Hilbert–Samuel function explained

M over a commutative Noetherian local ring A and a primary ideal I of A is the map \chi ^{I} _{M} : N → N such that, for all n\in N ,\chi ^{I} _{M} (n) = \ell(M/I ^{n} M)

where

\ell denotes the length over A . It is related to the Hilbert function of the associated graded module \operatorname{gr} _{I(M)} by the identity\chi ^{I} _{M} (n) = \sum ^{n} _{i =0} H(\operatorname{gr} _{I(M),i).}

For sufficiently large

n , it coincides with a polynomial function of degree equal to \dim(\operatorname{gr} _{I(M))} , often called the Hilbert-Samuel polynomial (or Hilbert polynomial).^[2]

Examples

For the ring of formal power series in two variables

k[[x,y]] taken as a module over itself and the ideal I generated by the monomials x² and y³ we have\chi( 1 ) =6 , \chi( 2 ) =18 , \chi( 3 ) =36 , \chi( 4 ) =60 , andingeneral \chi(n) =3 n(n +1 ) for n \geq 0 . ^[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

P _{I,} the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

Proof: Tensoring the given exact sequence with

R/I ^{n} and computing the kernel we get the exact sequence:0 \to (I ^{n} M \cap M')/I ^{n} M' \to M'/I ^{n} M' \to M/I ^{n} M \to M''/I ^{n} M'' \to 0 , which gives us:\chi ^{I(n -1 )} _{M} = \chi ^{I(n -1 )} _{M'} + \chi ^{I(n -1 )} _{M''} - \ell((I ^{n} M \cap M')/I ^{n} M') .The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large n and some k,I ^{n} M \cap M' = I ^{n-k} ((I ^{k} M) \cap M') \subset I ^{n-k} M'. Thus,\ell((I ^{n} M \cap M') / I ^{n} M') \le \chi ^{I} _{M'} (n -1 ) - \chi ^{I} _{M'} (n - k -1 ) .This gives the desired degree bound.

Multiplicity

If

A is a local ring of Krull dimension d , with m -primary ideal I , its Hilbert polynomial has leading term of the form (e)/(d!) ⋅ n ^{d} for some integer e . This integer e is called the multiplicity of the ideal I . When I = m is the maximal ideal of A , one also says e is the multiplicity of the local ring A .

The multiplicity of a point

x of a scheme X is defined to be the multiplicity of the corresponding local ring l{O} _{X,x} .

See also

• j-multiplicity

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.