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Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is 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. In Mathematics, combinatorial topology was an older name for Algebraic topology, dating from the time when topological invariants of spaces (for example Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In broadest terms Syzygy (ˈsɪzɪʤi is a kind of unity especially through coordination or alignment most commonly used in the Astronomical and/or Astrological Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most

The development of homological algebra was closely intertwined with the emergence of category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories The hidden fabric of mathematics is woven of chain complexes, which manifest themselves through their homology and cohomology. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain 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. In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. A powerful tool for doing this is provided by spectral sequences. In the area of Mathematics known as Homological algebra, especially in Algebraic topology and Group cohomology, a spectral sequence is a

From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of 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. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. In Functional analysis, an operator algebra is an algebra of continuous Linear operators on a Topological vector space with the multiplication Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i K-theory is an independent discipline which draws upon methods of homological algebra, as does noncommutative geometry of Alain Connes. In Mathematics, K-theory is a tool used in several disciplines Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the Alain Connes (born 1 April 1947 is a French Mathematician, currently Professor at the College de France, IHÉS and Vanderbilt University

Contents

Chain complexes and homology

Main article: Chain complex

The chain complex is the central notion of homological algebra. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. It 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 \to C_{n+1} \begin{matrix} d_{n+1} \\ \longrightarrow \\ \, \end{matrix} C_n \begin{matrix} d_n \\ \longrightarrow \\ \, \end{matrix} C_{n-1} \begin{matrix} d_{n-1} \\ \longrightarrow \\ \, \end{matrix} \cdots, \quad d_n \circ d_{n+1}=0.

The elements of Cn are called n-chains and the homomorphisms dn are called the boundary maps or differentials. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function The chain groups Cn may be endowed with extra structure; for example, they may be vector spaces or modules over a fixed ring R. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real 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. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships Mitchell's embedding theorem, also known as the Freyd-Mitchell theorem is a mathematical result about abelian categories; it states that these categories while rather abstractly In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist Every chain complex defines two further sequences of abelian groups, the cycles Zn = Ker dn and the boundaries Bn = Im dn+1, where Ker d and Im d denote the kernel and the image of d. In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage 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 nth homology group Hn(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. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G

Chain complexes arise in abundance in algebra and algebraic topology. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic For example, if X is a topological space then the singular chains Cn(X) are formal linear combinations of continuous maps from the n-sphere into X; if K is a simplicial complex then the simplicial chains Cn(K) are formal linear combinations of the n-simplices of X; 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 C1(A) = R, C0(A) = F, and Cn(A) = 0 for all other n defines a sequence of abelian groups. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle In Mathematics, one method of defining a group is by a presentation. In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in In all these cases, there are natural differentials dn making Cn 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. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

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.

Functoriality

A continuous map of topological spaces gives rise to a homomorphism between their nth homology groups for all n. In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is This basic fact of algebraic topology finds a natural explanation through certain properties of chain complexes. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic 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 Fn:Cn → Dn that commute with the differentials, in the sense that Fn -1 •  dnC = dnD • Fn for all n. A morphism of chain complexes induces a morphism H_\bullet(F) of their homology groups, consisting of the homomorphisms Hn(F): Hn(C) → Hn(D) for all n. A morphism F is called a quasi-isomorphism if it induces the identity map on the nth 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) from C_\bullet(X) to C_\bullet(Y), and moreover, the composition g • f of maps f: X → Y and g: Y → Z induces the morphism C(g • f) from C_\bullet(X) to C_\bullet(Z) that coincides with the composition C(g) • C(f). In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories 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\rightarrow L_\bullet \begin{matrix} f \\ \longrightarrow \\ \, \end{matrix} M_\bullet \begin{matrix} g \\ \longrightarrow \\ \, \end{matrix} N_\bullet \rightarrow 0,

if for any n, the sequence

0\rightarrow L_n \begin{matrix} f_n \\ \longrightarrow \\ \, \end{matrix} M_n \begin{matrix} g_n \\ \longrightarrow \\ \, \end{matrix} N_n \rightarrow 0

is a short exact sequence of abelian groups. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group By definition, this means that fn is an injection, gn is a surjection, and Im fn =  Ker gn. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every One of the most basic theorems of homological algebra states that, in this case, there is a long exact sequence in homology,

\ldots\rightarrow H_n(L) \begin{matrix} H_n(f) \\ \longrightarrow \\ \, \end{matrix} H_n(M) \begin{matrix} H_n(g) \\ \longrightarrow \\ \, \end{matrix} H_n(N)\, \begin{matrix} \delta_n \\ \rightarrow \\ \, \end{matrix} \, H_{n-1}(L) \begin{matrix} H_{n-1}(f) \\ \longrightarrow \\ \, \end{matrix} H_{n-1}(M) \rightarrow\ldots,

where the homology groups of L, M, and N cyclically follow each other, and δn are certain homomorphisms determined by f and g, called the connecting homomorphisms. In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the Topological manifestations of this theorem include the Mayer-Vietoris sequence and the long exact sequence for relative homology. In Algebraic topology and related branches of Mathematics, the Mayer–Vietoris sequence (named after Walther Mayer and Leopold Vietoris) is In Algebraic topology, a branch of Mathematics, the (singular homology of a topological space relative to a subspace is a construction in Singular

Foundational aspects

Cohomology theories have been defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C*-algebras. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. The study of modern algebraic geometry would be almost unthinkable without sheaf cohomology. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, sheaf cohomology is the aspect of Sheaf theory, concerned with sheaves of Abelian groups that applies Homological algebra to

Central to homological algebra is the notion of exact sequence; these can be used to perform actual calculations. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group A classical tool of homological algebra is that of derived functor; the most basic examples are functors Ext and Tor. In Mathematics, certain Functors may be derived to obtain other functors closely related to the original ones In Mathematics, the Ext functors of Homological algebra are Derived functors of Hom functors They were first used in Algebraic topology In higher Mathematics, the Tor functors of Homological algebra are the Derived functors of the Tensor product functor

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:

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. In the area of Mathematics known as Homological algebra, especially in Algebraic topology and Group cohomology, a spectral sequence is a 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). In Mathematics, a principal homogeneous space, or torsor, for a group G is a set X on which G acts freely and In Mathematics, Galois cohomology is the study of the Group cohomology of Galois modules that is the application of Homological algebra to

References

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