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.
Filtered vector spaces can have zero kernel and zero cokernel without satisfying AB2
Statement refuted
Zero kernel and zero cokernel are enough to force the coimage-image map to be an isomorphism.
Facts & Assumptions
Given: The filtered-vector-space category and the morphism from Filtered vector spaces can be additive with kernels and cokernels without being abelian.
In that example, has zero kernel and zero cokernel, with and , but is not an isomorphism (Filtered vector spaces can be additive with kernels and cokernels without being abelian).
Counterexample
The cited example already computes both endpoint objects explicitly: the coimage is and the image is , even though both kernel and cokernel vanish.
The canonical map from coimage to image is the morphism itself, and [L1] says that is not an isomorphism. So zero kernel and zero cokernel do not force AB2.
Depends on
Used by
Nothing in the library uses this result yet.
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, Section 12.3, Example 12.3.13 (standard reference, not scraped)