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 $\chi_^:\mathbb\rightarrow\mathbb$ such that, for all $n\in\mathbb$,

:$\chi_^(n)=\ell(M/I^M)$

where $\ell$ denotes the length over $A$. It is related to the Hilbert function of the associated graded module $\operatorname_I(M)$ by the identity

: $\chi_M^I (n)=\sum_^n H(\operatorname_I(M),i).$

For sufficiently large $n$, it coincides with a polynomial function of degree equal to $\dim(\operatorname_I(M))$, 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 ''x''² and ''y''³ we have

: $\chi(1)=6,\quad \chi(2)=18,\quad \chi(3)=36,\quad \chi(4)=60,\text \chi(n)=3n(n+1)\textn \geq 0.$

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 $P_$ the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

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:
:$\chi_M^I(n-1) = \chi_^I(n-1) + \chi_^I(n-1) - \ell((I^n M \cap M')/I^n M')$.
The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large ''n'' and some ''k'',
:$I^n M \cap M' = I^ ((I^k M) \cap M') \subset I^ M'.$
Thus,
:$\ell((I^n M \cap M') / I^n M') \le \chi^I_(n-1) - \chi^I_(n-k-1)$.
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 $\frac\cdot n^d$ 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 $\mathcal_$.

See also

* j-multiplicity

References

{{DEFAULTSORT:Hilbert-Samuel function
Commutative algebra
Algebraic geometry