Hilbert–Samuel function
Introduction In commutative algebra the '''Hilbert–Samuel function''', named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module  over a commutative Noetherian local ring  and a primary ideal  of  is the map  such that, for all ,:where  denotes the length over . 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).
Examples For the ring of formal power series in two variables  taken as a module over itself and the ideal  generated by the monomials ''x''2 and ''y''3 we have:
Degree bounds 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. We denote by  the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers. 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.
Multiplicity If  is a local ring of Krull dimension , with -primary ideal , its Hilbert polynomial has leading term of the form  for some integer . This integer  is called the '''multiplicity''' of the ideal . When  is the maximal ideal of , one also says  is the multiplicity of the local ring .The multiplicity of a point  of a scheme  is defined to be the multiplicity of the corresponding local ring .
See also *j-multiplicity
References