Zig-zag lemma

The zig-zag lemma states that a short exact sequence of chain complexes induces a long exact sequence in homology, with connecting maps built by lifting cycles and taking boundaries.

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Short exact sequence of chain complexes: A degreewise exact sequence of chain complexes whose maps commute with the differentials. This is the hypothesis from which the induced homology sequence is constructed.
Differential (homological algebra): A degree-lowering map in a chain complex whose composite with itself is zero. Applying this map to a lifted cycle produces the boundary used to define the connecting homomorphism.
Homology long exact sequence: The exact sequence of homology groups induced by a short exact sequence of chain complexes. This is the principal output used to relate the homology of all three complexes.
Homological algebra: The branch of mathematics that studies homology and cohomology through algebraic structures and derived constructions. The lemma is a basic method for transferring exactness into homology.
Connecting homomorphism: A map in a long exact sequence defined by lifting an element and applying a boundary or differential. It is the zig-zag step that links homology classes in adjacent complexes.
Cycle (homological algebra): An element of a chain complex annihilated by its differential. The construction begins with a homology class represented by a cycle.
Mayer–Vietoris sequence: A long exact sequence relating the homology or homotopy groups of spaces covered by two subspaces. Its homology form is commonly derived from a short exact sequence of chain complexes.
Long exact sequence: An exact sequence extending across multiple degrees, often arising from a short exact sequence or a pair of spaces. The lemma is a standard source of long exact sequences in homology.
Snake lemma: A homological algebra result that derives an exact sequence of kernels and cokernels from a commutative diagram with exact rows. The same lift-and-boundary pattern underlies the connecting map in both results.
Boundary (homological algebra): An element that is the differential of an element in the next higher degree. The differential of a lift lands in the preceding complex as a boundary representative.