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Injective comparison maps are unique up to cochain homotopy
Statement
Assume the Axiom of Dependent Choice.
Any two coaugmentation-preserving maps between injective resolutions extending the same object morphism are cochain-homotopic.
Facts & Assumptions
Given: Two maps between injective resolutions extending the same morphism .
Projective comparison maps are unique up to chain homotopy (Projective comparison maps are unique up to chain homotopy).
The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).
Proof
By [L2], pass to the opposite abelian category. There the two given maps become comparison maps between projective resolutions lifting the same morphism, so [L1] makes them chain-homotopic.
Translating the resulting chain homotopy back to the original category gives the required cochain homotopy.
Depends on
Used by
Dependency tree · two levels
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Sources
- Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
- Romyar Sharifi, Homological Algebra (standard reference, not scraped)