Editing Homological algebra (section)

==Standard tools==

===Exact sequences===
{{Main|Exact sequence}}
In the context of [[group theory]], a sequence
:<math>G_0 \;\xrightarrow{f_1}\; G_1 \;\xrightarrow{f_2}\; G_2 \;\xrightarrow{f_3}\; \cdots \;\xrightarrow{f_n}\; G_n</math>
of [[group (mathematics)|groups]] and [[group homomorphism]]s is called '''exact''' if the [[Image (mathematics)|image]] of each homomorphism is equal to the [[Kernel (algebra)|kernel]] of the next:
:<math>\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1}).\!</math>
Note that the sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain other [[algebraic structure]]s.  For example, one could have an exact sequence of [[vector space]]s and [[linear map]]s, or of [[module (mathematics)|modules]] and [[module homomorphism]]s.  More generally, the notion of an exact sequence makes sense in any [[category (mathematics)|category]] with [[kernel (category theory)|kernel]]s and [[cokernel]]s.

====Short====
<!-- :<math>A \;\xrightarrow{f}\; B \;\twoheadrightarrow\; C</math> -->
The most common type of exact sequence is the '''short exact sequence'''. This is an exact sequence of the form
:<math>A \;\overset{f}{\hookrightarrow}\; B \;\overset{g}{\twoheadrightarrow}\; C</math>
where &fnof; is a [[monomorphism]] and ''g'' is an [[epimorphism]].  In this case, ''A'' is a [[subobject]] of ''B'', and the corresponding [[quotient]] is [[isomorphism|isomorphic]] to ''C'':
:<math>C \cong B/f(A).</math> 
(where  ''f(A)'' = im(''f'')).

A short exact sequence of abelian groups may also be written as an exact sequence with five terms:
:<math>0 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C \;\xrightarrow{}\; 0</math>
where 0 represents the [[Initial and terminal objects|zero object]], such as the [[trivial group]] or a zero-dimensional vector space.  The placement of the 0's forces &fnof; to be a monomorphism and ''g'' to be an epimorphism (see below).

====Long====
A long exact sequence is an exact sequence indexed by the [[natural number]]s.

===Five lemma===
{{Main|Five lemma}}
Consider the following [[commutative diagram]] in any [[abelian category]] (such as the category of [[abelian group]]s or the category of [[vector space]]s over a given [[field (algebra)|field]]) or in the category of [[group (mathematics)|group]]s.

[[File:5 lemma.svg]]

The five lemma states that, if the rows are [[exact sequence|exact]], ''m'' and ''p'' are [[isomorphism]]s, ''l'' is an [[epimorphism]], and ''q'' is a [[monomorphism]], then ''n'' is also an isomorphism.

===Snake lemma===
{{Main|Snake lemma}}
In an [[abelian category]] (such as the category of [[abelian group]]s or the category of [[vector space]]s over a given [[field (algebra)|field]]), consider a [[commutative diagram]]:

[[File:Snake lemma origin.svg]]

where the rows are [[exact sequence]]s and 0 is the [[zero object]].
Then there is an exact sequence relating the [[kernel (category theory)|kernels]] and [[cokernel]]s of ''a'', ''b'', and ''c'':

:<math>\ker a \to \ker b  \to \ker c \overset{d}{\to} \operatorname{coker}a \to \operatorname{coker}b \to \operatorname{coker}c</math>

Furthermore, if the morphism ''f'' is a [[monomorphism]], then so is the morphism ker&nbsp;''a'' → ker&nbsp;''b'', and if ''g''' is an [[epimorphism]], then so is coker&nbsp;''b'' → coker&nbsp;''c''.

===Abelian categories===
{{Main|Abelian category}}
In [[mathematics]], an '''abelian category''' is a [[category (category theory)|category]] in which [[morphism]]s and objects can be added and in which [[kernel (category theory)|kernel]]s and [[cokernel]]s exist and have desirable properties. The motivating prototype example of an abelian category is the [[category of abelian groups]], '''Ab'''. The theory originated in a tentative attempt to unify several [[cohomology theory|cohomology theories]] by [[Alexander Grothendieck]]. Abelian categories are very ''stable'' categories, for example they are [[regular category|regular]] and they satisfy the [[snake lemma]]. The class of Abelian categories is closed under several categorical constructions, for example, the category of [[chain complex]]es of an Abelian category, or the category of [[functor]]s from a [[small category]] to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in [[algebraic geometry]], [[cohomology]] and pure [[category theory]]. Abelian categories are named after [[Niels Henrik Abel]].

More concretely, a category is '''abelian''' if
*it has a [[zero object]],
*it has all binary [[Product (category theory)|products]] and binary [[coproduct]]s, and
*it has all [[kernel (category theory)|kernels]] and [[cokernel]]s.
*all [[monomorphism]]s and [[epimorphism]]s are [[normal morphism|normal]].

===Derived functor===
{{Main|Derived functor}}
Suppose we are given a covariant [[left exact functor]] ''F'' : '''A''' → '''B''' between two [[abelian category|abelian categories]] '''A''' and '''B'''. If  0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in '''A''', then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if '''A''' is "nice" enough) there is one [[canonical form|canonical]] way of doing so, given by the right derived functors of ''F''. For every ''i''≥1, there is a functor ''R<sup>i</sup>F'': '''A''' → '''B''', and the above sequence continues like so: 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') → ''R''<sup>1</sup>''F''(''A'') → ''R''<sup>1</sup>''F''(''B'') → ''R''<sup>1</sup>''F''(''C'') → ''R''<sup>2</sup>''F''(''A'') → ''R''<sup>2</sup>''F''(''B'') → ... . From this we see that ''F'' is an exact functor if and only if ''R''<sup>1</sup>''F'' = 0; so in a sense the right derived functors of ''F'' measure "how far" ''F'' is from being exact.

===Ext functor===
{{Main|Ext functor}}
Let ''R'' be a [[ring (mathematics)|ring]] and let Mod<sub>''R''</sub> be the [[Category (mathematics)|category]] of [[module (mathematics)|modules]] over ''R''. Let ''B'' be in Mod<sub>''R''</sub> and set ''T''(''B'') = Hom<sub>''R''</sub>(''A,B''), for fixed ''A'' in Mod<sub>''R''</sub>. This is a [[left exact functor]] and thus has right [[derived functor]]s ''R<sup>n</sup>T''. The Ext functor is defined by

:<math>\operatorname{Ext}_R^n(A,B)=(R^nT)(B).</math>

This can be calculated by taking any [[injective resolution]]

:<math>0 \rightarrow B \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots, </math>

and computing

:<math>0 \rightarrow \operatorname{Hom}_R(A,I^0) \rightarrow \operatorname{Hom}_R(A,I^1) \rightarrow \cdots.</math>

Then (''R<sup>n</sup>T'')(''B'') is the [[chain complex|cohomology]] of this complex. Note that Hom<sub>''R''</sub>(''A,B'') is excluded from the complex.

An alternative definition is given using the functor ''G''(''A'')=Hom<sub>''R''</sub>(''A,B''). For a fixed module ''B'', this is a [[Covariance and contravariance of functors|contravariant]] [[left exact functor]], and thus we also have right [[derived functor]]s ''R<sup>n</sup>G'', and can define

:<math>\operatorname{Ext}_R^n(A,B)=(R^nG)(A).</math>

This can be calculated by choosing any [[projective resolution]]

:<math>\dots \rightarrow P^1 \rightarrow P^0 \rightarrow A \rightarrow 0, </math>

and proceeding dually by computing

:<math>0\rightarrow\operatorname{Hom}_R(P^0,B)\rightarrow  \operatorname{Hom}_R(P^1,B) \rightarrow \cdots.</math>

Then (''R<sup>n</sup>G'')(''A'') is the cohomology of this complex. Again note that Hom<sub>''R''</sub>(''A,B'') is excluded.

These two constructions turn out to yield [[isomorphic]] results, and so both may be used to calculate the Ext functor.

===Tor functor===
{{Main|Tor functor}}
Suppose ''R'' is a [[ring (mathematics)|ring]], and denoted by ''R''-'''Mod''' the [[category theory|category]] of [[module (mathematics)|left ''R''-modules]] and by '''Mod'''-''R'' the category of right ''R''-modules (if ''R'' is [[commutative ring|commutative]], the two categories coincide). Fix a module ''B'' in ''R''-'''Mod'''. For ''A'' in '''Mod'''-''R'', set ''T''(''A'') = ''A''&otimes;<sub>''R''</sub>''B''. Then ''T'' is a [[right exact functor]] from '''Mod'''-''R'' to the [[category of abelian groups]] '''Ab''' (in the case when ''R'' is commutative, it is a right exact functor from '''Mod'''-''R'' to '''Mod'''-''R'') and its [[derived functor|left derived functor]]s ''L<sub>n</sub>T'' are defined. We set

: <math>\mathrm{Tor}_n^R(A,B)=(L_nT)(A)</math>

i.e., we take a [[Projective module#Projective resolutions|projective resolution]]

: <math>\cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow A\rightarrow 0</math>

then remove the ''A'' term and tensor the projective resolution with ''B'' to get the complex

: <math>\cdots \rightarrow P_2\otimes_R B \rightarrow P_1\otimes_R B \rightarrow P_0\otimes_R B  \rightarrow 0</math>

(note that ''A''&otimes;<sub>''R''</sub>''B'' does not appear and the last arrow is just the zero map) and take the [[homology (mathematics)|homology]] of this complex.

===Spectral sequence===
{{Main|Spectral sequence}}
Fix an [[abelian category]], such as a category of modules over a ring.  A '''spectral sequence''' is a choice of a nonnegative integer ''r''<sub>0</sub> and a collection of three sequences:
# For all integers ''r'' ≥ ''r''<sub>0</sub>, an object ''E<sub>r</sub>'', called a ''sheet'' (as in a sheet of [[paper]]), or sometimes a ''page'' or a ''term'',
# Endomorphisms ''d<sub>r</sub>'' : ''E<sub>r</sub>'' → ''E<sub>r</sub>'' satisfying ''d<sub>r</sub>'' <small>o</small> ''d<sub>r</sub>'' = 0, called ''boundary maps'' or ''differentials'',
# Isomorphisms of ''E<sub>r+1</sub>'' with ''H''(''E<sub>r</sub>''), the homology of ''E<sub>r</sub>'' with respect to ''d<sub>r</sub>''.

[[Image:SpectralSequence.png|frame|The E<sub>2</sub> sheet of a cohomological spectral sequence]]
A doubly graded spectral sequence has a tremendous amount of data to keep track of, but there is a common visualization technique which makes the structure of the spectral sequence clearer.  We have three indices, ''r'', ''p'', and ''q''.  For each ''r'', imagine that we have a sheet of graph paper.  On this sheet, we will take ''p'' to be the horizontal direction and ''q'' to be the vertical direction.  At each lattice point we have the object <math>E_r^{p,q}</math>.

It is very common for ''n'' = ''p'' + ''q'' to be another natural index in the spectral sequence. ''n'' runs diagonally, northwest to southeast, across each sheet.  In the homological case, the differentials have bidegree (&minus;''r'',&nbsp;''r''&nbsp;&minus;&nbsp;1), so they decrease ''n'' by one.  In the cohomological case, ''n'' is increased by one.  When ''r'' is zero, the differential moves objects one space down or up.  This is similar to the differential on a chain complex.  When ''r'' is one, the differential moves objects one space to the left or right.  When ''r'' is two, the differential moves objects just like a [[knight (chess)|knight]]'s move in [[chess]].  For higher ''r'', the differential acts like a generalized knight's move.