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 ...

, especially

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

and other applications of

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 ...

theory, the five lemma is an important and widely used

lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), ...

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. The five lemma is not only valid for abelian categories but also works in the

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 ...

, for example. The five lemma can be thought of as a combination of two other theorems, the four lemmas, which are dual to each other.

Statements

Consider the following

in any

(such as the category of

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...

s or the category of

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

s over a given field) or 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. : file:5 lemma.svg The five lemma states that, if the rows are exact, ''m'' and ''p'' are

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

s, ''l'' is an

epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f ...

, and ''q'' is a

monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...

, then ''n'' is also an isomorphism. The two four-lemmas state:

Proof

The method of proof we shall use is commonly referred to as diagram chasing. We shall prove the five lemma by individually proving each of the two four lemmas. To perform diagram chasing, we assume that we are in a category of

modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...

over some ring, so that we may speak of ''elements'' of the objects in the diagram and think of the morphisms of the diagram as ''

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

s'' (in fact,

homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...

s) acting on those elements. Then a morphism is a monomorphism

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bic ...

it is

injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

, and it is an epimorphism if and only if it is

surjective 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 o ...

. Similarly, to deal with exactness, we can think of

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learn ...

s and

image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...

s in a function-theoretic sense. The proof will still apply to any (small) abelian category because of

Mitchell's embedding theorem Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete categor ...

, which states that any small abelian category can be represented as a category of modules over some ring. For the category of groups, just turn all additive notation below into multiplicative notation, and note that commutativity of abelian group is never used. So, to prove (1), assume that ''m'' and ''p'' are surjective and ''q'' is injective. Four lemma epic and zero

: file:4 lemma right.svg * Let ''c′'' be an element of ''C′''. * Since ''p'' is surjective, there exists an element ''d'' in ''D'' with ''p''(''d'') = ''t''(''c′''). * By commutativity of the diagram, ''u''(''p''(''d'')) = ''q''(''j''(''d'')). * Since im ''t'' = ker ''u'' by exactness, 0 = ''u''(''t''(''c′'')) = ''u''(''p''(''d'')) = ''q''(''j''(''d'')). * Since ''q'' is injective, ''j''(''d'') = 0, so ''d'' is in ker ''j'' = im ''h''. * Therefore, there exists ''c'' in ''C'' with ''h''(''c'') = ''d''. * Then ''t''(''n''(''c'')) = ''p''(''h''(''c'')) = ''t''(''c′''). Since ''t'' is a homomorphism, it follows that ''t''(''c′'' − ''n''(''c'')) = 0. * By exactness, ''c′'' − ''n''(''c'') is in the image of ''s'', so there exists ''b′'' in ''B′'' with ''s''(''b′'') = ''c′'' − ''n''(''c''). * Since ''m'' is surjective, we can find ''b'' in ''B'' such that ''b′'' = ''m''(''b''). * By commutativity, ''n''(''g''(''b'')) = ''s''(''m''(''b'')) = ''c′'' − ''n''(''c''). * Since ''n'' is a homomorphism, ''n''(''g''(''b'') + ''c'') = ''n''(''g''(''b'')) + ''n''(''c'') = ''c′'' − ''n''(''c'') + ''n''(''c'') = ''c′''. * Therefore, ''n'' is surjective. Then, to prove (2), assume that ''m'' and ''p'' are injective and ''l'' is surjective. Four lemma monic case

: file:4 lemma left.svg * Let ''c'' in ''C'' be such that ''n''(''c'') = 0. * ''t''(''n''(''c'')) is then 0. * By commutativity, ''p''(''h''(''c'')) = 0. * Since ''p'' is injective, ''h''(''c'') = 0. * By exactness, there is an element ''b'' of ''B'' such that ''g''(''b'') = ''c''. * By commutativity, ''s''(''m''(''b'')) = ''n''(''g''(''b'')) = ''n''(''c'') = 0. * By exactness, there is then an element ''a′'' of ''A′'' such that ''r''(''a′'') = ''m''(''b''). * Since ''l'' is surjective, there is ''a'' in ''A'' such that . * By commutativity, . * Since ''m'' is injective, ''f''(''a'') = ''b''. * So ''c'' = ''g''(''f''(''a'')). * Since the composition of ''g'' and ''f'' is trivial, ''c'' = 0. * Therefore, ''n'' is injective. Combining the two four lemmas now proves the entire five lemma.

Applications

The five lemma is often applied to

long 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 ...

s: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine the unknown homology groups, but if one can compare the original object and sub object to well-understood ones via morphisms, then a morphism between the respective long exact sequences is induced, and the five lemma can then be used to determine the unknown homology groups.

Notes

References

* * {{Citation, last=Massey, first=William S., author-link=William S. Massey, date=1991, title=A basic course in algebraic topology, edition=3rd, volume = 127 , series=Graduate texts in mathematics , publisher=Springer , isbn = 978-0-387-97430-9 Homological algebra Lemmas in category theory Articles containing proofs

Statements

Proof

Applications

See also

Notes

References