than that of
and having positive leading coefficient.
Proof: Tensoring the given exact sequence with
and computing the kernel we get the exact sequence: which gives us: The third term on the right can be estimated by Artin-Rees.
Indeed, by the lemma, for large n and some k, Thus, This gives the desired degree bound.
is a local ring of Krull dimension
, its Hilbert polynomial has leading term of the form
is called the multiplicity of the ideal
is the maximal ideal of
is the multiplicity of the local ring
The multiplicity of a point
is defined to be the multiplicity of the corresponding local ring