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Global sections do not determine a sheaf on P1

Statement refuted

Assume the Axiom of Choice, inherited from the two-affine construction of the projective line and from the affine equivalence (The Axiom of Choice).

False claim. For every scheme $X$ , a quasi-coherent $O_{X}$ -module is determined up to isomorphism by its module of global sections: if $F, G$ are quasi-coherent $O_{X}$ -modules with $Γ (X, F) ≅ Γ (X, G)$ , then $F ≅ G$ . In particular, on every scheme a quasi-coherent sheaf with vanishing global sections would be the zero sheaf, and every morphism of quasi-coherent sheaves inducing an isomorphism on global sections would be an isomorphism (Quasi-coherent module on a scheme).

For an affine scheme $X = Spec A$ this determination is a theorem: the comparison $\tilde{Γ (X, F)} \to F$ is an isomorphism for every quasi-coherent $F$ (Affine quasi-coherent sheaf determined by sections). The claim displayed above is its extension to arbitrary schemes, and that extension is false.

Counterexample. Let $k$ be a field and let $X = P_{k}^{1}$ be the two-affine projective line with charts $U_{0} = Spec k [t]$ , $U_{\infty} = Spec k [u]$ and $u = t^{- 1}$ on $W = U_{0} \cap U_{\infty} = Spec k [t, t^{- 1}]$ , and let $O (- 1)$ be the invertible sheaf with frames $e_{0}$ on $U_{0}$ , $e_{\infty}$ on $U_{\infty}$ and transition $e_{\infty} = t^{- 1} e_{0}$ (Two-affine projective line and its twists, Invertible sheaves). Comparing the two chart descriptions of a global section on $W$ shows that $Γ (X, O (- 1)) = 0$ , while $O (- 1)$ is nonzero because it is free of rank one with frame $e_{0}$ on $U_{0}$ (Locally free sheaves of finite rank). The zero $O_{X}$ -module $0$ also has $Γ (X, 0) = 0$ , and it is quasi-coherent, so the two quasi-coherent sheaves $O (- 1)$ and $0$ have equal (zero) modules of global sections but are not isomorphic. Consequently the zero morphism $0 \to O (- 1)$ induces an isomorphism on global sections without being an isomorphism itself. Since the affine determination above is a theorem, this failure on $P_{k}^{1}$ also shows that $P_{k}^{1}$ is not an affine scheme: the affine hypothesis cannot be dropped.

Facts & Assumptions

Given: A field $k$ ; the two-affine projective line $P_{k}^{1} = U_{0} \cup U_{\infty}$ with $U_{0} = Spec k [t]$ , $U_{\infty} = Spec k [u]$ and $u = t^{- 1}$ on the overlap $W = U_{0} \cap U_{\infty} = Spec k [t, t^{- 1}]$ ; the twist $O (- 1)$ with frames $e_{0}$ on $U_{0}$ and $e_{\infty}$ on $U_{\infty}$ ; the zero $O_{X}$ -module $0$ .

[F1]

The two-affine definition (Two-affine projective line and its twists): the two charts cover $P_{k}^{1}$ and have coordinate rings $k [t]$ and $k [u]$ with $t u = 1$ on $W$ ; for every $n \in Z$ the sheaf $O (n)$ is glued from the structure sheaves of the two charts with frames $e_{0}$ , $e_{\infty}$ related on $W$ by $e_{\infty} = t^{n} e_{0}$ , equivalently $e_{0} = t^{- n} e_{\infty}$ ; each $O (n)$ is free of rank one on each chart with the displayed frame, hence invertible, and $O (0) = O_{P_{k}^{1}}$ . On the overlap a section of $O (n)$ with $U_{0}$ -coordinate $a (t)$ has $U_{\infty}$ -coordinate $u^{n} a (u^{- 1})$ ; for $n = - 1$ this is $b (u) = u^{- 1} a (u^{- 1})$ , equivalently $u b (u) = a (u^{- 1})$ .

[F2]

Invertible, locally free and quasi-coherent (Invertible sheaves, Locally free sheaves of finite rank, Quasi-coherent module on a scheme, Modules on a ringed space): invertible means locally free of rank one; a locally free sheaf of finite rank is quasi-coherent; over a chart $U = Spec A$ a free rank-one module with generator $e$ has $e \neq 0$ because $A$ is a nonzero ring, so a free rank-one sheaf on a nonempty chart is not the zero sheaf; the zero $O_{X}$ -module is locally free of rank $0$ , hence quasi-coherent, and has zero sections on every open set, so $Γ (X, 0) = 0$ .

[F3]

Glued sections (Compatible local sheaves glue uniquely up to unique isomorphism, A gluing datum for sheaves on an open cover): for a sheaf $F$ glued from local sheaves $F_{0}$ on $U_{0}$ and $F_{\infty}$ on $U_{\infty}$ along an overlap identification $φ = φ_{0 \infty}$ , a section of $F$ on an open $V$ is a compatible pair $(s_{0}, s_{\infty}) \in F_{0} (V \cap U_{0}) \times F_{\infty} (V \cap U_{\infty})$ with $φ (s_{0} ∣_{V \cap W}) = s_{\infty} ∣_{V \cap W}$ ; in particular a global section of $F$ is exactly a pair $(s_{0}, s_{\infty}) \in F (U_{0}) \times F (U_{\infty})$ whose two restrictions to $W$ agree under $φ$ , and $F (U_{0}) ≅ F_{0} (U_{0}) = k [t] e_{0}$ , $F (U_{\infty}) ≅ F_{\infty} (U_{\infty}) = k [u] e_{\infty}$ for the twists of [F1].

[F4]

Polynomials and principal localisation (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, Principal localisation $R_{f} = {1, f, f^{2}, \dots}^{- 1} R$ , A localised module fraction is zero exactly when one denominator kills its numerator, A polynomial ring over an integral domain is an integral domain): the polynomial ring $k [u]$ is a set of finitely supported coefficient functions $N \to k$ , so two polynomials are equal exactly when their coefficient functions are equal; $k [u, u^{- 1}] = k [u]_{u}$ is the principal localisation at $u$ , whose elements are fractions $p / u^{m}$ ; a fraction $p / u^{m}$ is zero exactly when $u^{N} p = 0$ for some $N \geq 0$ ; and $k [u]$ is an integral domain with $u \neq 0$ , so $u$ is a nonzerodivisor and the canonical map $k [u] \to k [u, u^{- 1}]$ is injective. Consequently two polynomials in $k [u]$ that become equal in $k [u, u^{- 1}]$ are already equal in $k [u]$ .

[F5]

Affine determination of global sections (Affine quasi-coherent sheaf determined by sections, Module sheaf on an affine scheme, Quasi-coherent module on a scheme): for an affine scheme $X = Spec A$ and a quasi-coherent $F$ the comparison $κ_{F} : \tilde{Γ (X, F)} \to F$ is an isomorphism, and the associated sheaf of the zero $A$ -module is the zero sheaf. Hence on an affine scheme a quasi-coherent sheaf with vanishing global sections is the zero sheaf.

[F6]

The Axiom of Choice as inherited from the two-affine construction of [F1] and from the affine equivalence underlying [F5] (The Axiom of Choice).

Proof technique: direct; compute the global sections of $O (- 1)$ from its two chart descriptions, compare with the zero sheaf, and use the affine determination to conclude that $P_{k}^{1}$ is not affine.

Proof

1.1F1F3

A global section of $O (- 1)$ : by [F3] such a section is exactly a compatible pair $(a (t) e_{0}, b (u) e_{\infty})$ with $a \in k [t]$ , $b \in k [u]$ , and by the transition computation of [F1] for $n = - 1$ the two chart descriptions agree on $W$ precisely when $u b (u) = a (u^{- 1})$ in $k [u, u^{- 1}]$ .

2.1F4step 1.1

The only global section is zero: if $a = 0$ the identity $u b (u) = a (u^{- 1})$ of step 1.1 gives $u b (u) = 0$ , whence $b = 0$ because $u$ is a nonzerodivisor in the domain $k [u]$ ; assume therefore that $a \neq 0$ and let $d \geq 0$ be the largest index with $a_{d} \neq 0$ . Multiplying the identity of step 1.1 by $u^{d}$ gives $u^{d + 1} b (u) = \sum_{i = 0}^{d} a_{i} u^{d - i}$ , an equality in $k [u, u^{- 1}]$ between two polynomials in $k [u]$ , hence an equality in $k [u]$ by [F4]. The left side is divisible by $u^{d + 1}$ , so its coefficient function is supported in degrees $\geq d + 1$ , while the right side is supported in degrees $\leq d$ ; equal polynomials have equal coefficient functions by [F4], so the support is empty and $\sum_{i} a_{i} u^{d - i} = 0$ , which forces $a_{d} = 0$ , contradicting the choice of $d$ . Hence $a = 0$ , and then $b = 0$ as shown. Therefore the only global section of $O (- 1)$ is the zero section and $Γ (P_{k}^{1}, O (- 1)) = 0$ .

3.1F1F2step 2.1

The two sheaves have equal global sections but are not isomorphic: the zero $O_{X}$ -module is quasi-coherent with $Γ (X, 0) = 0$ and $O (- 1)$ is quasi-coherent and nonzero, since over $U_{0} = Spec k [t]$ it is free of rank one with frame $e_{0}$ and the section $e_{0} \in O (- 1) (U_{0})$ is nonzero; by step 2.1 also $Γ (X, O (- 1)) = 0$ . Thus the quasi-coherent sheaves $O (- 1)$ and $0$ have isomorphic (indeed equal zero) modules of global sections while not being isomorphic, so global sections do not determine a quasi-coherent sheaf on $P_{k}^{1}$ , and the false claim is refuted.

4.1step 3.1

The global-sections functor is not conservative here: the unique morphism $0 \to O (- 1)$ from the zero sheaf induces the identity map of the zero module on global sections, which is an isomorphism, while the morphism itself is an isomorphism only if its target $O (- 1)$ is the zero sheaf, which step 3.1 rules out. So a morphism can be invisible to global sections and the affine determination of morphisms cannot be extended to $P_{k}^{1}$ either.

4.2F5step 2.1step 3.1

Therefore $P_{k}^{1}$ is not affine: if $P_{k}^{1}$ were affine, [F5] applied to the quasi-coherent $O (- 1)$ would make $\tilde{Γ (P_{k}^{1}, O (- 1))} \to O (- 1)$ an isomorphism, and since $Γ (P_{k}^{1}, O (- 1)) = 0$ by step 2.1 that would force $O (- 1) = 0$ , contradicting step 3.1. Hence the affine hypothesis in the global-sections determination is essential, and the failure of determination recorded in step 3.1 occurs on the nonaffine scheme $P_{k}^{1}$ .

5.1F1F5F6step 1.1step 4.2∎

Choice accounting: the field $k$ , the two charts, the frames $e_{0}, e_{\infty}$ , the compatibility identity $u b (u) = a (u^{- 1})$ and the finite expansions $a = \sum_{i} a_{i} t^{i}$ and $b = \sum_{j} b_{j} u^{j}$ are fixed data, so no chart, frame or presentation is selected; the only Axiom of Choice is the inherited one recorded in [F6], used through the two-affine construction of [F1] and the affine equivalence underlying [F5].

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Sources

The Stacks Project, Schemes, §§26.5, 26.7, 26.24 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Chapters 6, 14, 17 (standard reference, not scraped)
Gao and Zhang, Lectures on Algebraic Geometry, projective-line gluing (standard reference, not scraped)