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Five lemma in an abelian category
Statement
In a commutative diagram with exact rows
if are isomorphisms, then is an isomorphism.
Facts & Assumptions
Given: The commutative exact-row diagram in the statement.
The sharp five lemma makes the middle map monic under one set of hypotheses and epic under the complementary one (Sharp five lemma in an abelian category).
In an abelian category, a morphism that is both monic and epic is an isomorphism (An abelian category is balanced).
Proof
Because are isomorphisms, they are in particular monic and epic. The first half of [L1] therefore makes monic, and the second half of [L1] makes epic.
Applying [L2] to now shows that is an isomorphism.
Hence the five lemma holds in every abelian category.
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
- Saunders Mac Lane, Categories for the Working Mathematician, Lemma VIII.4.4 (standard reference, not scraped)