Exact sequences Homological algebra

in context of group theory, sequence

g

0





→


f

1







g

1





→


f

2







g

2





→


f

3






⋯



→


f

n







g

n




{\displaystyle g_{0}\;{\xrightarrow {f_{1}}}\;g_{1}\;{\xrightarrow {f_{2}}}\;g_{2}\;{\xrightarrow {f_{3}}}\;\cdots \;{\xrightarrow {f_{n}}}\;g_{n}}

of groups , group homomorphisms called exact if image (or range) of each homomorphism equal kernel of next:

i
m

(

f

k


)
=

k
e
r

(

f

k
+
1


)
.



{\displaystyle \mathrm {im} (f_{k})=\mathrm {ker} (f_{k+1}).\!}

note sequence of groups , homomorphisms may either finite or infinite.

a similar definition can made other algebraic structures. example, 1 have exact sequence of vector spaces , linear maps, or of modules , module homomorphisms. more generally, notion of exact sequence makes sense in category kernels , cokernels.

short exact sequence

the common type of exact sequence short exact sequence. exact sequence of form

a



↪
f



b



↠
g



c


{\displaystyle a\;{\overset {f}{\hookrightarrow }}\;b\;{\overset {g}{\twoheadrightarrow }}\;c}

where ƒ monomorphism , g epimorphism. in case, subobject of b, , corresponding quotient isomorphic c:

c
≅
b

/

f
(
a
)
.


{\displaystyle c\cong b/f(a).}

(where f(a) = im(f)).

a short exact sequence of abelian groups may written exact sequence 5 terms:

0



→




a



→

f




b



→

g




c



→




0


{\displaystyle 0\;{\xrightarrow {}}\;a\;{\xrightarrow {f}}\;b\;{\xrightarrow {g}}\;c\;{\xrightarrow {}}\;0}

where 0 represents 0 object, such trivial group or zero-dimensional vector space. placement of 0 s forces ƒ monomorphism , g epimorphism (see below).

long exact sequence

a long exact sequence exact sequence indexed natural numbers.