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.
Cohomology of equivalent derived complexes
PDF p. 4: Property (4): equivalence of complexes
Statement
An isomorphism in the derived category of complexes of abelian groups induces an isomorphism for each integer . The derived category is the category in which quasi-isomorphisms, meaning cochain maps inducing isomorphisms on all cohomology groups, have been inverted.
Facts & Assumptions
Given: An isomorphism in the derived category and an integer .
Goal: Obtain the induced cohomology isomorphism.
Degree-i cohomology is a functor on the derived category. Cohomology descends to the derived category
Proof
Apply the functor in [F1] to the isomorphism and to its inverse.
Functoriality sends the two inverse composites to identity maps, so the resulting maps on cohomology are inverse isomorphisms.
Depends on
Used by · 1
Dependency tree · 3 levels
Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.