The Ext functor Homological algebra

let r ring , let modr category of modules on r. let b in modr , set t(b) = homr(a,b), fixed in modr. left exact functor , has right derived functors rt. ext functor defined by

ext

r


n


⁡
(
a
,
b
)
=
(

r

n


t
)
(
b
)
.


{\displaystyle \operatorname {ext} _{r}^{n}(a,b)=(r^{n}t)(b).}

this can calculated taking injective resolution

0
→
b
→

i

0


→

i

1


→
…
,


{\displaystyle 0\rightarrow b\rightarrow i^{0}\rightarrow i^{1}\rightarrow \dots ,}

and computing

0
→

hom

r


⁡
(
a
,

i

0


)
→

hom

r


⁡
(
a
,

i

1


)
→
…
.


{\displaystyle 0\rightarrow \operatorname {hom} _{r}(a,i^{0})\rightarrow \operatorname {hom} _{r}(a,i^{1})\rightarrow \dots .}

then (rt)(b) homology of complex. note homr(a,b) excluded complex.

an alternative definition given using functor g(a)=homr(a,b). fixed module b, contravariant left exact functor, , have right derived functors rg, , can define

ext

r


n


⁡
(
a
,
b
)
=
(

r

n


g
)
(
a
)
.


{\displaystyle \operatorname {ext} _{r}^{n}(a,b)=(r^{n}g)(a).}

this can calculated choosing projective resolution

⋯
→

p

1


→

p

0


→
a
→
0
,


{\displaystyle \dots \rightarrow p^{1}\rightarrow p^{0}\rightarrow a\rightarrow 0,}

and proceeding dually computing

0
→

hom

r


⁡
(

p

0


,
b
)
→

hom

r


⁡
(

p

1


,
b
)
→
…
.


{\displaystyle 0\rightarrow \operatorname {hom} _{r}(p^{0},b)\rightarrow \operatorname {hom} _{r}(p^{1},b)\rightarrow \dots .}

then (rg)(a) homology of complex. again note homr(a,b) excluded.

these 2 constructions turn out yield isomorphic results, , both may used calculate ext functor.