Exact sequence

In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a sequence is an ordered list of objects (or events In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism

1 Definition
2 Example
3 Special cases
4 Facts
5 Applications of exact sequences

Definition

To be precise, fix an abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or some other category with kernels and cokernels (such as the category of all groups). In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Choose an index set of consecutive integers. In Mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an index The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Then for each integer i in the index set, let A_i be an object in the category and let f_i be a morphism from A_i to A_i+1. This defines a sequence of objects and morphisms.

The sequence is exact at A_i if the image of f_i−1 is equal to the kernel of f_i:

im f_i−1 = ker f_i. Given a category C and a Morphism fX\rightarrow Y in C, the image of f is a Monomorphism hI\rightarrow In Category theory and its applications to other branches of Mathematics, kernels are a generalization of the kernels of Group homomorphisms and the kernels

The sequence itself is exact if it is exact at each object (except at the very first and the very last object, where exactness doesn't make sense).

Example

Consider the following sequence of abelian groups:

$0\to \Bbb{Z} \overset{2\cdot}{\to} \Bbb{Z} \to \Bbb{Z}/2\Bbb{Z}\to 0$

where 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z/2Z is given by reducing integers modulo 2. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In mathematics Two has many properties in Mathematics. An Integer is called Even if it is divisible by 2 In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers This is indeed an exact sequence:

the image of the map 0→Z is {0}, and the kernel of multiplication by 2 is also {0}, so the sequence is exact at the first Z.
the image of multiplication by 2 is 2Z, and the kernel of reducing modulo 2 is also 2Z, so the sequence is exact at the second Z.
the image of reducing modulo 2 is all of Z/2Z, and the kernel of the zero map is also all of Z/2Z, so the sequence is exact at the position Z/2Z

Special cases

To make sense of the definition, it is helpful to consider what it means in relatively simple cases where the sequence is finite and begins or ends with 0.

The sequence 0 → A → B is exact at A if and only if the map from A to B has kernel {0}, i. e. if and only if that map is a monomorphism. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.
Dually, the sequence B → C → 0 is exact at C if and only if the image of the map from B to C is all of C, i. e. if and only if that map is an epimorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which
A consequence of these last two facts is that the sequence 0 → X → Y → 0 is exact if and only if the map from X to Y is a bimorphism. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

When dealing with exact sequences of groups, it is common to write 1 instead of 0 for the trivial group with a single element.

Important are short exact sequences, which are exact sequences of the form

By the above, we know that for any such short exact sequence, f is a monomorphism and g is an epimorphism. Furthermore, the image of f is equal to the kernel of g. It is helpful to think of A as a subobject of B with f being the embedding of A into B, and of C as the corresponding factor object B/A, with the map g being the natural projection from B to B/A (whose kernel is exactly A).

Facts

The splitting lemma states that if the above short exact sequence admits a morphism t: B → A such that t o f is the identity on A or a morphism u: C → B such that g o u is the identity on C, then B is a twisted direct sum of A and C. In Mathematics, and more specifically in Homological algebra, the splitting lemma states that in any Abelian category, the following statements for (For groups, a twisted direct sum is a semidirect product; in an abelian category, every twisted direct sum is an ordinary direct sum. In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction ) In this case, we say that the short exact sequence splits.

The snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also The nine lemma is a special case. In Mathematics, the nine lemma is a statement about Commutative diagrams and Exact sequences valid in any Abelian category, as well as in the category

The five lemma gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the short five lemma is a special case thereof applying to short exact sequences. In Mathematics, especially Homological algebra and other applications of Abelian category theory the five lemma is an important and widely used lemma In Mathematics, especially Homological algebra and other applications of Abelian category theory the short five lemma is a special case of the Five

The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence

$A_1\to A_2\to A_3\to A_4\to A_5\to A_6 \,\!$

and define

$C_k = \ker (A_k\to A_{k+1})= \operatorname{im} (A_{k-1}\to A_k) = \operatorname{coker} (A_{k-2}\to A_{k-1})$

Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.

Applications of exact sequences

In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about sub- and factor objects.

The extension problem is essentially the question, given the end terms A and C of a short exact sequence, what possibilities exist for the middle term B? In the category of groups, this is equivalent to the question, what groups B have A as a normal subgroup and C as the corresponding factor group? This problem is important in the classification of groups. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. See also Outer automorphism group. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner

Notice that in an exact sequence, the composition f_i+1 o f_i maps A_i to 0 in A_i+2, so every exact sequence is a chain complex. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. Furthermore, only f_i-images of elements of A_i are mapped to 0 by f_i+1, so the homology of this chain complex is trivial. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is More succinctly:

Exact sequences are precisely those chain complexes which are acyclic.

Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.

If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (i. e. an exact sequence indexed by the natural numbers) by repeated application of the snake lemma. In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the This is explained in the article on homology. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is It comes up in algebraic topology in the study of relative homology; the Mayer-Vietoris sequence is another example. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Algebraic topology, a branch of Mathematics, the (singular homology of a topological space relative to a subspace is a construction in Singular In Algebraic topology and related branches of Mathematics, the Mayer–Vietoris sequence (named after Walther Mayer and Leopold Vietoris) is Long exact sequences induced by short exact sequences are also characteristic of derived functors. In Mathematics, certain Functors may be derived to obtain other functors closely related to the original ones

Exact functors are functors that transform exact sequences into exact sequences. In Homological algebra, an exact functor is a Functor, from some category to another which preserves Exact sequences Exact functors are very In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

Dictionary

-noun

(algebra) A sequence of groups with adjacent groups connected by homomorphisms such that the image of one homomorphism is the kernel of the next.

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