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A nonseparated affine cover can have nonaffine intersection

Statement refuted

Assume the Axiom of Choice (The Axiom of Choice). The following implication is false: if a scheme $X$ is covered by finitely many affine open subschemes, then every finite intersection of members of that cover is affine.

Explicitly, let $k$ be a field and let $A_{k}^{2} = Spec k [x, y]$ be the affine plane (The underlying space of an affine spectrum); put $U = D (x) \cup D (y) = A_{k}^{2} ∖ {0},$ the punctured plane, and let $X$ be the scheme obtained by gluing two copies $U_{1}, U_{2} ≅ A_{k}^{2}$ along the identity of $U$ (Gluing affine schemes along compatible open isomorphisms). Then $X$ has the affine two-open cover $X = U_{1} \cup U_{2}$ , its intersection $U_{1} \cap U_{2} ≅ U$ is not affine, and $X$ is not separated (Separated morphism of schemes). The obstruction is cohomological: $H^{1} (U, O_{U}) \neq 0,$ where $H^{1}$ is sheaf cohomology (Sheaf cohomology as right derived global sections); the class of the Laurent monomial $x^{- 1} y^{- 1}$ in the Čech quotient $k [x^{\pm 1}, y^{\pm 1}] / (k [x^{\pm 1}, y] + k [x, y^{\pm 1}])$ under the cover ${D (x), D (y)}$ is nonzero and defines a nonzero class in $H^{1} (U, O_{U})$ . This is exactly the point at which separatedness enters the Čech comparison theorem Cech cohomology computes quasi-coherent cohomology on a separated scheme, whose proof derives the affineness of the intersections from separatedness.

Facts & Assumptions

Given: The Axiom of Choice, a field $k$ , the affine plane $X_{0} = Spec k [x, y]$ , the open subscheme $U = D (x) \cup D (y)$ and the two-open cover ${D (x), D (y)}$ of $U$ .

[F1]

Principal opens of the affine plane: $D (x) = Spec k [x, y]_{x}$ and $D (y) = Spec k [x, y]_{y}$ are affine, with $D (x) \cap D (y) = D (x y)$ and ring $k [x, y]_{x y}$ ; more generally $Γ (D (f), O) = k [x, y]_{f}$ (Principal distinguished subsets of the prime spectrum, Multiplicative subsets and the localisation $S^{- 1} R$ as equivalence classes of fractions, Principal localisation $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$ , A principal localization identifies its spectrum with a distinguished open, Sections and restrictions on distinguished opens of an affine scheme). A prime $p \subseteq k [x, y]$ lies in $D (x) \cup D (y)$ if and only if $x \notin p$ or $y \notin p$ , i.e. if and only if $p \neq (x, y)$ ; hence $U$ is the complement of the origin, and the intersection $D (x) \cap D (y) = D (x y)$ is nonempty (it contains the zero ideal).

[F2]

Laurent monomials: every element of $k [x, y]_{x y}$ has a unique finite expansion $\sum_{m, n \in Z} c_{m, n} x^{m} y^{n}$ ; the monomials $x^{m} y^{n}$ are $k$ -linearly independent, because an equation between finitely many of them becomes, after multiplication by a high power of $x y$ , the unique expansion of a polynomial; so they form a $k$ -basis of $k [x, y]_{x y}$ . The subring $k [x, y]_{x}$ is spanned by those monomials with $n \geq 0$ , and $k [x, y]_{y}$ by those with $m \geq 0$ (Monomials, coefficients, degree in each variable and total degree in $F [x_{1}, \dots, x_{n}]$ , Principal localisation $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$ ).

[F3]

Gluing of affine schemes: affine schemes equipped with open subschemes and isomorphisms on overlaps satisfying the identity and cocycle conditions glue to a scheme, uniquely up to unique isomorphism, and the given affine schemes become an open affine cover. (Gluing affine schemes along compatible open isomorphisms)

[F4]

Leray acyclic-cover comparison: under the Axiom of Choice, for an open cover indexed by a linearly ordered set that is $F$ -acyclic, i.e. every nonempty finite intersection $W$ satisfies $H^{q} (W, F ∣_{W}) = 0$ for all $q > 0$ , the canonical Čech-to-sheaf comparison ${\overset{ˇ}{H}}^{p} (U, F) \to H^{p} (X, F)$ is an isomorphism for every $p \geq 0$ (Leray acyclic-cover comparison, Acyclic open cover for a sheaf). The ordered Čech complex of a two-member cover $U_{0}, U_{1}$ is $0 \to F (U_{0}) \oplus F (U_{1}) \overset{δ^{0}}{\to} F (U_{0} \cap U_{1}) \to 0$ with $δ^{0} (s_{0}, s_{1}) = s_{1} ∣_{U_{0} \cap U_{1}} - s_{0} ∣_{U_{0} \cap U_{1}}$ (Ordered Čech cochain complex of a cover, Fixed-cover Čech cohomology).

[F5]

Affine vanishing: under the Axiom of Choice, on an affine scheme every quasi-coherent module has vanishing higher cohomology (Affine acyclicity of quasi-coherent sheaves). The structure sheaf $O_{U}$ is quasi-coherent as an $O_{U}$ -module, and so are its restrictions to the open affine subschemes $D (x), D (y), D (x y)$ (Quasi-coherent module on a scheme, Modules on a ringed space).

[F6]

Separatedness criterion: for a morphism $f : X \to S$ and affine opens $U^{'}, V^{'} \subseteq X$ lying over one and the same affine open of $S$ , if $f$ is separated then $U^{'} \cap V^{'}$ is affine. In particular, in a separated scheme any two affine opens lying over a common affine open of the base have affine intersection (Affine-overlap criterion for separatedness, Separated morphism of schemes, Schemes and morphisms over a base).

Counterexample

technique · direct: the punctured plane is covered by the two affine charts $D(x),D(y)$ with affine overlap $D(xy)$, so the cover is acyclic for the structure sheaf and the Leray comparison computes $H^1$ as the Čech cokernel; the Laurent monomial $x^{-1}y^{-1}$ has negative exponents in both variables and hence survives. Gluing two affine planes along the punctured plane gives the nonseparated scheme with affine two-open cover

1.1F1

The cover ${D (x), D (y)}$ of $U$ is a finite affine open cover: $D (x)$ and $D (y)$ are affine by [F1] and they cover $U$ by definition of $U$ .

1.2F1

Its nonempty finite intersections are $D (x)$ , $D (y)$ and $D (x) \cap D (y) = D (x y)$ , all affine by [F1].

1.3F51.2

The cover is $O_{U}$ -acyclic: on each of these three affine open subschemes the restriction of the quasi-coherent module $O_{U}$ is quasi-coherent, so its higher cohomology vanishes by [F5].

1.4F41.3

By the Leray comparison [F4] applied to this two-member ordered cover and the sheaf $O_{U}$ , the canonical map ${\overset{ˇ}{H}}^{1} ({D (x), D (y)}, O_{U}) \to H^{1} (U, O_{U})$ is an isomorphism.

1.5F1F4

Computing the Čech group: by [F4], ${\overset{ˇ}{H}}^{1} = k [x, y]_{x y} / im δ^{0}$ with $C^{0} = k [x, y]_{x} \oplus k [x, y]_{y}$ , $C^{1} = k [x, y]_{x y}$ and $δ^{0} (f, g) = g - f$ , so $im δ^{0} = k [x, y]_{x} + k [x, y]_{y}$ as a subgroup of $k [x, y]_{x y}$ .

1.6F21.41.5

The monomial $x^{- 1} y^{- 1}$ is not in this image: by [F2] the monomials $x^{m} y^{n}$ form a $k$ -basis of $k [x, y]_{x y}$ , the subspace $k [x, y]_{x}$ is spanned by those with $n \geq 0$ and $k [x, y]_{y}$ by those with $m \geq 0$ , so their sum is spanned by the monomials with $m \geq 0$ or $n \geq 0$ and does not contain $x^{- 1} y^{- 1}$ , whose exponents are both $- 1$ . Hence $x^{- 1} y^{- 1}$ has nonzero class in ${\overset{ˇ}{H}}^{1}$ , and by 1.4 a nonzero class in $H^{1} (U, O_{U})$ ; in particular $H^{1} (U, O_{U}) \neq 0$ .

1.7F51.6

Consequently $U$ is not affine: if $U$ were affine, the quasi-coherent module $O_{U}$ would have $H^{1} (U, O_{U}) = 0$ by [F5], contradicting 1.6.

1.8F1F3

Construction of $X$ : take two copies $U_{1}, U_{2}$ of $A_{k}^{2}$ with open subschemes corresponding to $U$ , glued along the identity isomorphism; the identity and cocycle conditions are automatic, so by [F3] there is a scheme $X$ with open affine subschemes $U_{1}, U_{2}$ covering $X$ and $U_{1} \cap U_{2} ≅ U$ .

1.9F61.71.8

$X$ is not separated: if $X \to Spec Z$ were separated, then by the criterion [F6] applied to the two affine opens $U_{1}, U_{2}$ of $X$ , which both lie over the affine open $Spec Z$ of the base, their intersection $U_{1} \cap U_{2}$ would be affine; this contradicts 1.7.

2.1F3F4F51.61.8∎

Boundary and choice accounting. The field $k$ is arbitrary, including $k = F_{2}$ ; the zero ideal is a prime in $D (x y)$ , so the displayed rings are nonzero, while the origin is the distinct closed point $V (x, y) = {(x, y)}$ . The Axiom of Choice is a hypothesis and is consumed exactly through the Leray comparison [F4] and affine vanishing [F5], which are stated under AC; the gluing of 1.8, the Čech computation of 1.5-1.6 and the criterion application of 1.9 make no further choices.

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Sources

The Stacks Project, Cohomology of Schemes, Chapter 30, Sections 30.2-30.22 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Sections 19.1, 19.6, 19.9, 28.1-28.2 (standard reference, not scraped)