Five lemma

The five lemma is a result in homological algebra: in a commutative diagram with exact rows, certain isomorphism and injectivity or surjectivity conditions on four vertical maps force the middle map to be an isomorphism.

Commutative diagram: A diagram of mathematical objects and maps in which every indicated route between fixed objects gives the same map. The lemma’s hypotheses and conclusion are arranged as a commutative diagram.
Kernel (algebra): The set of elements mapped to the identity by a group homomorphism or to zero by a linear map. Exactness identifies each row’s intermediate image with a kernel.
Homological algebra: The study of homology and related algebraic structures using exact sequences and derived constructions. The five lemma is a foundational diagram lemma in this field.
Short five lemma: A special form of the five lemma applied to a commutative diagram of short exact sequences. It uses shorter exact rows and is often stated with simpler conditions than the general theorem.
Exact sequence: A sequence of maps in which the image of each map equals the kernel of the next. Exactness in the two rows supplies the kernel and image relations used in the proof.
Image (mathematics): The set of outputs attained by a function on its domain. The equality of an image and the next kernel is the row condition the lemma uses.
Alexander Grothendieck: A French mathematician whose work transformed algebraic geometry and influenced modern homological methods. Grothendieck’s categorical framework helped make diagram lemmas usable across broad areas of mathematics.
Four-term exact sequence: A sequence of four objects and three maps that is exact at its two interior objects. Exactness alone in shorter rows does not provide the full five-object setup of the theorem.
Diagram chasing: A proof technique that follows elements through a commutative diagram using map properties and exactness. The standard proof establishes injectivity and surjectivity by tracing elements across the rows.
Group homomorphism: A function between groups that preserves the group operation. In abelian groups, the vertical maps can be group homomorphisms satisfying the lemma’s hypotheses.

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Nine lemmaRelated: Both lemmas use exact diagrams to infer an unknown property from neighboring maps.