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.
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- 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.
Naturality of the cohomology connecting morphism
Statement
A morphism of short exact sequences of cochain complexes induces a commutative square for every .
Facts & Assumptions
Given: A morphism of short exact sequences of cochain complexes.
A cochain complex is read as a chain complex by the grading-reversal convention (Cochain complex in an abelian category).
The homology connecting morphism is natural under morphisms of short exact sequences (Naturality of the homology connecting morphism).
The cohomology connecting maps are the reindexed homology connecting maps (The long exact sequence in cohomology).
Proof
Apply [L1] to both rows of the given morphism. This turns the cochain ladder into a morphism of short exact sequences of chain complexes.
The resulting lower-index connecting square commutes by [L2]. Translating back with [L3] gives the upper-index square in the statement.