NationMaster - Encyclopedia: Group cohomology

In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors Hⁿ. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... This article or section does not cite its references or sources. ... Group theory is that branch of mathematics concerned with the study of groups. ... This picture illustrates how the hours on a clock form a group under modular addition. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Motivation

A general paradigm in group theory is that a group G should be studied via its group representations. A slight generalization of those representations are the G-modules: a G-module is an abelian group M together with a group action of G on M, with every element of G acting as an endomorphism of M. In the sequel we will write G multiplicatively and M additively. Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ...

Given such a G-module M, it is natural to consider the subgroup of G-invariant elements:

Now, if N is a submodule of M (i.e. a subgroup of M mapped to itself by the action of G), it isn't in general true that the invariants in M/N are found as the quotient of the invariants in M by the invariants in N: being invariant 'up to something in N ' is broader. The first group cohomology H¹(G,N) precisely measures the difference. The group cohomology functors Hⁿ in general measure the extent to which taking invariants doesn't respect exact sequences. This is expressed by a long exact sequence. In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

Formal constructions

In this article, G is a finite group. The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). This category of G-modules is an abelian category with enough injectives (since it is isomorphic to the category of all modules over the group ring ℤ[G]). In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described...

Sending each module M to the group of invariants M^G yields a functor from this category to the category $mathfrak{Ab}$ of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors; their values are abelian groups and they are denoted by Hⁿ(G,M), "the n-th cohomology group of G with coefficients in M". H⁰(G,M) is identified with M^G. Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In homological algebra, an exact functor is one which preserves exact sequences. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

Long exact sequence of cohomology

In practice, one often computes the cohomology groups using the following fact: if

is a short exact sequence of G-modules, then a long exact sequence In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

Cochain complexes

Rather than using the machinery of derived functors, we can also define the cohomology groups more concretely, as follows. For $n geq 0$ , we let

C n (G, M)

be the group of all functions from Gⁿ to M: Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...

This is an abelian group; its elements are called the (inhomogeneous) n-cochains. We further define group homomorphisms

These are known as the coboundary homomorphisms. The crucial thing to check here is

thus we have a chain complex and we can compute cohomology. For $n geq 0$ define the group of n-cocycles as: In mathematics, a chain complex is a construct originally used in the field of algebraic topology. ...

The functors Extⁿ

Yet another approach is to treat G-modules as modules over the group ring ℤ[G] and use Ext functors: In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. ...

Here ℤ is treated as the trivial G-module: every element of G acts as the identity. These Ext groups can also be computed via a projective resolution of ℤ, the advantage being that such a resolution only depends on G and not on M.

Group homology

Dually to the construction of group cohomology there is the following definition of group homology: given a G-module M, set DM to be the submodule generated by elements of the form g·m-m, g∈G, m∈M. Assigning to M its so-called co-invariants, the quotient In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

is a right exact functor. Its left derived functors are by definition the group homology In homological algebra, an exact functor is one which preserves exact sequences. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

The functor which assigns M_G to M is isomorphic to the functor which sends M to $mathbb{Z}otimes_{mathbb{Z}[G]}M$ , where $mathbb{Z}$ is endowed with the trivial G-action. Hence one also gets an expression for group homology in terms of the Tor functors, In mathematics, the Tor functors of homological algebra are the derived functors of the tensor product functor. ...

Group homology and cohomology can be treated uniformly for some groups, especially finite groups, in terms of complete resolutions and the Tate cohomology groups. In mathematics, a finite group is a group which has finitely many elements. ... In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. ...

Non-abelian group cohomology

Using the G-invariants and the 1-cochains, one can construct the zeroth and first group cohomology for a group G with coefficients in a non-abelian group. Specifically, a G-group is a (not necessarily abelian) group A together with an action by G.

The first cohomology of G with coefficents in A is defined as above using 1-cocycles and 1-coboundaries. However, it is generally not a group when A is non-abelian. It instead has the structure of a pointed set. In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ...

Using explicit calculations, one still obtains a truncated long exact sequence in cohomology. Specifically, let

be a short exact sequence of G-groups, then there is an exact sequence of pointed sets

Connections with topological cohomology theories

Group cohomology can be related to topological cohomology theories: to the topological group G there is an associated classifying space BG. (If G has no topology about which we care, then we assign the discrete topology to G. In this case, BG is an Eilenberg-MacLane space K(G,1), whose fundamental group is G and whose higher homotopy groups vanish). The n-th cohomology of BG, with coefficients in M (in the topological sense), is the same as the group cohomology of G with coefficients in M. This will involve a local coefficient system unless M is a trivial G-module. The connection holds because the total space EG is contractible, so its chain complex forms a projective resolution of M. In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In mathematics, an Eilenberg-MacLane space is a special kind of topological space that is important in many contexts in algebraic topology, including stage-by-stage constructions of spaces, computations of homotopy groups of spheres, and definition of cohomological operations. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ... In mathematics, local coefficients is an idea from algebraic topology, a kind of half-way stage between homology theory or cohomology theory with coefficients in the usual sense, in a fixed abelian group A, and general sheaf cohomology which, roughly speaking, allows coefficients to vary from point to point in...

When M is a ring with trivial G-action, we inherit good properties which are familiar from the topological context: in particular, there is a cup product under which In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. ...

is a graded module, and a Künneth formula applies. In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... In mathematics, the KÃ¼nneth theorem of algebraic topology describes the singular homology of the cartesian product X Ã— Y of two topological spaces, in terms of singular homology groups Hi(X, R) and Hj(Y, R). ...

If, furthermore, M=k is a field, then

H * (G; k)

is a graded k-algebra. In this case, the Künneth formula yields

For example, let G be the group with two elements, under the discrete topology. The real projective space ℝP^∞ is a classifying space for G. Let k=F₂, the field of two elements. Then This article does not cite its references or sources. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

a polynomial k-algebra on a single generator, since this is the cellular cohomology ring of ℝP^∞. In algebraic topology, singular homology refers to the usual homology functor from the category of topological spaces and continuous mappings to the category of graded abelian groups and group homomorphisms. ...

Hence, as a second example, if G is an elementary abelian 2-group of rank r, and k=F₂, then the Künneth formula gives In group theory an elementary Abelian group is a finite Abelian group, where every nontrivial element has order p where p is a prime. ...

Properties

Group cohomology depends contravariantly on the group G, in the following sense: if f : G → H is a group homomorphism and M is an H-module, then we have a naturally induced morphism Hⁿ(H,M) → Hⁿ(G,M) (where in the latter case, M is treated as a G-module via f). Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

If M is a trivial G-module (i.e. the action of G on M is trivial), the second cohomology group H²(G,M) is in one-to-one correspondence with the set of central extensions of G by M (up to a natural equivalence relation). More generally, if the action of G on M is nontrivial, H²(G,M) classifies the isomorphism classes of all extensions of G by M in which the induced action of G on M by inner automorphisms agrees with the given action. In group theory, a central extension of a group G is an exact sequence of groups such that A is in Z(E), the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to... In mathematics, for G a group or algebra over a field, or other algebraic structure, G′ is an extension of G if there is an exact sequence . See also central extension, extension problem, field extension. ... In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ...

The Hochschild-Serre spectral sequence relates the cohomology of a normal subgroup N of G and the quotient G/N to the cohomology of the group G (for (pro-)finite groups G). In mathematics, especially in the fields of group cohomology, homological algebra and number theory the Lyndon spectral sequence or Hochschild-Serre spectral sequence is a spectral sequence relating the group cohomology of a normal subgroup N and the quotient group G/N to the cohomology of the total group G...

History and relation to other fields

Early recognition of group cohomology came in the Noether's equations of Galois theory (an appearance of cocycles for H¹), and the factor sets of the extension problem for groups (Issai Schur's multiplicator) and in simple algebras (Richard Brauer, the Brauer group), both of these latter being connected with H². The first theorem of the subject can be identified as Hilbert's Theorem 90. Amalie Emmy Noether [1] (March 23, 1882 â€“ April 14, 1935) was a German-born mathematician, said by Einstein in eulogy to be [i]n the judgment of the most competent living mathematicians, [...] the most significant creative mathematical genius thus far produced since the higher education of women began. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group theory. ... In group theory, if the factor group G/K is isomorphic to H, one says that G is an extension of H by K. To consider some examples, if G = H Ã— K, then G is an extension of both H and K. More generally, if G is a semidirect product... Issai Schur (January 10, 1875 in Mogilyov - January 10, 1941 in Tel Aviv) was a mathematician who worked in Germany for most of his life. ... In mathematics, more specifically in group theory, the Schur multiplier (sometimes multiplicator), named after Issai Schur, is the second homology group of a group G with coefficients in the integers, . If the group is presented in terms of a free group F on a set of generators, and a normal... In mathematics, an algebra is simple if it contains no non-trivial ideals. ... Richard Dagobert Brauer (February 10, 1901 - April 17, 1977) was a leading German and American mathematician. ... In mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K. It is an abelian group with elements isomorphism classes of division algebras over K, such that the center is exactly K. The group is named for the algebraist Richard Brauer. ... In number theory, Hilberts Theorem 90 tells us that if L/K is a cyclic extension of number fields generated by an element s and if Î± is an element of L of relative norm 1, then then there exists Î² in L such that Î± = Î²/Î²s. ...

Some general theory was supplied by Mac Lane and Lyndon; from a module-theoretic point of view this was integrated into the Cartan-Eilenberg theory, and topologically into an aspect of the construction of the classifying space BG for G-bundles. Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ... Henri Cartan (born July 8, 1904) is a son of Ã‰lie Cartan, and is, as his father was, a distinguished and influential French mathematician. ... Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ... In mathematics, a classifying space in homotopy theory of a discrete group G is, roughly speaking, a path connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ...

In 1942, while studying $H_2(G,mathbb{Z})$ (which plays a special role in groups), Hopf discovered what is now called Hopf's integral homology formula (Hopf 1942), which is identical to Schur's formula for the Schur multiplier of a finite, finitely presented group: Heinz Hopf (November 19, 1894 – June 3, 1971) was a mathematician born in Gräbschen, Germany. ... In group theory, the Schur multiplier is the second homology group of a group G with coefficients in the integers, . If the group is presented in terms of a free group F on a set of generators, and a normal subgroup R generated by a set of relations on the...

The recognition that these formulas were the same has been said to have led Eilenberg and Mac Lane to the creation of cohomology of groups (Rotman 1995, p. 358). In general, $H_2(G,mathbb{Z}) cong bigl( H^2(G,mathbb{C}^times) bigr)^*$ where the star denotes the algebraic dual group, and when G is finite, there is an unnatural isomorphism $bigl( H^2(G,mathbb{C}^times) bigr)^* cong H^2(G,mathbb{C}^times)$ . Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ... Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ... In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

The application in algebraic number theory to class field theory provided theorems valid for general Galois extensions (not just abelian extensions). The cohomological part of class field theory was axiomatized as the theory of class formations. Galois cohomology is a large field, and now basic in the theories of algebraic groups and étale cohomology (which builds on it). This article or section does not cite its references or sources. ... In mathematics, class field theory is a major branch of algebraic number theory. ... In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois. ... In abstract algebra, an abelian extension is a field extension for which the associated Galois group is abelian. ... In mathematics, a class formation is a structure used to organize the various Galois groups and modules that appear in class field theory. ... In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ... In mathematics, the Ã©tale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. ...

Some refinements in the theory post-1960 have been made, such as continuous cocycles and Tate's redefinition, but the basic outlines remain the same. You may be looking for John Tate (boxer) John Torrence Tate, born March 13, 1925 in Minneapolis, Minnesota, is an American mathematician, distinguished for many fundamental contributions in algebraic number theory and related areas in algebraic geometry. ... In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. ...

The analogous theory for Lie algebras, called Lie algebra cohomology and largely developed after early papers in the late 1940s, by Jean-Louis Koszul, is formally similar, starting with the corresponding definition of invariant. It is much applied in representation theory, and is closely connected with the BRST quantization of theoretical physics. 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 algebra cohomology is a cohomology theory for Lie algebras. ... Jean-Louis Koszul (born January 3, 1920 in Strasbourg, France) is a mathematician best known for studying geometry and discovering the Koszul complex. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In theoretical physics, the BRST formalism is a method of implementing first class constraints. ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ...

References

Categories: Articles needing additional references from September 2007 | Group theory | Algebraic number theory | Homological algebra Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... Jean-Pierre Serre (born September 15, 1926) is one of the leading mathematicians of the twentieth century, active in algebraic geometry, number theory and topology. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... The Springer-Verlag (pronounced SHPRING er FAIR lahk) was a worldwide publishing company base in Germany. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ... The Princeton University Press is a publishing house, a division of Princeton University, that is highly respected in academic publishing. ... Mathematical Reviews is a scientific journal edited by the American Mathematical Society offering reviews of recent mathematical papers. ...

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