A diagram-chasing result in homological algebra: given a commutative diagram of two exact rows A1→A2→A3→A4→A5 and B1→B2→B3→B4→B5 with vertical maps f_i: A_i→B_i, if f1, f2, f4, f5 are isomorphisms and the middle maps satisfy the appropriate surjectivity/injectivity hypotheses, then f3 is an isomorphism.

Principle

Exactness plus commutativity allow local information about four surrounding morphisms to force invertibility of the central morphism by chasing images and kernels across the diagram.

Demonstration

In algebraic topology, compare the long exact sequences of a pair (X,A) and (Y,B) induced by a map (X,A)→(Y,B). If the outer four induced maps on homology groups are isomorphisms and the required surjectivity/injectivity holds at adjacent degrees, the Five Lemma yields that the induced map in the middle degree is an isomorphism.

Misapplication

Applying the lemma when rows are not exact, when the diagram fails to commute, or when the assumed maps are only bijections on underlying sets (not group/module homomorphisms) — any of these invalidates the kernel/image chase and can lead to false conclusions.

Consequence

One can promote local isomorphisms to a global isomorphism in the middle of an exact sequence, simplifying proofs that certain derived or homology objects coincide under a map.

Reversal

The converse is false in general: having the middle map an isomorphism does not imply the outer four maps are isomorphisms. A reversal that does hold in related situations is the Four Lemma, which gives injectivity or surjectivity of one map from injectivity/surjectivity of others but not full isomorphism.

Boundary

Requires abelian-category-style exactness (or at least exact sequences in groups/modules), strict commutativity of the diagram, and the specific surjectivity/injectivity hypotheses; it does not apply verbatim in non-abelian settings without extra hypotheses.

Semantic Tension

Closely connected to the Snake Lemma and the Nine Lemma; the Five Lemma targets isomorphism in the middle, while the Snake Lemma produces connecting homomorphisms and exact sequences from kernels and cokernels — they overlap in technique but answer different structural questions.

Synthesis

The Five Lemma is a diagram-chasing instrument: under exactness and commutativity, the pattern of isomorphisms and one-sided maps around a central morphism forces that central morphism to be an isomorphism, letting one deduce equivalence of middle objects from equivalences at the boundaries.