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

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projective zariski topology

Statement

Projective algebraic sets are the closed sets of the projective Zariski topology on $P_{k}^{n}$ . Its standard opens are $D_{+} (x_{i}) = {[a] : a_{i} \neq 0}$ .

Proof

Given: Homogeneous ideals $I, J, J_{α} \subseteq k [x_{0}, \dots, x_{n}]$ .

1.1

The empty and whole sets are $V_{+} ((1))$ and $V_{+} ((0))$ .

given

1.2

Direct evaluation gives $V_{+} (I J) = V_{+} (I) \cup V_{+} (J)$ .

givenalgebra

1.3

Direct evaluation gives $V_{+} (\sum_{α} J_{α}) = ⋂_{α} V_{+} (J_{α})$ .

givenalgebra

2.1

These identities prove the closed-set axioms, and the complement of $V_{+} ((x_{i}))$ is $D_{+} (x_{i})$ .

step 1.1step 1.2step 1.3∎

Depends on

projective algebraic set

Used by

standard projective opens are affine spaces Lemma

Dependency tree · two levels

4 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)