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The Baer sum is independent of extension representatives
Statement
The Baer sum depends only on the equivalence classes of the two input extensions.
Facts & Assumptions
Given: Two pairs of equivalent extensions representing the same two classes.
The Baer sum is defined by pullback over and pushout along addition on (Baer sum of abelian-kernel extensions).
Proof
An equivalence of extensions induces an isomorphism of the corresponding pullbacks over , because the pullback is defined by the universal condition that the two quotient maps agree.
Pushing out along the fixed homomorphism respects those pullback isomorphisms. Hence equivalent input extensions produce equivalent pushout extensions.
Therefore the Baer sum depends only on extension classes.
Depends on
Used by
Dependency tree · two levels
3 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
- Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
- Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)