right Rights are law, legal, social, or ethics, ethical principles of Liberty, freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convent ...

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the nine lemma (or 3×3 lemma) is a statement about

commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...

s and

exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...

s valid in the category of

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

s and any

abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...

. It states: if the diagram to the right is a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well. The nine lemma can be proved by direct

diagram chasing 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to th ...

, or by applying the

snake lemma The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance ...

(to the two bottom rows in the first case, and to the two top rows in the second case). Linderholm (p. 201) offers a satirical view of the nine lemma: :"Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares; put them at both ends of the horizontal and vertical lines. Make faces. You have now proved: ::(a) the Nine Lemma ::(b) the Sixteen Lemma ::(c) the Twenty-five Lemma..." There are two variants of nine lemma: sharp nine lemma and symmetric nine lemma (see Lemmas 3.3, 3.4 in Chapter XII of ).

References

*{{cite book , first = Carl , last = Linderholm , year = 1971 , title =

Mathematics Made Difficult ''Mathematics Made Difficult'' is a book by Carl E. Linderholm that uses advanced mathematical methods to prove results normally shown using elementary proofs. Although the aim is largely satirical, it also shows the non-trivial mathematics behin ...

, publisher = Wolfe , isbn = 0-7234-0415-1 Homological algebra Lemmas in category theory