Hilbert–Samuel function

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


 * $$\chi_{M}^{I}(n)=\ell(M/I^{n}M)$$

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


 * $$\chi_M^I (n)=\sum_{i=0}^n H(\operatorname{gr}_I(M),i).$$

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).

Examples
For the ring of formal power series in two variables $$kx,y$$ taken as a module over itself and the ideal $$I$$ generated by the monomials x2 and y3 we have


 * $$\chi(1)=6,\quad \chi(2)=18,\quad \chi(3)=36,\quad \chi(4)=60,\text{ and in general } \chi(n)=3n(n+1)\text{ for }n \geq 0.$$

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 $$P_{I, M}$$ 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:
 * $$\chi_M^I(n-1) = \chi_{M'}^I(n-1) + \chi_{M''}^I(n-1) - \ell((I^n M \cap M')/I^n M')$$.

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 $$\frac{e}{d!}\cdot n^d$$ 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 $$\mathcal{O}_{X,x}$$.