LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-27

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Affine morphisms are separated

Statement

Every affine morphism $f : X \to S$ is separated. No Noetherian, reducedness, finite-type, quasi-compactness or nonemptiness hypothesis is used, and the zero ring and empty scheme are allowed.

Facts & Assumptions

Given: An affine morphism of schemes $f : X \to S$ , its diagonal $Δ_{X / S}$ and the projections ${pr}_{1}, {pr}_{2} : X \times_{S} X \to X$ .

[F1]

A morphism $f : X \to S$ is affine when $f^{- 1} (U)$ is affine for every affine open subscheme $U \subseteq S$ . The empty scheme is affine, being $Spec 0$ , and affineness of a morphism does not require source or target to be affine. (Affine morphisms)

[F2]

A morphism $f : X \to S$ is separated when its diagonal $Δ_{X / S} : X \to X \times_{S} X$ is a closed immersion. (Separated morphism of schemes)

[F3]

The diagonal morphism is the unique $Δ_{X / S} : X \to X \times_{S} X$ with ${pr}_{1} Δ_{X / S} = {id}_{X} = {pr}_{2} Δ_{X / S}$ . (The diagonal morphism)

[F4]

For ring maps $A \to B$ , $A \to C$ , allowing the zero ring, $Spec B \times_{Spec A} Spec C ≅ Spec (B \otimes_{A} C)$ , with the projections $b \mapsto b \otimes 1$ and $c \mapsto 1 \otimes c$ . (Affine fibre products are spectra of tensor products)

[F5]

For a ring $A$ , closed immersions $Z \to Spec A$ are, up to unique isomorphism over $Spec A$ , precisely the morphisms $Spec (A / I) \to Spec A$ for ideals $I \subseteq A$ . (Closed immersions into affine schemes are quotient spectra)

[F6]

If opens $V \subseteq X$ and $W \subseteq Y$ of an $S$ -diagram map into an open $U \subseteq S$ , then ${pr}_{1}^{- 1} (V) \cap {pr}_{2}^{- 1} (W)$ is an open subscheme of $X \times_{S} Y$ representing $V \times_{U} W$ . (Restricting fibre products to open subschemes)

[F7]

A morphism $i : Z \to T$ is a closed immersion if and only if its restrictions to the members of an open cover of $T$ are closed immersions. (Closed immersions are local on the target)

Proof

technique · direct

1.1

For each affine open subscheme $W = Spec A \subseteq S$ the inverse image $U_{W} = f^{- 1} (W)$ is affine, say $U_{W} = Spec B_{W}$ , by [F1]; and $Q_{W} : = {pr}_{1}^{- 1} (U_{W}) \cap {pr}_{2}^{- 1} (U_{W})$ is, by [F6], the open subscheme of $X \times_{S} X$ representing $U_{W} \times_{W} U_{W}$ , which [F4] identifies with $Spec (B_{W} \otimes_{A} B_{W})$ .

F1F4F6given

1.2

By [F3] one has ${pr}_{j} Δ_{X / S} = {id}_{X}$ for $j = 1, 2$ .

2.1

The subschemes $Q_{W}$ cover $X \times_{S} X$ : for a point $z$ of the product, both ${pr}_{1} (z)$ and ${pr}_{2} (z)$ map to the same point $w \in S$ , so choosing an affine open $W$ with $w \in W$ gives ${pr}_{j} (z) \in f^{- 1} (W) = U_{W}$ for $j = 1, 2$ and hence $z \in Q_{W}$ .

givenstep 1.1

2.2

Fix an affine open $W = Spec A \subseteq S$ . Since ${pr}_{j} Δ_{X / S} = {id}_{X}$ , one has $Δ_{X / S}^{- 1} (Q_{W}) = U_{W}$ ; and under the identifications of step 1.1 the restriction $U_{W} \to Q_{W}$ corresponds to the morphism $Spec B_{W} \to Spec (B_{W} \otimes_{A} B_{W})$ induced by the multiplication $m_{W} : B_{W} \otimes_{A} B_{W} \to B_{W}$ , $b \otimes b^{'} \mapsto b b^{'}$ , which is surjective because $b = m_{W} (b \otimes 1)$ . By [F5] this restriction is a closed immersion; the cases $B_{W} = 0$ , $A = 0$ and $W = \emptyset$ are included, with $m_{W}$ again surjective.

F3F4F5step 1.1

3.1

The open subschemes $Q_{W}$ of step 2.1 form an open cover of $X \times_{S} X$ , and step 2.2 exhibits the restriction of $Δ_{X / S}$ to each of them as a closed immersion. By [F7] the diagonal $Δ_{X / S} : X \to X \times_{S} X$ is itself a closed immersion.

F7step 2.1step 2.2

4.1

By [F2] the morphism $f$ is separated, which is the assertion. Throughout, only affineness of $f$ and the tensor and quotient descriptions of affine fibre products were used, so no Noetherian, reducedness or finite-type hypothesis enters.

F2step 3.1∎

Depends on

Used by

Affine schemes and affine morphisms are separated Corollary
The graph of a polynomial map as a closed subscheme Example

Dependency tree · two levels

21 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

The Stacks Project, Schemes, Lemmas 26.21.1 and 26.21.15, printed pp.35, 42 (standard reference, not scraped)
Vakil, The Rising Sea, Section 11.3.4, printed p.308 (standard reference, not scraped)