Author: Eiko

Tags: hilbert series, hilbert polynoimal, algebraic geometry

Time: 2024-12-15 03:08:02 - 2024-12-15 03:08:02 (UTC)

Introduction To Hilbert Series And Polynomial

The Hilbert function, Hilbert series, and Hilbert polynomial are related notions associated to a graded algebra or module (and extendable to filtered ones). They measure the size or growth of the algebra or module just like complexity or dimension.

For example they are used with these things:

• Quotient of polynomial rings by homogeneous ideals, this is a graded algebra.

• Quotient by any ideal of a polynomial ring, filtered by total degree.

• Filtration of a local ring by powers of its maximal ideal, the Hilbert-Samuel polynomial.

They provide a useful invariant for families, as a flat family $X\to S$ has the same Hilbert polynomial for all fibers.

Definition

Let $S=\underset{i\ge 0}{⨁}{S}_i$ be a graded (commutative) algebra,

• Hilbert function is the graded dimension function

  $$h_S(n)={\dim }_k{S}_n$$

• Hilbert series is the generating function

  $$H_S(t)=\sum _{n\ge 0}{h}_S(n)t^n$$

• Hilbert polynomial is the polynomial $P_S(n)$ that agrees with the Hilbert function $h_S(n)$ for large $n$

Generators Of $S$ Reflected In Hilbert Series

If $S$ can be generated by several homogeneous elements of degrees $d_1,\cdots ,d_k$, then the Hilbert series is a rational function with known denominators

$$H_S(t)=\frac{f(t)}{(1-t^{d_1})\cdots (1-t^{d_k})}$$

where $f(t)$ is some integral polynomial representing initial conditions.

Hilbert Polynomial

For typical applications in algebraic geometry, all the generators are of degree $1$, let’s say we have $d$ generators (think of them as independent coordinate variables), we can write the Hilbert series as

$$H_S(t)=\frac{Q_S(t)}{(1-t{)}^d}$$

from which we can deduce

$$h_S(n)=\sum _{i\le min(\mathrm{deg}Q,n)}{q}_i(\genfrac{}{}{0ex}{}{n-i+d-1}{d-1})$$

we can see that if $n\ge \mathrm{deg}Q$, the Hilbert function is a polynomial of degree $d-1$.

Geometric Intuition On Hilbert Polynomial, Dimension And Degree

Imagine we are trying to measure certain properties of a geometric space $X$, for example dimension. We can do it via Krull dimension or utilizing Zariski topology.

An alternative way is try to measure number of functions or homogeneous forms of certain degree

$$h_X(m)=\chi ({\mathcal{O}}_X(m))$$

the idea is that

• On a point, the only functions are the constants, in the affine sense we would say there are no functions of degree $m>0$. But in projective sense, we would somehow think about $k[x_0{]}_m$ which has dimension $1$.

• For a line, the functions of degree $m$ are one dimensional, while the homogeneous forms we have $k[x_0,x_1{]}_m$ which has dimension $m+1\sim m$.

• For a plane, the functions of degree $m$ are $m+1$, while the homogeneous forms we have $k[x_0,x_1,x_2{]}_m$ which has dimension $(\genfrac{}{}{0ex}{}{m+2}{2})\sim \frac{m^2}{2}$.

Therefore we think of the asymptotic degree of homogeneous forms matches the dimension of the space.

• Moreover, consider the union of two lines, then we will have double amount of functions and homogeneous forms just like $\dim V_1\times V_2=\dim V_1+\dim V_2$.

• For example on points, for $n$ points the constant function space will have dimension $n$.

Therefore, the head coefficients of the Hilbert function or Hilbert polynomial in the multiple of $\frac{m^d}{d!}$ will reflect the degree of the space.

Properties

• Additivity On K-Group. For any exact sequence $0\to A\to B\to C\to 0$ of graded or filtered modules, the Hilbert polynomial/series/function of $B$ is the sum of those of $A$ and $C$.

  $$H_B(t)=H_A(t)+H_C(t).$$

• Quotient By Homogeneous Non-zero Divisor. If $f$ is a non-zero divisor of degree $d$, then

  $$0\to A[d]\stackrel{f}{\to }A\to A/(f)\to 0$$

  therefore

  $$H_{A/(f)}(t)=(1-t^d)H_A(t).$$

Example

Consider $A/f=k[x,y,z]/(xz-y^2)$, we have

$$H_{A/f}(t)=(1-t^2)H_A(t)=\frac{1-t^2}{(1-t{)}^3}=\frac{1+t}{(1-t{)}^2}.$$