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.
Kernel sheaves are objectwise, while cokernels and images are sheafified
Definition
Let be a morphism of sheaves of abelian groups on a topological space .
The kernel sheaf is the subsheaf
defined objectwise by
The cokernel presheaf is
and the cokernel sheaf is its sheafification.
The image presheaf is
and the image sheaf is its sheafification.
For sheaves of modules on a ringed space, the same formulas are taken in the corresponding module categories on each open set.
Depends on
Used by
- Euler characteristic in a proper flat family is locally constant Corollary
- Upper semicontinuity of fibre cohomology dimensions Corollary
- A global section of the quotient that does not lift, and its nonzero connecting class Counterexample
- The objectwise cokernel presheaf can fail to be a sheaf Counterexample
- Coherent module sheaves Definition
- Godement resolution of an abelian sheaf Definition
- Principal parts of an invertible sheaf on a curve Definition
- Analytic field sheaves have injective derived models Lemma
- Cartier divisor local equation equivalence Lemma
- Cofinal Čech vanishing implies derived acyclicity Lemma
- Derived tensor product of abelian sheaves Lemma
- Effective Cartier divisors give a short exact sequence Lemma
- Enough injective sheaves of modules Lemma
- Extension-by-zero generators detect sheaf-cohomology vanishing Lemma
- Extensions of line bundles on the projective line split after ordering Lemma
- Filtered colimits and sheaf cohomology on Noetherian spaces Lemma
- Finite filtration of a generated subsheaf of the constant integer sheaf Lemma
- Flasque kernel lifts quotient sections Lemma
- Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
- Flatness criteria and canonical epimorphisms from flat abelian sheaves Lemma
- Global sections are left exact but need not preserve epimorphisms Lemma
- Injective abelian sheaves are flasque Lemma
- Injective modules are flasque and Ext from the structure sheaf is cohomology Lemma
- Nonzero maps from an invertible sheaf to a locally free sheaf are injective Lemma
- Regular hyperplane step for coherent support induction Lemma
- The exact sequence for adding one point to a divisor Lemma
- A sequence of abelian sheaves is exact exactly when it is exact on every stalk Theorem
- Coherent sheaves on a locally Noetherian scheme Theorem
- Cohomology and base change for proper flat coherent families Theorem
- Conormal sequence for a closed immersion Theorem
- Formal unramifiedness iff Omega vanishes Theorem
- Godement terms are flasque and compute cohomology Theorem
- Kernels and cokernels of quasi-coherent modules Theorem
- Quasi-coherence of pushforward for qcqs morphisms Theorem
- Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories Theorem
- Sheaves of smooth-function modules are cohomologically acyclic Theorem
Dependency tree · two levels
8 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 and Section 6.26 (standard reference, not scraped)
- Ravi Vakil, The Rising Sea, Section 2.6 (standard reference, not scraped)