Alphabeta MathHow statement and proof provenance work
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Pullback and pushout descend to extension classes
Statement
Pullback along a morphism of quotient objects and pushout along a morphism of subobjects send equivalent short extensions to equivalent short extensions.
Facts & Assumptions
Given: Equivalent extensions of by , and a morphism into or out of .
Proof
Apply the pullback or pushout universal property from Pullback and pushout of an extension to the middle-object map of the equivalence. It produces a morphism between the induced short exact sequences that is the identity on their new endpoints.
This induced morphism is an isomorphism by A morphism of extensions is an isomorphism, hence it is an equivalence of the induced extensions.
Depends on
Used by
Dependency tree · two levels
7 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, Chapters 3–4 (standard reference, not scraped)