DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30

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Invertible sheaves

Definition

Let $X$ be a scheme. An $O_{X}$ -module $L$ is invertible if it is locally free of rank $1$ (Locally free sheaves of finite rank): every point $x \in X$ has an open neighbourhood $U$ with $L ∣_{U} ≅ O_{U} .$ Equivalently, $X$ is covered by open sets $U$ on which $L ∣_{U}$ admits a generator, that is, a section $s \in L (U)$ such that the morphism $O_{U} \to L ∣_{U}$ , $a \mapsto a \cdot s ∣_{U}$ , is an isomorphism. The structure sheaf $O_{X}$ is invertible.

Since local freeness of rank $1$ is a special case of local freeness of finite rank, an invertible sheaf is quasi-coherent, its rank function is the constant function $1$ , the zero sheaf is not invertible on a nonempty scheme, and restriction to an open subscheme preserves invertibility. The tensor product, inverse and dual of invertible sheaves are treated on this page; in particular $L^{\lor} = H o m_{O_{X}} (L, O_{X})$ is again invertible of rank one, and $L^{\lor} \otimes_{O_{X}} L$ is canonically $O_{X}$ .

Depends on

Locally free sheaves of finite rank

Used by

A degree-zero line bundle with a nonzero section is trivial Corollary
Blowing up the base ideal resolves a rational map to projective space Corollary
Degree is additive on invertible sheaves over a proper curve Corollary
Negative definiteness of the primitive part of the Neron-Severi space Corollary
Nontrivial degree-zero line bundles have no sections Corollary
The Picard group of the projective line Corollary
Twisting the exact sequence of an effective Cartier divisor Corollary
Global generation does not imply very ampleness Counterexample
Global sections do not determine a sheaf on P1 Counterexample
h0 differs from the Euler characteristic before vanishing Counterexample
The intersection form is not negative definite on all divisor classes Counterexample
Absolute ampleness by affine section opens Definition
Base points and base-point-free linear systems Definition
Canonical bundle and canonical divisors Definition
Cartier divisor Definition
Complete linear system Definition
Degree of an invertible sheaf on a proper one-dimensional scheme Definition
Exceptional curves of the first kind and their contractions Definition
G-linearizations of invertible sheaves on a complex G-variety Definition
Global generation by the evaluation map Definition
Hilbert function and Euler characteristic on a projective scheme Definition
Hyperelliptic curves and hyperelliptic maps Definition
Intersection numbers of Cartier divisors on a smooth projective surface Definition
Intersection with an invertible sheaf and the first Chern class Definition
Invariance of the blowup under invertible (fractional) rescaling of the ideal Definition
Invertible sheaf of cartier divisor Definition
Multiple test blow-ups, controlled transforms and resolutions of marked ideals Definition
Numerical equivalence and the Neron-Severi space of a surface Definition
Picard group of a scheme Definition
Principal parts of an invertible sheaf on a curve Definition
Rational section line bundle Definition
Relative very ampleness in the finite projective-space convention Definition
The residue pairing of a line bundle with the dual canonical twist Definition
The Riemann-Roch dimension l(D) Definition
The rigidified relative Picard functor and the dual abelian variety Definition
Twists of a quasi-coherent sheaf Definition
Zero scheme of a line-bundle section Definition
Adjunction on a smooth plane cubic: the canonical bundle is trivial Example
All twists on the projective line Example
Hilbert polynomial of projective space Example

…and 84 more results.

Dependency tree · two levels

6 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, §§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)