Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Modules on a ringed space
Definition
Let be a ringed space.
An -module is a sheaf of abelian groups on such that for every open set the section group is an -module, and for every inclusion the restriction map
is -linear after restricting scalars along .
A morphism of -modules is a morphism of the underlying sheaves of abelian groups whose components are -linear on every open set .
Depends on
Used by
- A non-quasi-coherent module with H1 on an affine scheme Counterexample
- A nonseparated affine cover can have nonaffine intersection Counterexample
- Finite type need not be locally free Counterexample
- Global sections do not determine a sheaf on P1 Counterexample
- Proper cohomology need not be finite for noncoherent sheaves Counterexample
- Quasi-separatedness in pushforward cannot be omitted Counterexample
- Algebraically constructible complexes of analytic field sheaves Definition
- Cohomology and base-change map Definition
- Equivariant constructible derived category of complex varieties Definition
- Fibre of a module sheaf at a point Definition
- Finite type and finitely presented module sheaves Definition
- Higher direct image of a sheaf Definition
- Linear systems, base loci, and general members Definition
- Locally free sheaves of finite rank Definition
- Principal parts of an invertible sheaf on a curve Definition
- Pullback of a module along a morphism of ringed spaces Definition
- Quasi-coherent module on a scheme Definition
- Relative cotangent and tangent spaces Definition
- Sheaf of relative Kähler differentials Definition
- Support of a module sheaf Definition
- Symmetric algebra of a quasi-coherent module Definition
- Tensor product of sheaves of modules Definition
- The internal Hom sheaf of two module sheaves Definition
- A coherent closed-point skyscraper Example
- An upper jump of h0 in a flat projective family Example
- The rank-zero bundle Example
- Twists on the two-affine projective line Example
- Units satisfying the cocycle law glue local rank-one free modules into a line bundle Example
- A section of an invertible sheaf has a canonical zero subscheme Lemma
- A vector bundle on the projective line has a line subbundle of maximal degree Lemma
- Analytic field sheaves have injective derived models Lemma
- Dual and base change for finite locally free sheaves Lemma
- Dual of a line bundle is its tensor inverse Lemma
- Enough injective sheaves of modules Lemma
- Euler characteristic is additive in short exact sequences Lemma
- Extensions of line bundles on the projective line split after ordering Lemma
- Finite simplicial constructible sheaves have an actual derived incidence model Lemma
- Global sections on module sheaves have ordinary sheaf cohomology as derived functors Lemma
- Higher direct images localize over an affine base Lemma
- Injective modules are flasque and Ext from the structure sheaf is cohomology Lemma
…and 27 more results.
Dependency tree · two levels
7 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, Section 6.26 (standard reference, not scraped)
- Ravi Vakil, The Rising Sea, Example 2.2.13 and Section 2.3 (standard reference, not scraped)