Tate cohomology group
Encyclopedia
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. They were invented by John Tate
, and are used in class field theory
.
and A a G-module
, then there is a natural map N from H0(G,A) to
H0(G,A) taking a representative a to Σ g(a) (the sum over all G-conjugates of a). The Tate cohomology groups
are defined by
is a short exact sequence of G-modules, then we get the usual long exact sequence of Tate cohomology groups:
If A is an induced G module then all Tate cohomology groups of A vanish.
The zeroth Tate cohomology group of A is/(Obvious fixed points of G acting on A)
where by the "obvious" fixed point we mean those of the form Σ g(a). In other words,
the zeroth cohomology group in some sense describes the non-obvious fixed points of G acting on A.
The Tate cohomology groups are characterized by the three properties above.
is as follows:
Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G
Then cup product with a is an isomorphism
for all n; in other words the graded Tate cohomology of A is isomorphic to
the Tate cohomology with integral coefficients, with the degree shifted by 2.
are isomorphic to the usual cohomology groups whenever n is greater than the virtual cohomological dimension of the group G. Finite groups have virtual cohomological dimension 0, and in this case Farrell's cohomology groups are the same as those of Tate.
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 H n. The study of fixed points of groups acting on modules and quotient modules...
of a finite group that combine homology and cohomology groups into one sequence. They were invented by John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
, and are used in class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
.
Definition
If G is a finite groupFinite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
and A a G-module
G-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...
, then there is a natural map N from H0(G,A) to
H0(G,A) taking a representative a to Σ g(a) (the sum over all G-conjugates of a). The Tate cohomology groups
are defined by
for n≥ 1.
cokernelCokernelIn mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
of N = quotient of H0(G,A) by norms
kernelKernel (algebra)In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
of N = subgroupSubgroupIn group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of norm 0 elements of H0(G,A)
for n≤ −2.
Properties
Ifis a short exact sequence of G-modules, then we get the usual long exact sequence of Tate cohomology groups:
If A is an induced G module then all Tate cohomology groups of A vanish.
The zeroth Tate cohomology group of A is/(Obvious fixed points of G acting on A)
where by the "obvious" fixed point we mean those of the form Σ g(a). In other words,
the zeroth cohomology group in some sense describes the non-obvious fixed points of G acting on A.
The Tate cohomology groups are characterized by the three properties above.
Tate's theorem
Tate's theorem gives conditions for multiplication by a cohomology class to be an isomorphism between cohomology groups. There are several slightly different versions of it; a version that is particularly convenient for class field theoryClass field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
is as follows:
Suppose that A is a module over a finite group G and a is an element of H2(G,A), such that for every subgroup E of G
- H1(E,A) is trivial, and
- H2(E,A) is generated by Res(a) which has order E.
Then cup product with a is an isomorphism
for all n; in other words the graded Tate cohomology of A is isomorphic to
the Tate cohomology with integral coefficients, with the degree shifted by 2.
Tate-Farrell cohomology
Farrell extended Tate cohomology groups to the case of all groups G of finite virtual cohomological dimension. In Farrell's theory, the groups are isomorphic to the usual cohomology groups whenever n is greater than the virtual cohomological dimension of the group G. Finite groups have virtual cohomological dimension 0, and in this case Farrell's cohomology groups are the same as those of Tate.