Editing Homological algebra

{{Short description|Branch of mathematics}}
[[File:Snake lemma origin.svg|thumb|350px|A diagram used in the [[snake lemma]], a basic result in homological algebra.]]

'''Homological algebra''' is the branch of [[mathematics]] that studies [[homology (mathematics)|homology]] in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in [[combinatorial topology]] (a precursor to [[algebraic topology]]) and [[abstract algebra]] (theory of [[module (mathematics)|modules]] and [[Syzygy (mathematics)|syzygies]]) at the end of the 19th century, chiefly by [[Henri Poincaré]] and [[David Hilbert]]. 

Homological algebra is the study of homological [[functor]]s and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of [[category theory]]. A central concept is that of [[chain complex]]es, which can be studied through their homology and [[cohomology]].

Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological [[invariant (mathematics)|invariants]] of [[ring (mathematics)|rings]], modules, [[topological space]]s, and other "tangible" mathematical objects. A [[spectral sequence]] is a powerful tool for this.

It has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includes [[commutative algebra]], [[algebraic geometry]], [[algebraic number theory]], [[representation theory]], [[mathematical physics]], [[operator algebra]]s, [[complex analysis]], and the theory of [[partial differential equation]]s. [[K-theory|''K''-theory]] is an independent discipline which draws upon methods of homological algebra, as does the [[noncommutative geometry]] of [[Alain Connes]].

==History==
Homological algebra began to be studied in its most basic form in the late 19th century as a branch of topology and in the 1940s became an independent subject with the study of objects such as the [[ext functor]] and the [[tor functor]], among others.<ref name="Weber">{{cite book|doi=10.1016/b978-044482375-5/50029-8|chapter=History of homological algebra|title=History of Topology|pages=797–836|year=1999|last1=Weibel|first1=Charles A.|isbn=9780444823755}}</ref>

== Chain complexes and homology ==
{{main|Chain complex}}
The notion of [[chain complex]] is central in homological algebra. An abstract '''chain complex''' is a sequence <math>  (C_\bullet, d_\bullet)</math> of [[abelian group]]s and [[group homomorphism]]s, 
with the property that the composition of any two consecutive [[map (mathematics)|map]]s is zero:
: <math> C_\bullet: \cdots \longrightarrow 
C_{n+1} \stackrel{d_{n+1}}{\longrightarrow}
C_n \stackrel{d_n}{\longrightarrow}
C_{n-1} \stackrel{d_{n-1}}{\longrightarrow}
\cdots, \quad d_n \circ d_{n+1}=0.</math> 
<!--
''d''<sub>''n''+1</sub> o ''d''<sub>''n''</sub> = 0 for all ''n''. 
-->
The elements of ''C''<sub>''n''</sub> are called ''n''-'''chains''' and the homomorphisms ''d''<sub>''n''</sub> are called the '''boundary maps''' or '''differentials'''. The '''chain groups''' ''C''<sub>''n''</sub> may be endowed with extra structure; for example, they may be [[vector space]]s or [[module (mathematics)|modules]] over a fixed [[ring (mathematics)|ring]] ''R''. The differentials must preserve the extra structure if it exists; for example, they must be [[linear map]]s or homomorphisms of ''R''-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the [[category (mathematics)|category]] '''Ab''' of abelian groups); a celebrated [[Mitchell's embedding theorem|theorem by Barry Mitchell]] implies the results will generalize to any [[abelian category]]. Every chain complex defines two further sequences of abelian groups, the '''cycles''' ''Z''<sub>''n''</sub>&nbsp;=&nbsp;Ker ''d''<sub>''n''</sub> and the '''boundaries''' ''B''<sub>''n''</sub>&nbsp;=&nbsp;Im ''d''<sub>''n''+1</sub>, where Ker&nbsp;''d'' and Im&nbsp;''d'' denote the [[kernel (algebra)|kernel]] and the [[image (mathematics)|image]] of ''d''. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as

: <math> B_n \subseteq Z_n \subseteq C_n. </math>

[[Subgroup]]s of abelian groups are automatically [[normal subgroup|normal]]; therefore we can define the ''n''th '''homology group''' ''H''<sub>''n''</sub>(''C'') as the [[factor group]] of the ''n''-cycles by the ''n''-boundaries,

: <math> H_n(C) = Z_n/B_n = \operatorname{Ker}\, d_n/ \operatorname{Im}\, d_{n+1}. </math>

A chain complex is called '''acyclic''' or an '''[[exact sequence]]''' if all its homology groups are zero.

Chain complexes arise in abundance in [[abstract algebra|algebra]] and [[algebraic topology]]. For example, if ''X'' is a [[topological space]] then the [[singular chain]]s ''C''<sub>''n''</sub>(''X'') are formal [[linear combination]]s of [[continuous map]]s from the standard ''n''-[[simplex]] into ''X''; if ''K'' is a [[simplicial complex]] then the [[Chain (algebraic topology)|simplicial chain]]s ''C''<sub>''n''</sub>(''K'') are formal linear combinations of the ''n''-simplices of ''K''; if ''A''&nbsp;=&nbsp;''F''/''R'' is a presentation of an abelian group ''A'' by [[Presentation of a group|generators and relations]], where ''F'' is a [[free abelian group]] spanned by the generators and ''R'' is the subgroup of relations, then letting ''C''<sub>1</sub>(''A'')&nbsp;=&nbsp;''R'', ''C''<sub>0</sub>(''A'')&nbsp;=&nbsp;''F'', and ''C''<sub>''n''</sub>(''A'')&nbsp;=&nbsp;0 for all other ''n'' defines a sequence of abelian groups. In all these cases, there are natural differentials ''d''<sub>''n''</sub> making  ''C''<sub>''n''</sub> into a chain complex, whose homology reflects the structure of the topological space ''X'', the simplicial complex ''K'', or the abelian group ''A''. In the case of topological spaces, we arrive at the notion of [[singular homology]], which plays a fundamental role in investigating the properties of such spaces, for example, [[manifold]]s.

On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects (topological spaces, simplicial complexes, ''R''-modules) contain a lot of valuable algebraic information about them, with the homology being only the most readily available part. On a technical level, homological algebra provides the tools for manipulating complexes and extracting this information. Here are two general illustrations.
*Two objects ''X'' and ''Y'' are connected by a map ''f  '' between them. Homological algebra studies the relation, induced by the map ''f'', between chain complexes associated with ''X'' and ''Y'' and their homology. This is generalized to the case of several objects and maps connecting them. Phrased in the language of [[category theory]], homological algebra studies the [[functor|functorial properties]] of various constructions of chain complexes and of the homology of these complexes.  
* An object ''X'' admits multiple descriptions (for example, as a topological space and as a simplicial complex) or the complex <math>C_\bullet(X)</math> is constructed using some 'presentation' of ''X'', which involves non-canonical choices. It is important to know the effect of change in the description of ''X'' on chain complexes associated with ''X''. Typically, the complex and its homology  <math>H_\bullet(C)</math> are functorial with respect to the presentation; and the homology (although not the complex itself) is actually independent of the presentation chosen, thus it is an [[invariant (mathematics)|invariant]] of ''X''.

==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.

== Functoriality ==
A [[continuous map]] of topological spaces gives rise to a homomorphism between their ''n''th [[homology group]]s for all ''n''. This basic fact of [[algebraic topology]] finds a natural explanation through certain properties of chain complexes. Since it is very common to study
several topological spaces simultaneously, in homological algebra one is led to simultaneous consideration of multiple chain complexes.

A '''morphism''' between two chain complexes, <math> F: C_\bullet\to D_\bullet,</math> is a family of homomorphisms of abelian groups <math>F_n: C_n \to D_n</math> that commute with the differentials, in the sense that <math>F_{n-1} \circ d_n^C = d_n^D \circ F_n</math> for all ''n''. A morphism of chain complexes induces a morphism <math> H_\bullet(F)</math> of their homology groups, consisting of the homomorphisms <math>H_n(F) : H_n(C) \to H_n(D)</math> for all ''n''. A morphism ''F'' is called a '''[[quasi-isomorphism]]''' if it induces an isomorphism on the ''n''th homology for all ''n''.

Many constructions of chain complexes arising in algebra and geometry, including [[singular homology]], have the following [[functor]]iality property: if two objects ''X'' and ''Y'' are connected by a map ''f'', then the associated chain complexes are connected by a morphism <math>F=C(f) : C_\bullet(X) \to C_\bullet(Y),</math> and moreover, the composition <math>g\circ f</math> of maps ''f'':&nbsp;''X''&nbsp;&rarr;&nbsp;''Y'' and  ''g'':&nbsp;''Y''&nbsp;&rarr;&nbsp;''Z'' induces the morphism <math>C(g\circ f): C_\bullet(X) \to C_\bullet(Z)</math> that coincides with the composition <math>C(g) \circ C(f).</math> It follows that the homology groups <math>H_\bullet(C)</math> are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.

The following definition arises from a typical situation in algebra and topology. A triple consisting of three chain complexes <math>L_\bullet, M_\bullet, N_\bullet</math> and two morphisms between them, <math>f:L_\bullet\to M_\bullet, g: M_\bullet\to N_\bullet,</math> is called an '''exact triple''', or a '''short exact sequence of complexes''', and written as

: <math> 0 \longrightarrow L_\bullet \overset{f}{\longrightarrow} M_\bullet \overset{g}{\longrightarrow} N_\bullet \longrightarrow 0,</math>

if for any ''n'', the sequence

: <math> 0 \longrightarrow L_n \overset{f_n}{\longrightarrow} M_n \overset{g_n}{\longrightarrow}
N_n \longrightarrow 0 </math>

is a [[short exact sequence]] of abelian groups. By definition, this means that ''f''<sub>''n''</sub> is an [[injection (mathematics)|injection]], ''g''<sub>''n''</sub> is a [[surjection]], and Im ''f''<sub>''n''</sub>&nbsp;=&nbsp; Ker ''g''<sub>''n''</sub>. One of the most basic theorems of homological algebra, sometimes known as the [[zig-zag lemma]], states that, in this case, there is a '''long exact sequence in homology'''

: <math> \cdots \longrightarrow H_n(L) \overset{H_n(f)}{\longrightarrow} H_n(M) \overset{H_n(g)}{\longrightarrow} H_n(N) \overset{\delta_n}{\longrightarrow} H_{n-1}(L) \overset{H_{n-1}(f)}{\longrightarrow} H_{n-1}(M) \longrightarrow \cdots, </math>

where the homology groups of ''L'', ''M'', and ''N'' cyclically follow each other, and ''&delta;''<sub>''n''</sub> are certain homomorphisms determined by ''f'' and ''g'', called the '''[[connecting homomorphism]]s'''.  Topological manifestations of this theorem include the [[Mayer–Vietoris sequence]] and the long exact sequence for [[relative homology]].
<!-- Requires more work
Functionality of ''homology'' is so fundamental a property that homological algebra takes it for granted. On the other hand, ''chain complexes'' may or may not be functorial in their 'arguments', the objects whose inner structure they are supposed to reflect. This difference serves as a source of constant tension in homological algebra. On the one hand, homology yields invariants of [[de Rham cohomology|smooth manifolds]], [[group cohomology|groups]], [[Hochschild homology|algebras]], and so on; and these invariants have extra algebraic structure, such as multiplication in the cohomology of manifolds, or the [[Mayer-Vietoris sequence]] relating homology of ''U'', ''V'', their union and their intersection. On the other hand, to ''describe'' or even ''define'' this structure, one must inevitably work with the complexes themselves. In particular, many constructions in homological algebra, such as [[connecting homomorphism]], involve intermediate non-canonical choices, which, however, do not affect the final outcome on the level of homology.
-->

== Foundational aspects ==
Cohomology theories have been defined for many different objects such as [[topological space]]s, [[sheaf (mathematics)|sheaves]], [[group (mathematics)|group]]s, [[ring (mathematics)|ring]]s, [[Lie algebra]]s, and [[C*-algebra]]s. The study of modern [[algebraic geometry]] would be almost unthinkable without [[sheaf cohomology]].

Central to homological algebra is the notion of [[exact sequence]]; these can be used to perform actual calculations. A classical tool of homological algebra is that of [[derived functor]]; the most basic examples are functors [[Ext functors|Ext]] and [[Tor functor|Tor]].

With a diverse set of applications in mind, it was natural to try to put the whole subject on a uniform basis. There were several attempts before the subject settled down. An approximate history can be stated as follows:

* [[Henri Cartan|Cartan]]-[[Samuel Eilenberg|Eilenberg]]: In their 1956 book "Homological Algebra", these authors used projective and injective [[resolution of a module|module resolutions]].
* 'Tohoku': The approach in a [[Grothendieck's Tôhoku paper|celebrated paper]] by [[Alexander Grothendieck]] which appeared in the Second Series of the ''[[Tohoku Mathematical Journal]]'' in 1957, using the [[abelian category]] concept (to include [[sheaf (mathematics)|sheaves]] of abelian groups).
* The [[derived category]] of [[Grothendieck]] and [[Jean-Louis Verdier|Verdier]].  Derived categories date back to Verdier's 1967 thesis.  They are examples of [[triangulated category|triangulated categories]] used in a number of modern theories.

These move from computability to generality.

The computational sledgehammer ''par excellence'' is the [[spectral sequence]]; these are essential in the Cartan-Eilenberg and Tohoku approaches where they are needed, for instance, to compute the derived functors of a composition of two functors.  Spectral sequences are less essential in the derived category approach, but still play a role whenever concrete computations are necessary.

There have been attempts at 'non-commutative' theories which extend first cohomology as ''[[torsor]]s'' (important in [[Galois cohomology]]).

==See also==
{{Wikiquote}}
* [[Abstract nonsense]], a term for homological algebra and  [[category theory]]
* [[Derivator]]
* [[Homotopical algebra]]
* [[List of homological algebra topics]]

== References ==
{{reflist}}
* [[Henri Cartan]], [[Samuel Eilenberg]], ''Homological Algebra''. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. xvi+390 pp. {{ISBN|0-691-04991-2}}
* {{cite journal|doi=10.2748/tmj/1178244839|author-link=Alexander Grothendieck|title=Sur quelques points d'algèbre homologique, I|journal=Tohoku Mathematical Journal|volume=9|issue=2|pages=119–221|year=1957|last1=Grothendieck|first1=Alexander|doi-access=free}}
* [[Saunders Mac Lane]], ''Homology''. Reprint of the 1975 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp. {{ISBN|3-540-58662-8}}
* [[Peter Hilton]]; Stammbach, U. ''A Course in Homological Algebra''. Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New York, 1997. xii+364 pp. {{ISBN|0-387-94823-6}}
* Gelfand, Sergei I.; [[Yuri Manin]], ''Methods of Homological Algebra''. Translated from Russian 1988 edition. Second edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. xx+372 pp. {{ISBN|3-540-43583-2}}
* Gelfand, Sergei I.; Yuri Manin, ''Homological Algebra''. Translated from the 1989 Russian original by the authors. Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences (''Algebra'', V, Encyclopaedia Math. Sci., 38, Springer, Berlin, 1994). Springer-Verlag, Berlin, 1999. iv+222 pp. {{ISBN|3-540-65378-3}} 
* {{Weibel IHA}}


{{Topology}}

{{Authority control}}
[[Category:Homological algebra| ]]