# Hilbert–Samuel function

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 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).[2]

## 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.

**Theorem** — Let be a Noethrian 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. In particular, if , then the degree of is strictly less than that of .

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

## References

- 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.