View source for Cohomology of groups

Historically, the earliest theory of a [[Cohomology of algebras|cohomology of algebras]].

With every pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231301.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231302.png" /> is a group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231303.png" /> a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231304.png" />-module (that is, a module over the integral group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231305.png" />), there is associated a sequence of Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231306.png" />, called the cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231307.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231308.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c0231309.png" />, which runs over the non-negative integers, is called the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313010.png" />. The cohomology groups of groups are important invariants containing information both on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313011.png" /> and on the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313012.png" />.

By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313013.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313015.png" /> is the submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313016.png" />-invariant elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313017.png" />. The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313019.png" />, are defined as the values of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313020.png" />-th [[Derived functor|derived functor]] of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313021.png" />. Let

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be some projective [[Resolution|resolution]] of the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313023.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313024.png" /> in the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313025.png" />-modules, that is, an exact sequence in which every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313026.png" /> is a projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313027.png" />-module. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313028.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313029.png" />-th cohomology group of the [[Complex|complex]]

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313031.png" /> is induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313032.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313033.png" />.

The homology groups of a group are defined using the dual construction, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313034.png" /> is replaced everywhere by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313035.png" />.

The set of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313037.png" /> is a cohomological functor (see [[Homology functor|Homology functor]]; [[Cohomology functor|Cohomology functor]]) on the category of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313038.png" />-modules.

A module of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313039.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313040.png" /> is an Abelian group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313041.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313042.png" /> according to the formula

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is said to be co-induced. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313044.png" /> is injective or co-induced, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313045.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313046.png" />. Every module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313047.png" /> is isomorphic to a submodule of a co-induced module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313048.png" />. The exact homology sequence for the sequence

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then defines isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313051.png" />, and an exact sequence

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Therefore, the computation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313053.png" />-dimensional cohomology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313054.png" /> reduces to calculating the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313055.png" />-dimensional cohomology group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313056.png" />. This device is called dimension shifting.

Dimension shifting enables one to give an axiomatic definition of cohomology groups, namely, they can be defined as a sequence of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313057.png" /> from the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313058.png" />-modules into the category of Abelian groups forming a cohomological functor and satisfying the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313060.png" />, for every co-induced module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313061.png" />.

The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313062.png" /> can also be defined as equivalence classes of exact sequences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313063.png" />-modules of the form

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with respect to a suitably defined equivalence relation (see [[#References|[1]]], Chapt. 3, 4).

To compute the cohomology groups, the standard resolution of the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313065.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313066.png" /> is generally used, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313067.png" /> and, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313068.png" />,

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where the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313070.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313071.png" /> means that the term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313072.png" /> is omitted. The cochains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313073.png" /> are the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313074.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313075.png" />. Changing variables according to the rules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313078.png" />, one can go over to inhomogeneous cochains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313079.png" />. The coboundary operation then acts as follows:

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For example, a one-dimensional cocycle is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313083.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313084.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313085.png" />, and a coboundary is a function of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313086.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313087.png" />. A one-dimensional cocycle is also said to be a crossed homomorphism and a one-dimensional coboundary a trivial crossed homomorphism. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313088.png" /> acts trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313089.png" />, crossed homomorphisms are just ordinary homomorphisms and all the trivial crossed homomorphisms are 0, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313090.png" /> in this case.

The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313091.png" /> can be interpreted as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313092.png" />-conjugacy classes of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313093.png" /> in the exact sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313094.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313095.png" /> is the [[Semi-direct product|semi-direct product]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313097.png" />. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313098.png" /> can be interpreted as classes of extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c02313099.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130100.png" />. Finally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130101.png" /> can be interpreted as obstructions to extensions of non-Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130102.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130103.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130104.png" /> (see [[#References|[1]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130105.png" />, there are no analogous interpretations known (1978) for the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130106.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130107.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130108.png" />, then restriction of cocycles from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130109.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130110.png" /> defines functorial restriction homomorphisms for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130111.png" />:

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For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130114.png" /> is just the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130115.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130116.png" /> is some quotient group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130117.png" />, then lifting cocycles from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130118.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130119.png" /> induces the functorial inflation homomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130120.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130121.png" /> be a homomorphism. Then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130122.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130123.png" /> can be regarded as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130124.png" />-module by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130125.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130126.png" />. Combining the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130128.png" /> gives mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130129.png" />. In this sense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130130.png" /> is a contravariant functor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130131.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130132.png" /> is a group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130133.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130134.png" /> can be given the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130135.png" />-module. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130136.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130137.png" />, the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130138.png" /> can be equipped with a natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130139.png" />-module structure. This is possible thanks to the fact that inner automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130140.png" /> induce the identity mapping on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130141.png" />. In particular, for a normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130142.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130143.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130144.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130145.png" /> be a subgroup of finite index in the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130146.png" />. Using the norm map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130147.png" />, one can use dimension shifting to define the functorial co-restriction mappings for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130148.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130149.png" /></td> </tr></table>

These satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130150.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130151.png" /> is a normal subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130152.png" /> then there exists the Lyndon spectral sequence with second term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130153.png" /> converging to the cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130154.png" /> (see [[#References|[1]]], Chapt. 11). In small dimensions it leads to the exact sequence

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130155.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130156.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130157.png" /> is the transgression mapping.

For a finite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130158.png" />, the norm map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130159.png" /> induces the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130160.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130161.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130162.png" /> is the ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130163.png" /> generated by the elements of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130165.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130166.png" /> can be used to unite the exact cohomology and homology sequences. More exactly, one can define modified cohomology groups (also called Tate cohomology groups) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130167.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130168.png" />. Here

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130169.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130170.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130171.png" /></td> </tr></table>

For these cohomology groups there exists an exact cohomology sequence that is infinite in both directions. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130172.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130173.png" /> is said to be cohomologically trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130174.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130175.png" /> and all subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130176.png" />. A module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130177.png" /> is cohomologically trivial if and only if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130178.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130179.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130180.png" /> for every subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130181.png" />. Every module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130182.png" /> is a submodule or a quotient module of a cohomologically trivial module, and this allows one to use dimension shifting both to raise and to lower the dimension. In particular, dimension shifting enables one to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130184.png" /> (but not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130185.png" />) for all integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130186.png" />. For a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130187.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130188.png" /> the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130189.png" /> are finite.

The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130190.png" /> are annihilated on multiplication by the order of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130191.png" />, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130192.png" />, induced by restrictions, is a monomorphism, where now <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130193.png" /> is a Sylow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130194.png" />-subgroup (cf. [[Sylow subgroup|Sylow subgroup]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130195.png" />. A number of problems concerning the cohomology of finite groups can be reduced in this way to the consideration of the cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130196.png" />-groups. The cohomology of cyclic groups has period 2, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130197.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130198.png" />.

For arbitrary integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130199.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130200.png" /> there is defined a mapping

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130201.png" /></td> </tr></table>

(called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130203.png" />-product, cup-product), where the tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130205.png" /> is viewed as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130206.png" />-module. In the special case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130207.png" /> is a ring and the operations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130208.png" /> are automorphisms, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130209.png" />-product turns <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130210.png" /> into a graded ring. The duality theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130212.png" />-products asserts that, for every divisible Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130213.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130214.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130215.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130216.png" />-product

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130217.png" /></td> </tr></table>

defines a group isomorphism between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130218.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130219.png" /> (see [[#References|[2]]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130220.png" />-product is also defined for infinite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130221.png" /> provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130222.png" />.

Many problems lead to the necessity of considering the cohomology of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130223.png" /> acting continuously on a topological module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130224.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130225.png" /> is a [[Profinite group|profinite group]] (the case nearest to that of finite groups) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130226.png" /> is a discrete Abelian group that is a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130227.png" />-module, one can consider the cohomology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130228.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130229.png" />, computed in terms of continuous cochains [[#References|[5]]]. These groups can also be defined as the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130230.png" /> with respect to the inflation mapping, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130231.png" /> runs over all open normal subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130232.png" />. This cohomology has all the usual properties of the cohomology of finite groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130233.png" /> is a pro-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130234.png" />-group, the dimension over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130235.png" /> of the first and second cohomology groups with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130236.png" /> are interpreted as the minimum number of generators and relations (between these generators) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130237.png" />, respectively.

See [[#References|[6]]] for different variants of continuous cohomology, and also for certain other types of cohomology groups. See [[Non-Abelian cohomology|Non-Abelian cohomology]] for cohomology with a non-Abelian coefficient group.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956) {{MR|0077480}} {{ZBL|0075.24305}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1967) {{MR|0215665}} {{ZBL|0153.07403}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0181643}} {{ZBL|0143.05901}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Koch, "Galoissche Theorie der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130238.png" />-Erweiterungen" , Deutsch. Verlag Wissenschaft. (1970)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> ''Itogi Nauk. Mat. Algebra. 1964'' (1966) pp. 202–235</TD></TR></table>



====Comments====
The norm map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130239.png" /> is defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130240.png" /> be a set of representatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130241.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130242.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130243.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130244.png" />. For a definition of the transgression relation in general spectral sequences cf. [[Spectral sequence|Spectral sequence]]; for the particular case of group cohomology, where this gives a relation, sometimes called connection, between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130245.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130246.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023130/c023130247.png" />, cf. also [[#References|[a1]]], Chapt. 11, Par. 9.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K.S. Brown, "Cohomology of groups" , Springer (1982) {{MR|0672956}} {{ZBL|0584.20036}} </TD></TR></table>