ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

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A nowhere-vanishing section has empty zero divisor

Example

Let $X$ be a scheme and let $1 \in Γ (X, O_{X})$ be the unit section of the structure sheaf, regarded as a section of the invertible sheaf $O_{X}$ (Zero scheme of a line-bundle section). Then the zero ideal of $1$ is the unit ideal sheaf $I_{1} = O_{X},$ and its zero scheme is empty: $Z (1) = V (I_{1}) = \emptyset .$ Moreover the empty closed subscheme is the effective Cartier divisor with unit local equation $1$ . This holds for every scheme $X$ , including $X = \emptyset$ and including schemes whose structure sheaf has nilpotents or zero divisors.

Facts & Assumptions

Given: A scheme $X$ , the invertible sheaf $O_{X}$ , its global unit section $1$ , and the contraction map $c_{s} : O_{X}^{- 1} \to O_{X}$ of Zero scheme of a line-bundle section.

[A1]

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

[F1]

For a global section $s \in Γ (X, L)$ of an invertible sheaf $L$ , the contraction map $c_{s} : L^{- 1} \to O_{X}$ is defined by $φ \mapsto φ (s)$ , its image $I_{s} = Im (c_{s})$ is a quasi-coherent sheaf of ideals, and the zero scheme is the closed subscheme $Z (s) = V (I_{s}) ↪ X$ ; on an affine open $U = Spec A$ on which $L$ is trivialised with local equation $f$ , the map $c_{s} ∣_{U}$ becomes multiplication by $f$ and $Z (s) \cap U = Spec (A / (f))$ . (Zero scheme of a line-bundle section)

[F2]

If the section $s$ vanishes nowhere, then each local equation $f$ is a unit, so $I_{s} = O_{X}$ and the induced closed immersion $Z (s) \to X$ is an isomorphism onto the empty subscheme; one says $Z (s) = \emptyset$ . Moreover $Z (s)$ is an effective Cartier divisor precisely when each local equation is a nonzerodivisor or a unit, and if every local equation is a unit then $Z (s) = \emptyset$ is the empty divisor. (Zero scheme of a line-bundle section)

Verification

technique · direct: compute the contraction map of the unit section in every trivialisation and evaluate the local chart formula $Z(s)\cap U=\operatorname{Spec}(A/(f))$ with $f=1$

1.1F1

The contraction is the identity. Take $L = O_{X}$ and $s = 1$ . The dual is $L^{- 1} = O_{X}$ and the pairing $O_{X} \otimes O_{X} \to O_{X}$ is multiplication, so $c_{1} : O_{X} \to O_{X}$ sends a local function $g$ to $g \cdot 1 = g$ ; that is, $c_{1} = {i d}_{O_{X}}$ . Consequently $I_{1} = Im (c_{1}) = O_{X}$ , the unit ideal sheaf.

2.1F1step 1.1

The zero scheme is empty. Let $U = Spec A \subseteq X$ be any affine open; over $U$ the structure sheaf is trivialised by the identity and the local equation of the unit section is $f = 1 \in A$ . By the local chart formula of [F1], $Z (1) \cap U = Spec (A / (1)) = Spec 0 = \emptyset$ . Since the affine opens cover $X$ , the closed subscheme $Z (1)$ has no points; it is the empty closed subscheme.

3.1F2step 1.1step 2.1

The empty divisor. In the trivialisation of step 2.1 the local equation $f = 1$ is a unit of $A$ , hence in particular a nonzerodivisor, and $I_{1} = O_{X}$ is an invertible sheaf of ideals; by the criterion of [F2] the zero scheme $Z (1) = \emptyset$ is an effective Cartier divisor, namely the empty divisor, whose local equation is the unit $1$ .

4.1

Conclusion and empty scheme. Steps 1.1, 2.1 and 3.1 give $I_{1} = O_{X}$ and $Z (1) = \emptyset$ with unit local equation, so the empty closed subscheme of $X$ is an effective Cartier divisor. If $X = \emptyset$ then $O_{X}$ is the zero sheaf, $Γ (X, O_{X}) = 0$ and the unit section is $1 = 0$ , the unit of the zero ring; the same computation gives $I_{1} = 0 = O_{X}$ and $Z (1) = \emptyset = X$ , and since $O_{X}$ is invertible (the zero sheaf is locally free of rank one on the empty scheme, where there is no point to test) the conclusion holds vacuously for the empty base as well. No hypothesis on $X$ beyond the trivialisations of $O_{X}$ enters; in particular nilpotents or zero divisors in $O_{X}$ do not affect the computation, which uses only multiplication by $1$ . The Axiom of Choice [A1] is inherited from the affine quotient and gluing suppliers of [F1]; no choice is made here. [A1, F1, F2, step 1.1, step 2.1, step 3.1, cases: empty X and units as local equations] \qed

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Constructions of Schemes, Sections 27.8-27.21 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Sections 4.5, 7.4, 9.3, 10.6, 17.4, 17.6, 18.2 (standard reference, not scraped)