**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.

- History of homological algebra
- Chain complexes and homology
- Standard tools
- Exact sequences
- The five lemma
- The snake lemma
- Abelian categories
- The Ext functor
- Tor functor
- Spectral sequence
- Derived functor
- Functoriality
- Foundational aspects
- See also
- References

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.

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.^{ [1] }

The notion of chain complex is central in homological algebra. An abstract **chain complex** is a sequence of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero:

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

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,

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 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 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*.

In the context of group theory, a sequence

of groups and group homomorphisms is called **exact** if the image of each homomorphism is equal to the kernel of the next:

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.

The most common type of exact sequence is the **short exact sequence**. This is an exact sequence of the form

where ƒ 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*:

(where *f(A)* = im(*f*)).

A short exact sequence of abelian groups may also be written as an exact sequence with five terms:

where 0 represents the zero object, such as the trivial group or a zero-dimensional vector space. The placement of the 0's forces ƒ to be a monomorphism and *g* to be an epimorphism (see below).

A long exact sequence is an exact sequence indexed by the natural numbers.

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.

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.

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*:

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*.

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.

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 *R ^{n}T*. The Ext functor is defined by

This can be calculated by taking any injective resolution

and computing

Then (*R ^{n}T*)(

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 *R ^{n}G*, and can define

This can be calculated by choosing any projective resolution

and proceeding dually by computing

Then (*R ^{n}G*)(

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

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 *L _{n}T* are defined. We set

i.e., we take a projective resolution

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

(note that *A*⊗_{R}*B* does not appear and the last arrow is just the zero map) and take the homology of this complex.

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*E*, called a_{r}*sheet*(as in a sheet of paper), or sometimes a*page*or a*term*, - Endomorphisms
*d*:_{r}*E*→_{r}*E*satisfying_{r}*d*o_{r}*d*= 0, called_{r}*boundary maps*or*differentials*, - Isomorphisms of
*E*with_{r+1}*H*(*E*), the homology of_{r}*E*with respect to_{r}*d*._{r}

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 .

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.

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 *R ^{i}F*:

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, is a family of homomorphisms of abelian groups that commute with the differentials, in the sense that for all *n*. A morphism of chain complexes induces a morphism of their homology groups, consisting of the homomorphisms 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 and moreover, the composition of maps *f*: *X* → *Y* and *g*: *Y* → *Z* induces the morphism that coincides with the composition It follows that the homology groups 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 and two morphisms between them, is called an **exact triple**, or a **short exact sequence of complexes**, and written as

if for any *n*, the sequence

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**

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.

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).

Wikiquote has quotations related to: Homological algebra |

- Abstract nonsense, a term for homological algebra and category theory
- Derivator
- Homotopical algebra
- List of homological algebra topics

The **snake lemma** is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called *connecting homomorphisms*.

An **exact sequence** is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

In mathematics, and more specifically in homological algebra, the **splitting lemma** states that in any abelian category, the following statements are equivalent for a short exact sequence

In mathematics, a **chain complex** is an algebraic structure that consists of a sequence of abelian groups and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of the next. Associated to a chain complex is its homology, which describes how the images are included in the kernels.

In mathematics, **homology** is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, **group cohomology** is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group *G* in an associated *G*-module*M* to elucidate the properties of the group. By treating the *G*-module as a kind of topological space with elements of representing *n*-simplices, topological properties of the space may be computed, such as the set of cohomology groups . The cohomology groups in turn provide insight into the structure of the group *G* and *G*-module *M* themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper. As in algebraic topology, there is a dual theory called *group homology*. The techniques of group cohomology can also be extended to the case that instead of a *G*-module, *G* acts on a nonabelian *G*-group; in effect, a generalization of a module to non-Abelian coefficients.

In mathematics, particularly algebraic topology and homology theory, the **Mayer–Vietoris sequence** is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into subspaces, for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces.

In algebraic topology, a branch of mathematics, **singular homology** refers to the study of a certain set of algebraic invariants of a topological space *X*, the so-called **homology groups** Intuitively, singular homology counts, for each dimension *n*, the *n*-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

In mathematics, certain functors may be *derived* to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

In algebra, a **module homomorphism** is a function between modules that preserves the module structures. Explicitly, if *M* and *N* are left modules over a ring *R*, then a function is called an *R*-*module homomorphism* or an *R*-*linear map* if for any *x*, *y* in *M* and *r* in *R*,

In algebra, a **flat module** over a ring *R* is an *R*-module *M* such that taking the tensor product over *R* with *M* preserves exact sequences. A module is **faithfully flat** if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact.

In homological algebra and algebraic topology, a **spectral sequence** is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by Jean Leray (1946), they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra.

In mathematics, the **derived category***D*(*A*) of an abelian category *A* is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on *A*. The construction proceeds on the basis that the objects of *D*(*A*) should be chain complexes in *A*, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

In algebraic topology, **universal coefficient theorems** establish relationships between homology groups with different coefficients. For instance, for every topological space X, its *integral homology groups*:

In mathematics, in the field of homological algebra, the **Grothendieck spectral sequence**, introduced by Alexander Grothendieck in his *Tôhoku* paper, is a spectral sequence that computes the derived functors of the composition of two functors , from knowledge of the derived functors of *F* and *G*.

In mathematics, **reduced homology** is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero. This change is required to make statements without some number of exceptional cases.

In homological algebra, the **hyperhomology** or **hypercohomology** of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. It is a sort of cross between the derived functor cohomology of an object and the homology of a chain complex.

In mathematics, particularly homological algebra, the **zig-zag lemma** asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.

In mathematics, and more specifically in homological algebra, a **resolution** is an exact sequence of modules, which is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a **finite resolution** is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object or the rightmost object is the zero-object.

- ↑ Weibel, Charles A. (1999). "History of Homological Algebra".
*History of Topology*. pp. 797–836. doi:10.1016/b978-044482375-5/50029-8. ISBN 9780444823755.

- 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 - Grothendieck, Alexander (1957). "Sur quelques points d'algèbre homologique, I".
*Tohoku Mathematical Journal*.**9**(2): 119–221. doi: 10.2748/tmj/1178244839 . - 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, Charles A. (1994).
*An introduction to homological algebra*. Cambridge Studies in Advanced Mathematics.**38**. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.

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