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 ...

, and more specifically in

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

, the splitting lemma states that in 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 ...

, the following statements are

equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equivale ...

for a

short 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 ...

0 \longrightarrow A \mathrel B \mathrel C \longrightarrow 0.

If any of these statements holds, the sequence is called a

split exact sequence In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. Equivalent characterizations A short exact sequence of abelian groups or of modules over a ...

, and the sequence is said to ''split''. In the above short exact sequence, where the sequence splits, it allows one to refine the

first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist f ...

, which states that: : (i.e., isomorphic to the

coimage In algebra, the coimage of a homomorphism :f : A \rightarrow B is the quotient :\text f = A/\ker(f) of the domain by the kernel. The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies. M ...

of or

cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the name: ...

of ) to: : where the first isomorphism theorem is then just the projection onto . It is a categorical generalization of the

rank–nullity theorem The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theor ...

(in the form in

linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...

Proof for the category of abelian groups

and

First, to show that 3. implies both 1. and 2., we assume 3. and take as the natural projection of the direct sum onto , and take as the natural injection of into the direct sum.

prove Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

that 1. implies 3., first note that any member of ''B'' is in the set (). This follows since for all in , ; is in , and is in , since : Next, the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...

of and is 0, since if there exists in such that , and , then ; and therefore, . This proves that is the direct sum of and . So, for all in , can be uniquely identified by some in , in , such that . By exactness . The subsequence implies that is

onto In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...

; therefore for any in there exists some such that . Therefore, for any ''c'' in ''C'', exists ''k'' in ker ''t'' such that ''c'' = ''r''(''k''), and ''r''(ker ''t'') = ''C''. If , then is in ; since the intersection of and , then . Therefore, the

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is an isomorphism; and is isomorphic to . Finally, is isomorphic to due to the exactness of ; so ''B'' is isomorphic to the direct sum of and , which proves (3).

To show that 2. implies 3., we follow a similar argument. Any member of is in the set ; since for all in , , which is in . The intersection of and is , since if and , then . By exactness, , and since is an

injection Injection or injected may refer to: Science and technology * Injective function, a mathematical function mapping distinct arguments to distinct values * Injection (medicine), insertion of liquid into the body with a syringe * Injection, in broadca ...

, is isomorphic to , so is isomorphic to . Since is a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...

, is an injection, and thus is isomorphic to . So is again the direct sum of and . An alternative "

abstract nonsense In mathematics, abstract nonsense, general abstract nonsense, generalized abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, "a ...

proof of the splitting lemma
may be formulated entirely in

category theoretic Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...

terms.

Non-abelian groups

In the form stated here, the splitting lemma does not hold in the full

category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...

, which is not an abelian category.

Partially true

It is partially true: if a short exact sequence of groups is left split or a direct sum (1. or 3.), then all of the conditions hold. For a direct sum this is clear, as one can inject from or project to the summands. For a left split sequence, the map gives an isomorphism, so is a direct sum (3.), and thus inverting the isomorphism and composing with the natural injection gives an injection splitting (2.). However, if a short exact sequence of groups is right split (2.), then it need not be left split or a direct sum (neither 1. nor 3. follows): the problem is that the image of the right splitting need not be

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

. What is true in this case is that is a

semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...

, though not in general a

direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...

Counterexample

To form a counterexample, take the smallest

non-abelian group In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ '' ...

, the

symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...

on three letters. Let denote the alternating subgroup, and let . Let and denote the inclusion map and the

sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...

map respectively, so that :

0 \longrightarrow A \mathrel B \mathrel C \longrightarrow 0

is a short exact sequence. 3. fails, because is not abelian, but 2. holds: we may define by mapping the generator to any

two-cycle A two-stroke (or two-stroke cycle) engine is a type of internal combustion engine that completes a power cycle with two strokes (up and down movements) of the piston during one power cycle, this power cycle being completed in one revolution of t ...

. Note for completeness that 1. fails: any map must map every two-cycle to the

identity Identity may refer to: * Identity document * Identity (philosophy) * Identity (social science) * Identity (mathematics) Arts and entertainment Film and television * ''Identity'' (1987 film), an Iranian film * ''Identity'' (2003 film), ...

because the map has to be a

group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...

, while the

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

of a two-cycle is 2 which can not be divided by the order of the elements in ''A'' other than the identity element, which is 3 as is the alternating subgroup of , or namely the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

3. But every

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

is a product of two-cycles, so is the trivial map, whence is the trivial map, not the identity.

References

Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftvill ...

: ''Homology''. Reprint of the 1975 edition, Springer Classics in Mathematics, , p. 16 *

Allen Hatcher Allen, Allen's or Allens may refer to: Buildings * Allen Arena, an indoor arena at Lipscomb University in Nashville, Tennessee * Allen Center, a skyscraper complex in downtown Houston, Texas * Allen Fieldhouse, an indoor sports arena on the Univer ...

: ''Algebraic Topology''. 2002, Cambridge University Press, , p. 147 {{DEFAULTSORT:Splitting Lemma Homological algebra Lemmas in category theory Articles containing proofs