The Hilbert Function Eventually Becomes a Polynomial

For a finitely generated graded module, the dimensions of graded pieces agree eventually with a polynomial. The point is to make the formal expression readable enough to audit line by line.

Statement

A graded ring $S=\bigoplus_{d\ge0}S_d$ produces $\operatorname{Proj}S$, whose points are homogeneous primes avoiding the irrelevant ideal $S_+$. Projective space is the basic example.

$$ H_M(d)=\dim_kM_d $$

The formulas should not be merged too early. The datum $\mathsf D$, the conclusion $\mathsf C$, and the bridge $\Longrightarrow$ have three different logical jobs.

$$ H_M(d)=P_M(d)\quad\text{for }d\gg0 $$

Worked algebra

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ S=k[x_0,\ldots,x_n],\qquad H_S(d)=\binom{n+d}{n},\qquad P_S(t)=\binom{n+t}{n} $$

Conceptual compression

A good test for understanding is to change the presentation while keeping the invariant fixed. The aligned form makes that comparison unusually easy.

$$ \begin{aligned} \mathsf{D}\;&:\quad H_M(d)=\dim_kM_d,\\[5pt] \mathsf{C}\;&:\quad H_M(d)=P_M(d)\quad\text{for }d\gg0. \end{aligned} $$

Caveat

Homogeneity is essential: a nonhomogeneous equation is not well defined under rescaling of projective coordinates.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] H_M(d)=P_M(d)\quad\text{for }d\gg0 \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Sat 29 September 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

The Hilbert Function Eventually Becomes a Polynomial by Vicfred

Statement

Worked algebra

Conceptual compression

Caveat