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.
Exact sequences of sheaves
Definition
Let be a topological space. A sequence of sheaves of abelian groups on is exact when it is exact in the abelian category of Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories, equivalently in the sense of Exact sequence and short exact sequence in an abelian category.
Thus a sequence
is exact at when the image sheaf of the first morphism equals the kernel sheaf of the second.
The same language applies to -modules on a ringed space.
Depends on
Used by
- Twisting the exact sequence of an effective Cartier divisor Corollary
- A global section of the quotient that does not lift, and its nonzero connecting class Counterexample
- Finite type and finitely presented module sheaves Definition
- A short exact sequence of abelian groups gives a short exact sequence of skyscraper sheaves Example
- Compensating h0 and h1 jumps with constant Euler characteristic Example
- Bounded fibrewise-soft resolutions from a uniform all-sheaf dimension bound Lemma
- Effective Cartier divisors give a short exact sequence Lemma
- Enough injective sheaves of modules Lemma
- Euler characteristic is additive in short exact sequences Lemma
- Extension by zero and the closed complement: a short exact sequence Lemma
- Extension-by-zero generators detect sheaf-cohomology vanishing Lemma
- Extensions of line bundles on the projective line split after ordering Lemma
- Finite filtration of a generated subsheaf of the constant integer sheaf Lemma
- Finite twisted locally free resolutions on projective space Lemma
- Flasque kernel lifts quotient sections Lemma
- Global sections are left exact but need not preserve epimorphisms Lemma
- Local-section formula for derived direct image Lemma
- Projective coherent finiteness and large twist vanishing Lemma
- The Euler characteristic of the structure sheaf is one minus the genus Lemma
- The exact sequence for adding one point to a divisor Lemma
- The point-divisor exact sequence and the Euler-characteristic step Lemma
- The quotient by a maximal line subbundle is locally free Lemma
- Vanishing compactly supported first cohomology on opens characterises c-softness Lemma
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk Theorem
- Cech--Dolbeault comparison for holomorphic line bundles on a compact Riemann surface Theorem
- Coherent sheaves on a locally Noetherian scheme Theorem
- Degree of the coherent Hilbert polynomial Theorem
- Euler characteristic is a Hilbert polynomial Theorem
- Extension by zero is left adjoint to restriction and is exact on abelian sheaves Theorem
- Flasque abelian sheaves are Γ-acyclic Theorem
- Grothendieck vanishing on a Noetherian space Theorem
Dependency tree · two levels
10 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 17.3 (standard reference, not scraped)
- Ravi Vakil, The Rising Sea, Section 2.6 (standard reference, not scraped)