TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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Projective morphisms are proper

Statement

Assume the Axiom of Choice. Every projective morphism in the finite-dimensional H-projective convention of this page is proper: if $f : X \to S$ factors over $S$ as $X \overset{i}{\to} P_{S}^{n} ⟶ S$ with $n \geq 0$ , $i$ a closed immersion and the second arrow the projection, then $f$ is proper. The empty source, the empty base and the case $n = 0$ are included, and no Noetherian, field, reducedness or nonemptiness hypothesis is imposed.

Facts & Assumptions

Given: The Axiom of Choice and a projective morphism $f : X \to S$ with a factorization $X \overset{i}{\to} P_{S}^{n} \overset{π}{\to} S$ over $S$ , where $n \geq 0$ , $i$ is a closed immersion and $π$ is the projection.

[F1]

A morphism $f : X \to S$ is projective on this page if for some $n \geq 0$ it factors over $S$ as $X \overset{i}{\to} P_{S}^{n} \to S$ with $i$ a closed immersion; the relative projective space and its projection are those of the cited chart construction, and $n = 0$ is allowed with $P_{S}^{0} ≅ S$ . (Projective morphisms before Proj)

[F2]

Assume AC. For every scheme $S$ and every $n \geq 0$ the projection $P_{S}^{n} \to S$ is proper. (Finite-dimensional projective space is proper over every base)

[F3]

Assume AC. Every closed immersion is finite, hence proper; the empty closed immersion is included. (Closed immersions are proper)

[F4]

Assume AC. A composite of proper morphisms is proper. (Properness survives composition)

[F5]

The Axiom of Choice states that every family of nonempty sets has a choice function. (The Axiom of Choice)

Proof

technique · direct: write $f$ as a closed immersion into a relative projective space followed by the projection, and compose the two proper morphisms

1.1F1

By [F1] the morphism $f$ admits a factorization $f = π \circ i$ over $S$ , where $n \geq 0$ , $i : X \to P_{S}^{n}$ is a closed immersion and $π : P_{S}^{n} \to S$ is the projection of the relative projective space.

2.1F3step 1.1

By the AC-qualified [F3] the closed immersion $i$ is finite, hence proper.

2.2F2step 1.1

By the AC-qualified [F2] the projection $π : P_{S}^{n} \to S$ is proper.

3.1F1F4step 2.1step 2.2

Since $f = π \circ i$ is a composite of the proper morphisms $i$ and $π$ , the AC-qualified [F4] shows that $f$ is proper.

4.1F1F2F3F5step 3.1∎

This proves the theorem for the finite-dimensional H-projective convention fixed in [F1]; the separate projective-bundle convention of the sources is not treated here and is not claimed. The Axiom of Choice [F5] is assumed and is used only through [F2], [F3] and [F4]. The degenerate cases are included: for $n = 0$ the projection is the isomorphism $P_{S}^{0} ≅ S$ and $f$ is the closed immersion $i$ ; if $S = \emptyset$ then $P_{S}^{n} = X = \emptyset$ and the morphisms are the empty ones; if $i$ is the identity then $f = π$ is the projection, proper by [F2]; and if $X = \emptyset$ then $i$ is the empty closed immersion, proper by [F3]. No Noetherian, field, reducedness or nonemptiness hypothesis is imposed.

Projective morphisms are proper

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources