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 x2 and y3 we have

[2]

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

  1. 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.
  2. Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969.
  3. This implies that and also have finite length.
  4. 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.
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