Wrapped distribution

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 $n\in \mathbb {N}$,

$\chi _{M}^{I}(n)=\ell (M/I^{n+1}M)$

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 $\dim(\operatorname {gr} _{I}(M))-1$.

Examples

For the ring of formal power series in two variables $k[[x,y]]$ taken as a module over itself and graded by the order and the ideal generated by the monomials x² and y³ we have

$\chi (1)=1,\quad \chi (2)=3,\quad \chi (3)=5,\quad \chi (4)=6{\text{ and }}\chi (k)=6{\text{ for }}k>4.$

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.

Let $(R,m)$ be a Noethrian local ring and I a 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.

References

• Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.