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Five lemma for a morphism of long exact sequences
Statement
Let and be long exact sequences in an abelian category, together with a morphism of these sequences. If the four comparison maps at are isomorphisms, then the comparison map is an isomorphism.
Facts & Assumptions
Given: The morphism of long exact sequences in the statement.
Every five-term exact window satisfies the sharp five lemma (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
Extract the five-term window and the corresponding window in the -sequence. Exactness of the long sequences makes both rows exact.
Because the four surrounding comparison maps are isomorphisms, they satisfy both halves of the hypotheses of [L1]. Hence the middle comparison map is both monic and epic.
Therefore that middle comparison map is an isomorphism by [L2].
Depends on
Used by
- The triangulated five lemma Corollary
- Two-out-of-three for quasi-isomorphisms in a short exact sequence diagram Corollary
- A collar constructs the relative orientation class and its boundary class Lemma
- A comparison isomorphism propagates over one skeleton stage Lemma
- A product collar deformation retracts onto its face Lemma
- Canonical twisted fundamental classes over compact subsets Lemma
- Cap duality for open subsets of Euclidean space Lemma
- Derived coherent duality on projective space Lemma
- Duality extends to finite unions of coordinate balls Lemma
- Relative derived duality on projective space over a regular local ring Lemma
- Two high relative cell layers have free homotopy bases and their cellular boundary matrix Lemma
- Alexander duality for compact locally contractible subsets of a sphere Theorem
- Fully relative Poincaré–Lefschetz duality Theorem
- Poincare duality with the orientation local system Theorem
- Poincaré–Lefschetz duality Theorem
- Poincare–Lefschetz duality with local coefficients Theorem
- Simplicial and singular homology agree Theorem
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, VIII.4 (standard reference, not scraped)