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= Homological algebra =
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Introduction
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Homological algebra is the branch of mathematics that studies 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 modules and syzygies) at the end of the 19th century,
chiefly by Henri Poincaré and David Hilbert.
The development of homological algebra was closely intertwined with
the emergence of category theory. By and large, homological algebra is
the study of homological functors and the intricate algebraic
structures that they entail. One quite useful and ubiquitous concept
in mathematics is that of chain complexes, which can be studied
through both their homology and cohomology. Homological algebra
affords the means to extract information contained in these complexes
and present it in the form of homological invariants of rings,
modules, topological spaces, and other 'tangible' mathematical
objects. A powerful tool for doing this is provided by spectral
sequences.
From its very origins, homological algebra 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
algebras, complex analysis, and the theory of partial differential
equations. K-theory is an independent discipline which draws upon
methods of homological algebra, as does the noncommutative geometry of
Alain Connes.
History of homological algebra
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Homological algebra began to be studied in its most basic form in the
1800s as a branch of topology, but it wasn't until the 1940s that it
became an independent subject with the study of objects such as the
ext functor and the tor functor, among others.
Chain complexes and homology
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The notion of chain complex is central in homological algebra. An
abstract chain complex is a sequence (C_\bullet, d_\bullet) of
abelian groups and group homomorphisms,
with the property that the composition of any two consecutive maps is
zero:
: 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.
The elements of 'C''n' are called 'n'-chains and the homomorphisms
'd''n' are called the boundary maps or differentials. The chain groups
'C''n' may be endowed with extra structure; for example, they may be
vector spaces or modules over a fixed ring 'R'. The differentials must
preserve the extra structure if it exists; for example, they must be
linear maps or homomorphisms of 'R'-modules. For notational
convenience, restrict attention to abelian groups (more correctly, to
the category Ab of abelian groups); a celebrated 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''n' = Ker 'd''n' and the boundaries 'B''n' = Im
'd''n'+1, where Ker 'd' and Im 'd' denote the kernel and the image of
'd'. Since the composition of two consecutive boundary maps is zero,
these groups are embedded into each other as
: B_n \subseteq Z_n \subseteq C_n.
Subgroups of abelian groups are automatically normal; therefore we can
define the 'n'th homology group 'H''n'('C') as the factor group of the
'n'-cycles by the 'n'-boundaries,
: H_n(C) = Z_n/B_n = \operatorname{Ker}\, d_n/ \operatorname{Im}\,
d_{n+1}.
A chain complex is called acyclic or an exact sequence if all its
homology groups are zero.
Chain complexes arise in abundance in algebra and algebraic topology.
For example, if 'X' is a topological space then the singular chains
'C''n'('X') are formal linear combinations of continuous maps from the
standard 'n'-simplex into 'X'; if 'K' is a simplicial complex then the
simplicial chains 'C''n'('K') are formal linear combinations of the
'n'-simplices of 'K'; if 'A' = 'F'/'R' is a presentation of an abelian
group 'A' by generators and relations, where 'F' is a free abelian
group spanned by the generators and 'R' is the subgroup of relations,
then letting 'C'1('A') = 'R', 'C'0('A') = 'F', and 'C''n'('A') = 0 for
all other 'n' defines a sequence of abelian groups. In all these
cases, there are natural differentials 'd''n' making 'C''n' 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, manifolds.
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 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
C_\bullet(X) 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 H_\bullet(C) 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 of 'X'.
Exact sequences
=================
In the context of group theory, a sequence
:G_0 \;\xrightarrow{f_1}\; G_1 \;\xrightarrow{f_2}\; G_2
\;\xrightarrow{f_3}\; \cdots \;\xrightarrow{f_n}\; G_n
of groups and group homomorphisms is called exact if the image (or
range) of each homomorphism is equal to the kernel of the next:
:\mathrm{im}(f_k) = \mathrm{ker}(f_{k+1}).\!
Note that the sequence of groups and homomorphisms may be either
finite or infinite.
A similar definition can be made for certain other algebraic
structures. For example, one could have an exact sequence of vector
spaces and linear maps, or of modules and module homomorphisms. More
generally, the notion of an exact sequence makes sense in any category
with kernels and cokernels.
Short exact sequence
======================
The most common type of exact sequence is the short exact sequence.
This is an exact sequence of the form
:A \;\overset{f}{\hookrightarrow}\; B
\;\overset{g}{\twoheadrightarrow}\; C
where � is a monomorphism and 'g' is an epimorphism. In this case,
'A' is a subobject of 'B', and the corresponding quotient is
isomorphic to 'C':
:C \cong B/f(A).
(where 'f(A)' = im('f')).
A short exact sequence of abelian groups may also be written as an
exact sequence with five terms:
:0 \;\xrightarrow{}\; A \;\xrightarrow{f}\; B \;\xrightarrow{g}\; C
\;\xrightarrow{}\; 0
where 0 represents the zero object, such as the trivial group or a
zero-dimensional vector space. The placement of the 0's forces � to
be a monomorphism and 'g' to be an epimorphism (see below).
Long exact sequence
=====================
A long exact sequence is an exact sequence indexed by the natural
numbers.
The five lemma
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Consider the following commutative diagram in any abelian category
(such as the category of abelian groups or the category of vector
spaces over a given field) or in the category of groups.
image:FiveLemma.png
The five lemma states that, if the rows are exact, 'm' and 'p' are
isomorphisms, 'l' is an epimorphism, and 'q' is a monomorphism, then
'n' is also an isomorphism.
The snake lemma
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In an abelian category (such as the category of abelian groups or the
category of vector spaces over a given field), consider a commutative
diagram:
where the rows are exact sequences and 0 is the zero object.
Then there is an exact sequence relating the kernels and cokernels of
'a', 'b', and 'c':
:\ker a \to \ker b \to \ker c \overset{d}{\to} \operatorname{coker}a
\to \operatorname{coker}b \to \operatorname{coker}c
Furthermore, if the morphism 'f' is a monomorphism, then so is the
morphism ker 'a' � ker 'b', and if 'g is an epimorphism, then so is
coker 'b' � coker 'c'.
Abelian categories
====================
In mathematics, an abelian category is a category in which morphisms
and objects can be added and in which kernels and cokernels 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 theories
by Alexander Grothendieck. Abelian categories are very 'stable'
categories, for example they are regular and they satisfy the snake
lemma. The class of Abelian categories is closed under several
categorical constructions, for example, the category of chain
complexes of an Abelian category, or the category of functors 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 products and binary coproducts, and
*it has all kernels and cokernels.
*all monomorphisms and epimorphisms are normal.
The Ext functor
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Let 'R' be a ring and let Mod'R' be the category of modules over 'R'.
Let 'B' be in Mod'R' and set 'T'('B') = Hom'R'('A,B'), for fixed 'A'
in Mod'R'. This is a left exact functor and thus has right derived
functors 'RnT'. The Ext functor is defined by
:\operatorname{Ext}_R^n(A,B)=(R^nT)(B).
This can be calculated by taking any injective resolution
:0 \rightarrow B \rightarrow I^0 \rightarrow I^1 \rightarrow \dots,
and computing
:0 \rightarrow \operatorname{Hom}_R(A,I^0) \rightarrow
\operatorname{Hom}_R(A,I^1) \rightarrow \dots.
Then ('RnT')('B') is the homology of this complex. Note that
Hom'R'('A,B') is excluded from the complex.
An alternative definition is given using the functor
'G'('A')=Hom'R'('A,B'). For a fixed module 'B', this is a
contravariant left exact functor, and thus we also have right derived
functors 'RnG', and can define
:\operatorname{Ext}_R^n(A,B)=(R^nG)(A).
This can be calculated by choosing any projective resolution
:\dots \rightarrow P^1 \rightarrow P^0 \rightarrow A \rightarrow 0,
and proceeding dually by computing
:0\rightarrow\operatorname{Hom}_R(P^0,B)\rightarrow
\operatorname{Hom}_R(P^1,B) \rightarrow \dots.
Then ('RnG')('A') is the homology of this complex. Again note that
Hom'R'('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
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Suppose 'R' is a ring, and denoted by 'R'-Mod the category of left
'R'-modules and by Mod-'R' the category of right 'R'-modules (if 'R'
is commutative, the two categories coincide). Fix a module 'B' in
'R'-Mod. For 'A' in Mod-'R', set 'T'('A') = 'A'�'R''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 left derived functors 'LnT' are defined.
We set
: \mathrm{Tor}_n^R(A,B)=(L_nT)(A)
i.e., we take a projective resolution
: \cdots\rightarrow P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow
A\rightarrow 0
then remove the 'A' term and tensor the projective resolution with 'B'
to get the complex
: \cdots \rightarrow P_2\otimes_R B \rightarrow P_1\otimes_R B
\rightarrow P_0\otimes_R B \rightarrow 0
(note that 'A'�'R''B' does not appear and the last arrow is just the
zero map) and take the homology of this complex.
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'0 and a
collection of three sequences:
# For all integers 'r' � 'r'0, an object 'Er', called a 'sheet' (as in
a sheet of paper), or sometimes a 'page' or a 'term',
# Endomorphisms 'dr' : 'Er' � 'Er' satisfying 'dr' o 'dr' = 0, called
'boundary maps' or 'differentials',
# Isomorphisms of 'Er+1' with 'H'('Er'), the homology of 'Er' with
respect to 'dr'.
The E2 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 E_r^{p,q}.
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 (�'r', 'r' � 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's move in
chess. For higher 'r', the differential acts like a generalized
knight's move.
Derived functor
=================
Suppose we are given a covariant left exact functor 'F' : A � B
between two 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 way of doing so, given by the right derived functors of 'F'.
For every 'i'�1, there is a functor 'RiF': A � B, and the above
sequence continues like so: 0 � 'F'('A') � 'F'('B') � 'F'('C') �
'R'1'F'('A') � 'R'1'F'('B') � 'R'1'F'('C') � 'R'2'F'('A') �
'R'2'F'('B') � ... . From this we see that 'F' is an exact functor if
and only if 'R'1'F' = 0; so in a sense the right derived functors of
'F' measure "how far" 'F' is from being exact.
Functoriality
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A continuous map of topological spaces gives rise to a homomorphism
between their 'n'th homology groups 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, F: C_\bullet\to D_\bullet, is
a family of homomorphisms of abelian groups F_n: C_n \to D_n that
commute with the differentials, in the sense that F_{n-1} \circ d_n^C
= d_n^D \circ F_n for all 'n'. A morphism of chain complexes induces a
morphism H_\bullet(F) of their homology groups, consisting of the
homomorphisms H_n(F) : H_n(C) \to H_n(D) 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 functoriality
property: if two objects 'X' and 'Y' are connected by a map 'f', then
the associated chain complexes are connected by a morphism F=C(f) :
C_\bullet(X) \to C_\bullet(Y), and moreover, the composition g\circ f
of maps 'f': 'X' � 'Y' and 'g': 'Y' � 'Z' induces the morphism
C(g\circ f): C_\bullet(X) \to C_\bullet(Z) that coincides with the
composition C(g) \circ C(f). It follows that the homology groups
H_\bullet(C) 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 L_\bullet,
M_\bullet, N_\bullet and two morphisms between them, f:L_\bullet\to
M_\bullet, g: M_\bullet\to N_\bullet, is called an exact triple, or a
short exact sequence of complexes, and written as
: 0 \longrightarrow L_\bullet \overset{f}{\longrightarrow} M_\bullet
\overset{g}{\longrightarrow} N_\bullet \longrightarrow 0,
if for any 'n', the sequence
: 0 \longrightarrow L_n \overset{f_n}{\longrightarrow} M_n
\overset{g_n}{\longrightarrow}
N_n \longrightarrow 0
is a short exact sequence of abelian groups. By definition, this means
that 'f''n' is an injection, 'g''n' is a surjection, and Im 'f''n' =
Ker 'g''n'. 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
: \ldots \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
\ldots,
where the homology groups of 'L', 'M', and 'N' cyclically follow each
other, and 'δ''n' are certain homomorphisms determined by 'f' and 'g',
called the connecting homomorphisms. Topological manifestations of
this theorem include the Mayer-Vietoris sequence and the long exact
sequence for relative homology.
Foundational aspects
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Cohomology theories have been defined for many different objects such
as topological spaces, sheaves, groups, rings, Lie algebras, and
C*-algebras. 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 and 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:
* Cartan-Eilenberg: In their 1956 book "Homological Algebra", these
authors used projective and injective module resolutions.
* 'Tohoku': The approach in a 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 sheaves of abelian groups).
* The derived category of Grothendieck and Verdier. Derived
categories date back to Verdier's 1967 thesis. They are examples of
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 'torsors' (important in Galois cohomology).
See also
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* Abstract nonsense, a term for homological algebra and category
theory
* Derivator
* Homotopical algebra
* List of homological algebra topics
References
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* 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.
*
* Saunders Mac Lane, 'Homology'. Reprint of the 1975 edition. Classics
in Mathematics. Springer-Verlag, Berlin, 1995. x+422 pp.
* Peter Hilton; Stammbach, U. 'A course in homological algebra'.
Second edition. Graduate Texts in Mathematics, 4. Springer-Verlag, New
York, 1997. xii+364 pp.
* 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.
* 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.
*
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