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 ,
For sufficiently large , it coincides with a polynomial function of degree equal to , often called the Hilbert-Samuel polynomial (or Hilbert polynomial).
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,
This gives the desired degree bound.
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 .
- H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203.
- Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969.
- This implies that and also have finite length.
- Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8. Lemma 12.3.