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 $\dim(\operatorname {gr} _{I}(M))$ , often called the Hilbert-Samuel polynomial (or Hilbert polynomial).[2]

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.

[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 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,[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.

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

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.
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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[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.

[ica-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.

Hilbert–Samuel function

Examples

Degree bounds

Multiplicity

See also

References