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.
Restriction of a sheaf to an open subspace
Definition
Let be the inclusion of an open subspace, and let be a sheaf on . The restriction of to is the inverse image sheaf
For an open set , one may identify with : because is open in , the set is also open in , and in the neighbourhood-colimit defining the open set itself is a neighbourhood of .
Depends on
Used by
- A gluing datum for sheaves on an open cover Definition
- Acyclic open cover for a sheaf Definition
- Extension by zero for abelian sheaves on an open subspace Definition
- Gamma-acyclic abelian sheaf Definition
- Locally constant sheaves of finite rank Definition
- The internal Hom sheaf of two module sheaves Definition
- Direct image along an open immersion is restriction-compatible intersection Example
- Two-affine Mayer–Vietoris on the projective line Example
- A sheaf with no stalks off a closed subset is a pushforward Lemma
- Affine pushforward algebra localizes Lemma
- Bounded fibrewise-soft resolutions from a uniform all-sheaf dimension bound Lemma
- Compact-support calculus along open and closed embeddings Lemma
- Extension by zero and the closed complement: a short exact sequence Lemma
- Finite simplicial constructible sheaves have an actual derived incidence model Lemma
- Flatness criteria and canonical epimorphisms from flat abelian sheaves Lemma
- Vanishing compactly supported first cohomology on opens characterises c-softness Lemma
- Cech--Dolbeault comparison for holomorphic line bundles on a compact Riemann surface Theorem
- Coherent sheaves on a locally Noetherian scheme Theorem
- Cohomology of a finite disjoint union Theorem
- Extension by zero is left adjoint to restriction and is exact on abelian sheaves Theorem
- Grothendieck vanishing on a Noetherian space Theorem
- Kernels and cokernels of quasi-coherent modules Theorem
- Mayer–Vietoris sequence for sheaf cohomology Theorem
- Quasi-coherence of pushforward for qcqs morphisms Theorem
- Sheaves of smooth-function modules are cohomologically acyclic Theorem
Dependency tree · two levels
3 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, Lemma 6.21.9 (standard reference, not scraped)
- Ravi Vakil, The Rising Sea, Exercise 2.7.D (standard reference, not scraped)