Exact sequence

The concept of the exact sequence or exact sequence plays a central role in the mathematical subfield of homological algebra . The short, exact sequences are particularly important .

definition

One sequence

${\displaystyle A'\longrightarrow A\longrightarrow A''}$

of objects and morphisms in a suitable category means exactly at the point if ${\displaystyle A}$

${\displaystyle \mathrm {im} (A'\to A)=\ker(A\to A'')}$

holds, d. H. when the image of one arrow is equal to the core of the next. A longer sequence

${\displaystyle A_{1}\longrightarrow A_{2}\longrightarrow A_{3}\longrightarrow A_{4}\longrightarrow A_{5}}$

means exact , if it is exactly at the places , and (analogous for shorter or longer sequences). ${\displaystyle A_{2}}$ ${\displaystyle A_{3}}$ ${\displaystyle A_{4}}$

In this sense, a category is apparently only suitable if one can meaningfully speak of the core and the image. This is the case for all Abelian categories , but also, for example, for the category Grp of groups and group homomorphisms .

Examples

• If there is a homomorphism between Abelian groups , then is and . The sequence is therefore exactly at the point if is. ${\displaystyle f\colon A'\to A}$ ${\displaystyle \operatorname {im} (A'\to A)=\operatorname {Bild} (f):=\{f(a')|a'\in A'\}}$ ${\displaystyle \operatorname {ker} (A'\to A)=\operatorname {Kern} (f):=\{a'|a'\in A',f(a')=0\}}$ ${\displaystyle A'{\overset {f}{\longrightarrow }}A{\overset {g}{\longrightarrow }}A''}$ ${\displaystyle A}$ ${\displaystyle \operatorname {Bild} (f)=\operatorname {Kern} (g)}$
• A sequence is exact if and only if a monomorphism , i. H. is injective . Using a hook arrow, this can also be written with 2 terms: ${\displaystyle 0\longrightarrow A'\;{\overset {f}{\longrightarrow }}\;A}$ ${\displaystyle f\colon A'\to A}$ ${\displaystyle A'\;{\overset {f}{\hookrightarrow }}\;A}$
• One sequence

${\displaystyle A\;{\overset {g}{\longrightarrow }}\;A''\longrightarrow 0}$is exact if and only if an epimorphism , i. H. is surjective . Using a double-headed arrow, this can also be written with 2 terms: ${\displaystyle g\colon A\to A''}$

${\displaystyle A\;{\overset {g}{\twoheadrightarrow }}\;A''}$

• For every homomorphism of vector spaces ( Abelian groups , modules , every morphism of an Abelian category) there is an exact sequence as follows: ${\displaystyle f\colon A\to B}$

${\displaystyle 0\longrightarrow \ker f\longrightarrow A\longrightarrow B\longrightarrow \mathrm {coker} \,f\longrightarrow 0}$

In Grp , however, the sequence is only exact if the image is a normal divider in . Even in additive but not Abelian categories, the exactness is not necessarily given. This denotes the coke core of . ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle B}$ ${\displaystyle \operatorname {coker} f}$ ${\displaystyle f}$

• Be for a group ${\displaystyle G}$
  • ${\displaystyle Z(G)\,}$the center ,
  • ${\displaystyle \mathrm {Aut} \,G}$the group of automorphisms ,
  • ${\displaystyle \mathrm {Inn} \,G}$ the group of internal automorphisms and
  • ${\displaystyle \mathrm {Out} \,G=\mathrm {Aut} \,G/\mathrm {Inn} \,G}$ the group of external automorphisms

from . Then there is the sequence ${\displaystyle G}$

${\displaystyle 1\longrightarrow Z(G)\longrightarrow G\longrightarrow \mathrm {Aut} \,G\longrightarrow \mathrm {Out} \,G\longrightarrow 1}$

exactly. The middle arrow is through

${\displaystyle g\mapsto (h\mapsto ghg^{-1})\in \mathrm {Inn} \,G\subseteq \mathrm {Aut} \,G}$

given.

Short exact sequences

definition

An exact sequence of form

${\displaystyle 0\longrightarrow A'\longrightarrow A\longrightarrow A''\longrightarrow 0}$

is called a short exact sequence .

Disintegrating short exact sequences

A short exact sequence disintegrates when has a cut . Occasionally, instead of disintegrating , the term splits is also used, which is due to an incorrect translation of the English term split . ${\displaystyle A\to A''}$

In an additive category it also follows from this that a retraction has the resulting sequence ${\displaystyle A'\to A}$

${\displaystyle 0\longleftarrow A'\longleftarrow A\longleftarrow A''\longleftarrow 0}$

is also exact and that these sequences are isomorphic to

${\displaystyle 0\longrightarrow A'\longrightarrow A'\oplus A''\longrightarrow A''\longrightarrow 0}$

or.

${\displaystyle 0\longleftarrow A'\longleftarrow A'\oplus A''\longleftarrow A''\longleftarrow 0}$

are.

If a short exact sequence falls apart in the category of groups, this only results in an operation of on , and that is a semi-direct product of and with regard to this operation . For example, the cyclic group is a subgroup of the symmetric group , which is the short exact sequence ${\displaystyle A''}$ ${\displaystyle A'}$ ${\displaystyle A}$ ${\displaystyle A'}$ ${\displaystyle A''}$ ${\displaystyle \mathbb {Z} /3\mathbb {Z} }$ ${\displaystyle S_{3}}$

${\displaystyle 0\longrightarrow \mathbb {Z} /3\mathbb {Z} \longrightarrow S_{3}\longrightarrow \mathbb {Z} /2\mathbb {Z} \longrightarrow 0}$

results; by mapping the non-neutral element of to an element of order 2 in , a split is obtained. ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle S_{3}}$

Splitting a long exact sequence

Every long exact sequence can be broken down into short exact sequences by inserting cores and coke cores : is

${\displaystyle A_{1}\longrightarrow A_{2}\longrightarrow A_{3}\longrightarrow A_{4}\longrightarrow A_{5}}$

an exact sequence, so be

${\displaystyle Z_{n}:=\ker(A_{n}\to A_{n+1})=\mathrm {im} (A_{n-1}\to A_{n})=\mathrm {coker} (A_{n-2}\to A_{n-1}).}$

Then there are short exact sequences

${\displaystyle 0\longrightarrow Z_{n}\longrightarrow A_{n}\longrightarrow Z_{n+1}\longrightarrow 0.}$

If there is a chain complex , the exactness of all this short sequence is equivalent to the exactness of the long sequence. ${\displaystyle (A_{*})}$

Extensions

In the context of a short, exact sequence

${\displaystyle 0\longrightarrow A'\longrightarrow A\longrightarrow A''\longrightarrow 0}$

one also says that an extension of is through . ${\displaystyle A}$ ${\displaystyle A''}$ ${\displaystyle A'}$

If, for example, there is a normal divisor in the group and the factor group , a short, exact sequence is obtained ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle G/N}$

${\displaystyle 0\longrightarrow N\longrightarrow G\longrightarrow G/N\longrightarrow 0}$,

where the second arrow is the embedding of in and the third is the quotient map . This is an extension of and and one can ask the question of a classification of all possible extensions of and . Corresponding questions can be found in the category of rings or modules over a solid ring. This leads to mathematical terms like Ext or group cohomology . ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle G/N}$ ${\displaystyle N}$ ${\displaystyle G/N}$

See also

• Exact functor
• Chain complex

literature

• Siegfried Bosch : Linear Algebra . Springer Verlag, 2008, ISBN 978-3-540-76437-3 , pp. 77-79.