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

This gives the desired degree bound.

References

↑ 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.
1 2 Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969.
↑ This implies that $M'/IM'$ and $M''/IM''$ also have finite length.
↑ 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.

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Hilbert–Samuel function

Examples

Degree bounds

See also

References