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 $k[[x,y]]$ taken as a module over itself and the ideal $I$ generated by the monomials x² and y³ we have

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

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.

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

$0\to M'\to M\to M''\to 0$

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

$P_{I,M}=P_{I,M'}+P_{I,M''}-F$

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

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,

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

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 ${\mathcal {O}}_{X,x}$.

See also

• j-multiplicity