Hilbert–Samuel function

In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel,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. 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).Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969.

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.

{{math_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 $M/I M$ has finite length,This implies that $M'/IM'$ and $M$ /IM also have finite length. then we have: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.

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

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.

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

Category:Commutative algebra

Category:Algebraic geometry

Hilbert–Samuel function

Examples

Degree bounds

Multiplicity

See also

References