Hilbert–Samuel function

In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,^[1] 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 $\dim(\operatorname{gr}_I(M))$.^[2]

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

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

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 Noethrian 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 $M/I M$ has finite length,^[3] then we have:^[4]

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

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,

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

This gives the desired degree bound.

See also

• j-multiplicity

References

1. H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203.
2. Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969.
3. This implies that $M'/IM'$ and $M''/IM''$ also have finite length.
4. Eisenbud, David, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8. Lemma 12.3.