Hilbert–Samuel function

In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module [math]\displaystyle{ M }[/math] over a commutative Noetherian local ring [math]\displaystyle{ A }[/math] and a primary ideal [math]\displaystyle{ I }[/math] of [math]\displaystyle{ A }[/math] is the map [math]\displaystyle{ \chi_{M}^{I}:\mathbb{N}\rightarrow\mathbb{N} }[/math] such that, for all [math]\displaystyle{ n\in\mathbb{N} }[/math],

[math]\displaystyle{ \chi_{M}^{I}(n)=\ell(M/I^{n}M) }[/math]

where [math]\displaystyle{ \ell }[/math] denotes the length over [math]\displaystyle{ A }[/math]. It is related to the Hilbert function of the associated graded module [math]\displaystyle{ \operatorname{gr}_I(M) }[/math] by the identity

[math]\displaystyle{ \chi_M^I (n)=\sum_{i=0}^n H(\operatorname{gr}_I(M),i). }[/math]

For sufficiently large [math]\displaystyle{ n }[/math], it coincides with a polynomial function of degree equal to [math]\displaystyle{ \dim(\operatorname{gr}_I(M)) }[/math], often called the Hilbert-Samuel polynomial (or Hilbert polynomial).

Examples

For the ring of formal power series in two variables [math]\displaystyle{ kx,y }[/math] taken as a module over itself and the ideal [math]\displaystyle{ I }[/math] generated by the monomials x² and y³ we have

[math]\displaystyle{ \chi(1)=6,\quad \chi(2)=18,\quad \chi(3)=36,\quad \chi(4)=60,\text{ and in general } \chi(n)=3n(n+1)\text{ for }n \geq 0. }[/math]

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 [math]\displaystyle{ P_{I, M} }[/math] the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

Theorem — Let [math]\displaystyle{ (R, m) }[/math] be a Noetherian local ring and I an m-primary ideal. If

[math]\displaystyle{ 0 \to M' \to M \to M'' \to 0 }[/math]

is an exact sequence of finitely generated R-modules and if [math]\displaystyle{ M/I M }[/math] has finite length, then we have:

[math]\displaystyle{ P_{I, M} = P_{I, M'} + P_{I, M''} - F }[/math]

where F is a polynomial of degree strictly less than that of [math]\displaystyle{ P_{I, M'} }[/math] and having positive leading coefficient. In particular, if [math]\displaystyle{ M' \simeq M }[/math], then the degree of [math]\displaystyle{ P_{I, M''} }[/math] is strictly less than that of [math]\displaystyle{ P_{I, M} = P_{I, M'} }[/math].

Proof: Tensoring the given exact sequence with [math]\displaystyle{ R/I^n }[/math] and computing the kernel we get the exact sequence:

[math]\displaystyle{ 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, }[/math]

which gives us:

[math]\displaystyle{ \chi_M^I(n-1) = \chi_{M'}^I(n-1) + \chi_{M''}^I(n-1) - \ell((I^n M \cap M')/I^n M') }[/math].

The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large n and some k,

[math]\displaystyle{ I^n M \cap M' = I^{n-k} ((I^k M) \cap M') \subset I^{n-k} M'. }[/math]

Thus,

[math]\displaystyle{ \ell((I^n M \cap M') / I^n M') \le \chi^I_{M'}(n-1) - \chi^I_{M'}(n-k-1) }[/math].

This gives the desired degree bound.

Multiplicity

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

The multiplicity of a point [math]\displaystyle{ x }[/math] of a scheme [math]\displaystyle{ X }[/math] is defined to be the multiplicity of the corresponding local ring [math]\displaystyle{ \mathcal{O}_{X,x} }[/math].

See also

• j-multiplicity