Alphabeta MathHow statement and proof provenance work
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Morphisms of schemes are local on compatible open covers
Statement
Compatible morphisms of schemes on an open cover of a scheme glue uniquely to a morphism from ; two morphisms out of are equal if their restrictions to an open cover are equal. Both assertions may be checked after affine-open refinement of source and target.
Facts & Assumptions
Given: An open cover of and compatible local scheme morphisms.
Proof
The underlying continuous maps agree on overlaps and therefore glue to a unique continuous map from .
Compatibility of the maps of structure sheaves on overlaps glues their sheaf maps, and locality on stalks is local on ; thus the glued map is a scheme morphism.
Restricting a hypothetical equality to the cover proves necessity, while uniqueness in the gluing construction proves sufficiency; refining by affine opens preserves the same argument.
Depends on
Used by
- A proper nonprojective scheme from glued projective spaces Counterexample
- Formally etale morphism Definition
- A section of an invertible sheaf has a canonical zero subscheme Lemma
- Gluing fibre products along open covers Lemma
- Immersions and affine localizations are monomorphisms Lemma
- Proper normal curve rational function map Lemma
- Rational maps from a smooth curve to a proper scheme are morphisms Lemma
- An unramified morphism has an open diagonal Theorem
- Generating line-bundle sections define a morphism to projective space Theorem
- Maps to projective space equal generating line-bundle data Theorem
- Normalization of a reduced curve is finite Theorem
- Normalization of an integral finite-type curve by gluing affine integral closures Theorem
- Projective bundle represents line quotients Theorem
- Scheme-theoretic image of a quasi-compact morphism Theorem
- Smooth proper curves, dominant morphisms and function fields Theorem
- Tangent vectors as dual-number points Theorem
- Universal property of scheme reduction Theorem
- Universal property of the blowup Theorem
Dependency tree · two levels
9 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, Section 14 (standard reference, not scraped)