Nine lemma

The Nine lemma states that in a commutative diagram with three exact rows, exactness of two columns implies exactness of the third.

Snake lemma: A theorem deriving an exact sequence of kernels and cokernels from a commutative diagram with exact rows. Its kernel-and-cokernel argument is a standard route to proving the Nine lemma.
Commutative diagram: A diagram of objects and maps in which every pair of paths with the same endpoints gives the same composite map. Commutativity lets maps across rows and columns constrain one another.
Five lemma: A theorem that gives conditions under which the middle map in a commutative diagram of exact sequences is an isomorphism. Both lemmas use exact diagrams to infer an unknown property from neighboring maps.
Alexander Grothendieck: A French mathematician whose work reshaped algebraic geometry and introduced foundational categorical methods. The lemma is commonly associated with the categorical style of homological algebra developed in his mathematical era.
Diagram chase: A proof technique that follows elements through maps in a commutative diagram to establish algebraic claims. Element tracking makes the surjectivity and injectivity steps of the lemma explicit.
Abelian category: A category with kernels and cokernels in which every monomorphism and epimorphism behaves as in the category of abelian groups. The Nine lemma holds in abelian categories, where exactness and diagram arguments are well behaved.
Four lemma: A result that gives sufficient conditions for a map in a commutative diagram of exact sequences to be injective or surjective. It supplies one-sided exactness arguments closely related to the Nine lemma's column conclusion.
Homological algebra: The study of homology and cohomology methods applied to algebraic structures and their maps. The Nine lemma is a basic diagram result within this broader subject.
Exact sequence: A sequence of maps in which the image of each map equals the kernel of the next. The lemma's conclusion is exactness of the third vertical sequence.
Monomorphism: A morphism that can be cancelled from the left in a composition equality. The leftmost maps in short exact columns must be monomorphisms.

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