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

1.1given

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

1.2givenalgebra

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

1.3givenalgebra

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

2.1step 1.1step 1.2step 1.3∎

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

Depends on

projective algebraic set

projective zariski topology

Statement

Proof

Depends on

Dependency tree · 2 levels

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