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 ${\chi }_M^I:\mathbb{\mathbb{N}}\to \mathbb{\mathbb{N}}$ such that, for all $n\in \mathbb{\mathbb{N}}$,

${\chi }_M^I(n)=\ell (M/I^{n}M)$

where $\ell$ denotes the length over $A$. It is related to the Hilbert function of the associated graded module ${\mathrm{gr}}_I(M)$ by the identity

${\chi }_M^I(n)=\sum _{i=0}^{n}H({\mathrm{gr}}_I(M),i).$

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

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

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

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.

Theorem — Let $(R,m)$ be a Noetherian local ring and I an m-primary ideal. If

$0\to M^{\prime }\to M\to M^{″}\to 0$

is an exact sequence of finitely generated R-modules and if $M/IM$ has finite length, then we have:

$P_{I,M}=P_{I,M^{\prime }}+P_{I,M^{″}}-F$

where F is a polynomial of degree strictly less than that of $P_{I,M^{\prime }}$ and having positive leading coefficient. In particular, if $M^{\prime }\simeq M$, then the degree of $P_{I,M^{″}}$ is strictly less than that of $P_{I,M}=P_{I,M^{\prime }}$.

Proof: Tensoring the given exact sequence with $R/I^n$ and computing the kernel we get the exact sequence:

$0\to (I^{n}M\cap M^{\prime })/I^n{M}^{\prime }\to M^{\prime }/I^n{M}^{\prime }\to M/I^{n}M\to M^{″}/I^n{M}^{″}\to 0,$

which gives us:

${\chi }_M^I(n-1)={\chi }_{M^{\prime }}^I(n-1)+{\chi }_{M^{″}}^I(n-1)-\ell ((I^{n}M\cap M^{\prime })/I^n{M}^{\prime })$.

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

$I^{n}M\cap M^{\prime }=I^{n-k}((I^{k}M)\cap M^{\prime })\subset I^{n-k}{M}^{\prime }.$

Thus,

$\ell ((I^{n}M\cap M^{\prime })/I^n{M}^{\prime })\le {\chi }_{M^{\prime }}^I(n-1)-{\chi }_{M^{\prime }}^I(n-k-1)$.

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 ${{𝒪}}_{X,x}$.

See also

• j-multiplicity