ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-27

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A quadratic-cubic plane complete intersection has eventual Hilbert value six

Example

Let $k$ be a field and let $S = k [x_{0}, x_{1}, x_{2}] / (x_{0}^{2}, x_{1}^{3})$ carry the standard grading. Then ${HS}_{S} (t) = \frac{(1 + t) (1 + t + t^{2})}{1 - t},$ and the Hilbert function of $S$ is $1, 3, 5, 6, 6, 6, \dots$ in degrees $0, 1, 2, 3, 4, 5, \dots$ : it is constantly $6 = 2 \cdot 3$ from degree $3$ on. The quotient $S$ is not a finite-dimensional $k$ -algebra; only its graded pieces are computed here.

Facts & Assumptions

Given: A field $k$ and the standard graded quotient $S = k [x_{0}, x_{1}, x_{2}] / (x_{0}^{2}, x_{1}^{3})$ .

[L1]

The Hilbert function of a graded module records $\dim_{k} M_{n}$ , and its Hilbert series is the formal power series $\sum_{n} \dim_{k} M_{n} t^{n}$ , whose coefficients are read off by coefficient extraction (The Hilbert function and formal Hilbert series of a graded module with finite-length pieces, Formal power series over a commutative ring and the coefficient-extraction functional $[x^{n}]$ , Nonnegatively graded rings and modules, homogeneous elements, and twists).

[L2]

For two plane forms $F, G$ with no common nonconstant factor, of positive degrees $d$ and $e$ , the pair $(F, G)$ is a regular sequence and ${HS}_{k [x_{0}, x_{1}, x_{2}] / (F, G)} (t) = (1 + t + \dots + t^{d - 1}) (1 + t + \dots + t^{e - 1}) / (1 - t)$ , with Hilbert function constantly $d e$ in every degree $n \geq d + e - 2$ (Coprime positive-degree plane forms form a regular sequence, Hilbert series and eventual Hilbert value of a two-form plane complete intersection, homogeneous polynomial and homogeneous ideal).

[L3]

The monomials of $k [x_{0}, x_{1}, x_{2}]$ form a $k$ -basis and are graded by total degree (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ ).

[L4]

Assuming the Axiom of Choice, for coprime plane forms of degrees $d, e$ , the total length of $Proj (k [x_{0}, x_{1}, x_{2}] / (F, G))$ equals $d e$ (Two coprime projective plane forms meet in total length equal to their degree product, Total length of a zero-dimensional projective scheme).

Verification

technique · direct

1.1

The two forms $x_{0}^{2}$ and $x_{1}^{3}$ are coprime in $k [x_{0}, x_{1}, x_{2}]$ because they involve distinct variables, and have degrees $2$ and $3$ , so [L2] gives ${HS}_{S} (t) = (1 - t^{2}) (1 - t^{3}) / (1 - t)^{3} = (1 + t) (1 + t + t^{2}) / (1 - t)$ , using $(1 - t^{2}) = (1 - t) (1 + t)$ and $(1 - t^{3}) = (1 - t) (1 + t + t^{2})$ .

L2algebra

1.2

Equivalently, the monomials $x_{0}^{a} x_{1}^{b} x_{2}^{c}$ with $0 \leq a \leq 1$ , $0 \leq b \leq 2$ , $c \geq 0$ form a $k$ -basis of $S$ by [L3], and their generating function by total degree is $(1 + t) (1 + t + t^{2}) (1 + t + t^{2} + \dots) = (1 + t) (1 + t + t^{2}) / (1 - t)$ .

L3algebra

2.1

By step 1.2, $\dim_{k} S_{n} = # {(a, b) : 0 \leq a \leq 1, 0 \leq b \leq 2, a + b \leq n}$ , which is $1, 3, 5, 6, 6, 6, \dots$ for $n = 0, 1, 2, 3, 4, 5, \dots$ ; equivalently these are the partial sums of the coefficients $1, 2, 2, 1$ of $(1 + t) (1 + t + t^{2})$ , in agreement with [L1].

L1step 1.2algebra

3.1

Since $S_{n} \neq 0$ for every $n$ by step 2.1, the $k$ -vector space $S$ is infinite-dimensional, so $S$ is not a finite-dimensional $k$ -algebra; no Artinian claim is made.

step 2.1

4.1

If the Axiom of Choice is assumed, then as a consistency check [L4] gives ${len}_{k} (Proj S) = 2 \cdot 3 = 6$ , which is exactly the eventual value of the Hilbert function computed in step 2.1. The Hilbert-series and Hilbert-function claims above hold over every field without this additional assumption.

L4step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

A. Gathmann, Algebraic Geometry class notes (2002), Theorem 6.2.1, p. 96 (standard reference, not scraped)