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A morphism of short exact sequences with invertible outer maps is invertible
Statement
In a morphism of short exact sequences in an abelian category, if the left and right vertical maps are isomorphisms, then the middle vertical map is an isomorphism.
Facts & Assumptions
Given: A morphism of short exact sequences whose outer vertical maps are isomorphisms.
The short five lemma makes the middle map both monic and epic (Short five lemma in an abelian category).
Every morphism that is both monic and epic in an abelian category is an isomorphism (An abelian category is balanced).
Proof
Because the outer maps are isomorphisms, they are monic and epic. Therefore [L1] shows that the middle map is monic and epic.
Applying [L2] to that middle map shows that it is an isomorphism.
Hence a morphism of short exact sequences with invertible outer maps has invertible middle map as well.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
12 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
- Saunders Mac Lane, Categories for the Working Mathematician, Lemma VIII.4.1 (standard reference, not scraped)
- The Stacks Project, Section 12.5, Lemma 12.5.2 (standard reference, not scraped)