Standard tools Homological algebra

1 standard tools

1.1 exact sequences

1.1.1 short exact sequence
1.1.2 long exact sequence


1.2 5 lemma
1.3 snake lemma
1.4 abelian categories
1.5 ext functor
1.6 tor functor
1.7 spectral sequence
1.8 derived functor

standard tools
exact sequences

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.

the 5 lemma

consider following commutative diagram in abelian category (such category of abelian groups or category of vector spaces on given field) or in category of groups.

the 5 lemma states that, if rows exact, m , p isomorphisms, l epimorphism, , q monomorphism, n isomorphism.

the snake lemma

in abelian category (such category of abelian groups or category of vector spaces on given field), consider commutative diagram:

where rows exact sequences , 0 0 object. there exact sequence relating kernels , cokernels of a, b, , c:

ker
⁡
a



⟶


ker
⁡
b



⟶


ker
⁡
c



⟶
d


coker
⁡
a



⟶


coker
⁡
b



⟶


coker
⁡
c


{\displaystyle \ker a\;{\color {gray}\longrightarrow }\ker b\;{\color {gray}\longrightarrow }\ker c\;{\overset {d}{\longrightarrow }}\operatorname {coker} a\;{\color {gray}\longrightarrow }\operatorname {coker} b\;{\color {gray}\longrightarrow }\operatorname {coker} c}

furthermore, if morphism f monomorphism, morphism ker a → ker b, , if g epimorphism, coker b → coker c.

abelian categories

in mathematics, abelian category category in morphisms , objects can added , in kernels , cokernels exist , have desirable properties. motivating prototype example of abelian category category of abelian groups, ab. theory originated in tentative attempt unify several cohomology theories alexander grothendieck. abelian categories stable categories, example regular , satisfy snake lemma. class of abelian categories closed under several categorical constructions, example, category of chain complexes of abelian category, or category of functors small category abelian category abelian well. these stability properties make them inevitable in homological algebra , beyond; theory has major applications in algebraic geometry, cohomology , pure category theory. abelian categories named after niels henrik abel.

more concretely, category abelian if

it has 0 object,
it has binary products , binary coproducts, and
it has kernels , cokernels.
all monomorphisms , epimorphisms normal.

the ext functor

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.

tor functor

suppose r ring, , denoted r-mod category of left r-modules , mod-r category of right r-modules (if r commutative, 2 categories coincide). fix module b in r-mod. in mod-r, set t(a) = a⊗rb. t right exact functor mod-r category of abelian groups ab (in case when r commutative, right exact functor mod-r mod-r) , left derived functors lnt defined. set

t
o
r


n


r


(
a
,
b
)
=
(

l

n


t
)
(
a
)


{\displaystyle \mathrm {tor} _{n}^{r}(a,b)=(l_{n}t)(a)}

i.e., take projective resolution

⋯
→

p

2


→

p

1


→

p

0


→
a
→
0


{\displaystyle \cdots \rightarrow p_{2}\rightarrow p_{1}\rightarrow p_{0}\rightarrow a\rightarrow 0}

then remove term , tensor projective resolution b complex

⋯
→

p

2



⊗

r


b
→

p

1



⊗

r


b
→

p

0



⊗

r


b
→
0


{\displaystyle \cdots \rightarrow p_{2}\otimes _{r}b\rightarrow p_{1}\otimes _{r}b\rightarrow p_{0}\otimes _{r}b\rightarrow 0}

(note a⊗rb not appear , last arrow 0 map) , take homology of complex.

spectral sequence

fix abelian category, such category of modules on ring. spectral sequence choice of nonnegative integer r0 , collection of 3 sequences:

a doubly graded spectral sequence has tremendous amount of data keep track of, there common visualization technique makes structure of spectral sequence clearer. have 3 indices, r, p, , q. each r, imagine have sheet of graph paper. on sheet, take p horizontal direction , q vertical direction. @ each lattice point have object




e

r


p
,
q




{\displaystyle e_{r}^{p,q}}

.

it common n = p + q natural index in spectral sequence. n runs diagonally, northwest southeast, across each sheet. in homological case, differentials have bidegree (−r, r − 1), decrease n one. in cohomological case, n increased one. when r zero, differential moves objects 1 space down or up. similar differential on chain complex. when r one, differential moves objects 1 space left or right. when r two, differential moves objects knight s move in chess. higher r, differential acts generalized knight s move.

derived functor

suppose given covariant left exact functor f : → b between 2 abelian categories , b. if 0 → → b → c → 0 short exact sequence in a, applying f yields exact sequence 0 → f(a) → f(b) → f(c) , 1 ask how continue sequence right form long exact sequence. strictly speaking, question ill-posed, since there numerous different ways continue given exact sequence right. turns out (if nice enough) there 1 canonical way of doing so, given right derived functors of f. every i≥1, there functor rf: → b, , above sequence continues so: 0 → f(a) → f(b) → f(c) → rf(a) → rf(b) → rf(c) → rf(a) → rf(b) → ... . see f exact functor if , if rf = 0; in sense right derived functors of f measure how far f being exact.