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The affine line with doubled origin is not separated

Statement

Let $k$ be a field and let $D$ be the affine line with doubled origin over $k$ , obtained by gluing two copies of $A_{k}^{1}$ by the identity on the complement of the origin. Then $D \to Spec k$ is quasi-separated but not separated. Moreover the two origins give two distinct lifts of one valuative diagram: the morphism $Spec k (t) \to D$ with image in the shared $G_{m}$ extends to $Spec k [t]_{(t)} \to D$ over $Spec k$ in two different ways.

Facts & Assumptions

Given: A field $k$ , the two affine charts $U = Spec k [x]$ and $V = Spec k [y]$ , the identity isomorphism $D (x) ≅ D (y)$ between the complements of the origin, and the scheme $D = U \cup V$ obtained by gluing $U$ and $V$ along it.

[F1]

Gluing the two charts by the identity on the complement of the origin produces a scheme $D$ having $U$ and $V$ as open subschemes with $U \cap V = D (x) ≅ D (y)$ ; the two copies of every nonzero point are identified, while the two closed points $0_{1} \in U$ and $0_{2} \in V$ remain distinct because the gluing isomorphism identifies only the open complements. This is the affine line with doubled origin. (Gluing affine schemes along compatible open isomorphisms)

[F2]

For affine opens $U = Spec B$ , $V = Spec C$ over a common affine $W = Spec A$ of the base, separatedness of $f : X \to S$ is equivalent to requiring for every such pair that: $U \cap V$ is affine and $B \otimes_{A} C \to Γ (U \cap V, O_{X})$ is surjective. (Affine-overlap criterion for separatedness)

[F3]

$D$ is quasi-separated over $k$ if and only if its diagonal is quasi-compact; $Δ$ is quasi-compact as soon as some affine open cover of $D \times_{k} D$ has quasi-compact inverse images. (Quasi-separatedness and the diagonal, Quasi-compact and quasi-separated morphisms)

[F4]

A discrete valuation ring is the ring of nonnegative values of a surjective integer-valued valuation on a field. (Discrete valuation rings)

Proof

technique · direct

1.1

Take the affine open cover of $D \times_{k} D$ by the four products $U \times_{k} U$ , $U \times_{k} V$ , $V \times_{k} U$ , $V \times_{k} V$ , all of which are affine; their inverse images under $Δ_{D / k}$ are $U$ , the overlap $U \cap V$ , $U \cap V$ and $V$ respectively, and each of these is affine, hence quasi-compact.

F3given

1.2

By [F1] the glued overlap $U \cap V = D (x) ≅ D (y)$ is the affine scheme $Spec k [x, x^{- 1}] = Spec k [t, t^{- 1}]$ ; its coordinate ring contains $t^{- 1}$ , which is not in the image of $k [x] \otimes_{k} k [y] \to k [t, t^{- 1}]$ induced by $x \mapsto t$ , $y \mapsto t$ .

F1given

1.3

For the valuative statement, let $R = k [t]_{(t)}$ and $K = k (t)$ . On $K^{\times}$ , define $v (f / g) = {ord}_{t} (f) - {ord}_{t} (g)$ , and put $v (0) = \infty$ . Factorization by the largest power of $t$ shows independence of the fraction representation, additivity under multiplication and $v (a + b) \geq \min (v (a), v (b))$ ; every nonzero fraction has finite value, and $v (t) = 1$ gives surjectivity onto $Z$ . Its nonnegative-value ring is precisely $R$ , whose fraction field is $K$ , so $R$ is a DVR by [F4]. Let $Spec K \to D$ be induced by $k [t] \to R \to K$ and either chart, and let $Spec R \to Spec k$ be the structure morphism; the two chart inclusions $U ↪ D$ and $V ↪ D$ restrict to the same morphism on $Spec K$ because the generic point lies in the glued $G_{m}$ , and they differ at the closed point, which maps to the two distinct origins.

F1F4given

2.1

Step 1.1 shows that $Δ_{D / k}$ is quasi-compact, so $D \to Spec k$ is quasi-separated by [F3].

F3step 1.1

2.2

Step 1.2 gives an affine pair $U, V$ over the affine base $Spec k$ whose intersection is affine, so the first clause of [F2] holds for it; but the map $k [x] \otimes_{k} k [y] \to Γ (U \cap V, O_{D}) = k [t, t^{- 1}]$ is not surjective, since $t^{- 1}$ has no preimage.

F2step 1.2

3.1

By the second clause of [F2] the failure in step 2.2 shows that $D \to Spec k$ is not separated; together with step 2.1 this gives a quasi-separated nonseparated morphism.

F2step 2.1step 2.2

4.1

Steps 3.1 and 1.3 prove that $D$ is quasi-separated but not separated and that the displayed valuative diagram has two distinct lifts, as asserted.

step 3.1step 1.3∎

Depends on

Used by

The doubled-origin diagonal is not closed Counterexample
Two DVR lifts of one diagram over the doubled-origin line Counterexample

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 26.21.7-8 and Example 26.22.2, printed pp.41-45 (standard reference, not scraped)
Vakil, The Rising Sea, Sections 11.3.I and 13.7.C, printed pp.309, 382 (standard reference, not scraped)