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.
FALSE: the horseshoe connecting map is independent without a comparison proof
Statement
False. Once the left derived connecting map has been written using a horseshoe resolution, its independence from the chosen horseshoe is automatic and needs no proof.
Facts & Assumptions
Given: Two choices of horseshoe resolution for the same short exact sequence.
Item 9 defines the left derived connecting map from a chosen horseshoe resolution and comparison isomorphisms (The connecting map for left derived functors).
Item 10 is the statement that different horseshoe choices give the same map (The left derived connecting map is independent of the horseshoe resolution and lifts).
Refutation
Before [L2] is proved, item 9 produces one map from each chosen horseshoe resolution. Those maps are a priori attached to different auxiliary choices, so there is no equality available merely from the definition in [L1].
The assertion of [L2] is exactly the missing comparison statement. Thus the independence is not automatic; it is a separate mathematical obligation that must be proved.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
11 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
- Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)