TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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ideal projective closure saturation

Statement

If $A = V (I) \subseteq A_{k}^{n}$ and $I = I (A)$ , write $I^{h} = (f^{h} : f \in I)$ for the homogeneous ideal generated by the homogenizations of elements of $I$ . Then $I_{+} ({\overline{A}}^{p r o j}) = I^{h} : x_{0}^{\infty}$ .

Proof

Given: $A = V (I)$ with $I = I (A)$ , $I^{h} = (f^{h} : f \in I)$ , and a homogeneous polynomial $G$ .

1.1

Dehomogenization on $D_{+} (x_{0})$ identifies $V_{+} (I^{h})$ there with $A$ .

givenalgebra

2.1

$G$ vanishes on this chart exactly when $G (1, x) \in I$ ; homogenizing this condition is exactly $x_{0}^{r} G \in I^{h}$ for some $r$ .

step 1.1algebra

3.1

Homogeneous equations vanish on a set exactly when they vanish on its closure, hence the ideal is $I^{h} : x_{0}^{\infty}$ .

step 2.1∎

Depends on

Used by

naive homogenization adds component Counterexample
projective closure parabola Example

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Algebraic Geometry, Chapter 6 (standard reference, not scraped)
Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)