Alphabeta MathHow statement and proof provenance work
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FALSE: universality removes the need for supplied resolution data
Statement
False. Because derived functors are universal delta functors, one never needs to supply projective or injective resolution data in their definition.
Facts & Assumptions
Given: The derived-functor construction and its universality theorem.
Left and right derived objects are defined relative to supplied projective or injective resolution data (Supplied projective resolution data, Supplied injective resolution data).
The previous page proves well-definedness only after comparing different supplied data by natural isomorphism (Two supplied projective resolution data define naturally isomorphic left derived functors, Two supplied injective resolution data define naturally isomorphic right derived functors).
Universality is a later comparison principle for already constructed delta functors (Derived functors are universal delta functors, A morphism between universal delta functors is determined in degree zero).
Refutation
The construction of the derived objects starts with the chosen resolution data in [L1]. Without those data there is no deleted resolution to which the functor can be applied.
Item [L2] shows that even the basic well-definedness claim is a separate comparison theorem about different supplied data. Only after that construction work is in place does [L3] compare the resulting delta functors abstractly. Therefore universality does not remove the need for supplied resolution data at the definition stage.
Depends on
- Supplied projective resolution data
- Supplied injective resolution data
- Two supplied projective resolution data define naturally isomorphic left derived functors
- Two supplied injective resolution data define naturally isomorphic right derived functors
- Derived functors are universal delta functors
- A morphism between universal delta functors is determined in degree zero
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
28 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)