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<!-- {{about|homology and cohomology ''of'' a group|homology or cohomology groups of a space or other object|Homology (mathematics)}} -->
{{about|群のコホモロジー|位相空間などのホモロジー群・コホモロジー群|ホモロジー群}}
In [[mathematics]] (more specifically, in [[homological algebra]]), '''group cohomology''' is a set of mathematical tools used to study [[group (mathematics)|group]]s using [[cohomology theory]], a technique from [[algebraic topology]]. Analogous to [[group representation]]s, group cohomology looks at the [[group action]]s of a group ''G'' in an associated [[G-module|''G''-module]] ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of <math>G^n</math> representing ''n''-[[Simplex|simplices]], topological properties of the space may be computed, such as the set of cohomology groups <math>H^n(G,M)</math>. 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 cohomology#Group homology|''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 group|non-Abelian]] coefficients.
These algebraic ideas are closely related to topological ideas. The group cohomology of a discrete group ''G'' is the [[singular cohomology]] of a suitable space having ''G'' as its [[fundamental group]], namely the corresponding [[Eilenberg–MacLane space]]. Thus, the group cohomology of '''Z''' can be thought of as the singular cohomology of the circle '''S'''<sup>1</sup>, and similarly for '''Z'''/2'''Z''' and '''P'''<sup>∞</sup>('''R''').
A great deal is known about the cohomology of groups, including interpretations of low-dimensional cohomology, functoriality, and how to change groups. The subject of group cohomology began in the 1920s, matured in the late 1940s, and continues as an area of active research today.
[[数学]]、とくに[[ホモロジー代数学]]において、'''群のコホモロジー'''（{{lang-en-short|group cohomology}}）とは[[代数的トポロジー]]に由来する技法である[[コホモロジー論]]を使って[[群 (数学)|群]]を研究するために使われる数学的な道具立てである。[[群の表現]]のように、群のコホモロジーは群 {{mvar|G}} の [[群上の加群|{{mvar|G}} 加群]]への[[群作用|作用]]をみることで、その群の性質を明らかにする。{{mvar|G}} 加群を {{mvar|G<sup>n</sup>}} の元が [[単体 (数学)|{{mvar|n}} 単体]]を表す[[位相空間]]のように扱うことで、コホモロジー群 {{math|''H''<sup>''n''</sup>(''G'', ''M'')}} などの位相的な性質が計算できる。コホモロジー群は群 {{mvar|G}} や {{mvar|G}} 加群 {{mvar|M}} の構造に関する洞察を与える。群のコホモロジーは加群や空間への群作用の固定点や群作用に関する商加群や商空間を研究において一定の役割を果たす。群のコホモロジーは純[[群論]]への応用はもちろん、[[抽象代数]]・[[ホモロジー代数]]・[[代数的トポロジー]]・[[代数的整数論]]などの分野でも用いられている。代数的トポロジーには、群のホモロジーと呼ばれる双対理論がある。<!-- nonabelian G-group??
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 group|non-Abelian]] coefficients.
群のコホモロジーについては非常に多くのこと——低次元コホモロジーの解釈・関手性・群の変更——が知られている。群のコホモロジーに関する主題は1920年代に始まり、1940年代後半に発達し、現在でも活発に研究が続いている。
== Motivation ==
A general paradigm in [[group theory]] is that a [[group (mathematics)|group]] ''G'' should be studied via its [[group representation]]s. A slight generalization of those representations are the [[G-module|''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 [[automorphism]] of ''M''. We will write ''G'' multiplicatively and ''M'' additively.
Given such a ''G''-module ''M'', it is natural to consider the submodule of [[G-invariant|''G''-invariant]] elements:
:<math> M^{G} = \lbrace x \in M \ | \ \forall g \in G : \ gx=x \rbrace. </math>
Now, if ''N'' is a ''G''-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 those in ''N'': being invariant 'modulo ''N'' ' is broader. The purpose of the first group cohomology ''H''<sup>1</sup>(''G'',''N'') is to precisely measure this difference.
The group cohomology functors ''H*'' in general measure the extent to which taking invariants doesn't respect [[exact sequence]]s. This is expressed by a [[long exact sequence]].
== 動機 ==
[[群 (数学)|群]] {{mvar|G}} はその[[群の表現|表現]]を通じて研究されるべきであるという[[群論]]における一般的なパラダイムがある。このような表現をわずかに一般化したものに [[群上の加群|{{mvar|G}} 加群]]がある：{{mvar|G}} 加群とは群 {{mvar|G}} の各元が[[自己同型]]として[[群作用|作用]]する[[アーベル群]] {{mvar|M}} である。われわれは {{mvar|G}} は乗法的に、 {{mvar|M}} は加法的に書くことにする。
一般に群のコホモロジー関手 {{math|''H''<sup>∗</sup>}} は不変な元をとる関手がどれほど[[完全列|完全]]でないかを測っている。これは[[長完全列]]によって表される。
== Definitions ==
The collection of all ''G''-modules is a [[category theory|category]] (the morphisms are group homomorphisms ''f'' with the property ''f''(''gx'') = ''g''(''f''(''x'')) for all ''g'' in ''G'' and ''x'' in ''M'').
Sending each module ''M'' to the group of invariants ''M<sup>G</sup>'' yields a [[functor]] from the category of ''G''-modules to the category '''Ab''' of abelian groups. This functor is [[left exact functor|left exact]] but not necessarily right exact. We may therefore form its right [[derived functor]]s.<ref>This uses that the category of ''G''-modules has enough [[injective object|injectives]], since it is isomorphic to the category of all [[module (mathematics)|modules]] over the [[group ring]] '''Z'''[''G''].</ref>
Their values are abelian groups and they are denoted by ''H<sup>n</sup>''(''G'', ''M''), "the ''n''-th cohomology group of ''G'' with coefficients in ''M''". ''H''<sup>0</sup>(''G'', ''M'') is identified with ''M<sup>G</sup>''.
== 定義 ==
すべての {{mvar|G}} 加群からなるクラスは[[圏 (数学)|圏]]である。（その射は群準同型 {{math|''f'' : ''M'' → ''N''}} であって、すべての {{math|''g'' ∈ ''G''}} と {{math|''x'' ∈ ''M''}} に対して {{math|''f''(''gx'') {{=}} ''g''(''f''(''x''))}} を満たすものである。）各 {{mvar|G}} 加群 {{mvar|M}} に {{mvar|M<sup>G</sup>}} を対応させることで {{mvar|G}} 加群の圏から[[アーベル群の圏]] {{math|'''Ab'''}} への[[関手]]が得られる。この関手は[[左完全関手|左完全]]であるが右完全とは限らない。したがって右[[導来関手]]をとることができる<ref>これは {{mvar|G}} 加群の圏が[[群環]] {{math|'''Z'''''G''}} 上の加群圏と同値なので[[十分多くの入射対象]]をもつことを使っている。</ref>。その値は[[アーベル群]]であり、{{math|''H''<sup>''n''</sup>(''G'', ''M'')}} と表され、'''{{mvar|M}} に係数をもつ群の {{mvar|n}} 次コホモロジー群'''と呼ばれる。
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===Cochain complexes===
The definition using derived functors is conceptually very clear, but for concrete applications, the following computations, which some authors also use as a definition, are often helpful.<ref>Page 62 of [[#Reference-Mil2008|Milne 2008]] or section VII.3 of [[#Reference-Se1979|Serre 1979]]</ref> For ''n'' ≥ 0, let ''C<sup>n</sup>''(''G'', ''M'') be the group of all [[function (mathematics)|function]]s from ''G<sup>n</sup>'' to ''M''. This is an abelian group; its elements are called the (inhomogeneous) ''n''-cochains. The coboundary homomorphisms
:<math>\begin{cases}
d^{n+1} : C^{n} (G,M) \to C^{n+1}(G,M) \\
\left(d^{n+1}\varphi\right) (g_1, \ldots, g_{n+1}) = g_1\varphi(g_2, \ldots, g_{n+1}) + \sum_{i=1}^n (-1)^{i} \varphi \left (g_1,\ldots, g_{i-1},g_i g_{i+1}, g_{i+2}, \ldots, g_{n+1} \right ) + (-1)^{n+1} \varphi(g_1,\ldots, g_n)
\end{cases}</math>
One may check that <math>d^{n+1} \circ d^n = 0,</math> so this defines a [[cochain complex]] whose cohomology can be computed. It can be shown that the above-mentioned definition of group cohomology in terms of derived functors is isomorphic to the cohomology of this complex
:<math>H^n(G,M) = Z^n(G,M)/B^n(G,M).</math>
Here the groups of ''n''-cocycles, and ''n''-coboundaries, respectively, are defined as
:<math>Z^n(G,M) = \ker(d^{n+1}) </math>
:<math>B^n(G,M) = \begin{cases} 0 & n = 0 \\ \operatorname{im}(d^{n}) & n \geqslant 1 \end{cases}</math>
=== 双対鎖複体 ===
[[導来関手]]を使った定義は概念的には極めて明快であるが、実際に利用するには一部の著者が定義としている、次の計算法が役に立つことが多い<ref>{{harvtxt|Milne|20082007|p=62}} あるいは {{harvtxt|Serre|1979|loc=Section VII.3}} 参照。</ref>。{{math|''n'' ≥ 0}} に対して {{math|''C''<sup>''n''</sup>(''G'', ''M'')}} を {{mvar|G}} から {{mvar|M}} への関数全体からなる群とする。これは[[アーベル群]]であり、その元を（非斉次）{{mvar|n}} 次の双対鎖という。双対境界作用素を
:<math> d^{n+1} \colon C^n(G, M) \to C^{n+1}(G, M),\ \varphi \mapsto d^{n+1}\varphi; </math>
:<math> d^{n+1}\varphi(g_1, \dotsc, g_{n+1}) = \sum_{i = 1}^{n+1} \varphi(g_1, \dotsc, \widehat{g_i}, \dotsc, g_{n+1}) </math>
</math>
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===The functors Ext<sup>''n''</sup> and formal definition of group cohomology===
Interpreting ''G''-modules as modules over the [[group ring]] <math>\Z[G],</math> one can note that
:<math>H^{0}(G,M) = M^G = \operatorname{Hom}_{\Z [G]}(\Z ,M),</math>
i.e., the subgroup of ''G''-invariant elements in ''M'' is identified with the group of homomorphisms from <math>\Z</math>, which is treated as the trivial ''G''-module (every element of ''G'' acts as the identity) to ''M''.
Therefore, as [[Ext functor]]s are the derived functors of [[Hom functor|Hom]], there is a natural isomorphism
:<math>H^{n}(G,M) = \operatorname{Ext}^{n}_{\Z [G]}(\Z ,M).</math>
These Ext groups can also be computed via a projective resolution of <math>\Z</math>, the advantage being that such a resolution only depends on ''G'' and not on ''M''.
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=== 関手 {{math|Ext<sup>''n''</sup>}} と群のコホモロジーの形式的な定義 ===
{{mvar|G}} 加群を[[群環]] {{math|'''Z'''[''G'']}} 上の加群とみると
</math>
となる。
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==Group homology ==
Group cohomology theory also has a direct application in condensed matter physics. Just like group theory being the mathematical foundation of [[spontaneous symmetry breaking]] phases, group cohomology theory is the mathematical foundation of a class of quantum states of matter—short-range entangled states with symmetry. Short-range entangled states with symmetry are also known as [[symmetry protected topological order|symmetry protected topological states]].
== 注釈 ==
{{reflist}}
==Notes 参考文献 ==
<references/>
==References==
*{{Citation |last1= Adem |first1= Alejandro|author1-link= Alejandro Adem | first2=R. James | last2=Milgram |title= Cohomology of Finite Groups |publisher= [[Springer-Verlag]] |year= 2004 |isbn= 3-540-20283-8 |series=Grundlehren der Mathematischen Wissenschaften |mr=2035696 |volume=309 | zbl=1061.20044 | edition=2nd }}
* {{Citation | first=Kenneth S. | last1=Brown | title=Cohomology of Groups | authorlink=Kenneth Brown (mathematician) | publisher=[[Springer Verlag]] | year=1972 | isbn=0-387-90688-6 | series=[[Graduate Texts in Mathematics]] | mr=0672956 | volume=87 }}
* {{Citation | url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002053403 | first=Heinz | last=Hopf|title=Fundamentalgruppe und zweite Bettische Gruppe| journal=Comment. Math. Helv.|volume=14 | issue=1|year=1942|pages=257–309 | mr=6510 | doi=10.1007/BF02565622 | zbl=0027.09503 | jfm=68.0503.01 }}
*{{Citation |last1= Knudson |first1= Kevin P.|title= Homology of Linear Groups |publisher= [[Birkhäuser Verlag]] |year= 2001|series=Progress in Mathematics|volume=193|zbl=0997.20045}}
* Chapter II of {{Citation | ref=Reference-Mil2008 | last1=Milne | first1=James | year=2007 | title=Class Field Theory | date=5/2/2008 | volume=v4.00 | url=http://www.jmilne.org/math| accessdate=8/9/2008}}
* {{Citation | last1=Rotman | first1=Joseph | year=1995 | title=An Introduction to the Theory of Groups | edition=4th | series=[[Graduate Texts in Mathematics]] | volume=148 | publisher=[[Springer-Verlag]] | isbn=978-0-387-94285-8 | mr = 1307623 }}
* Chapter VII of {{Citation | ref=Reference-Se1979 | last1=Serre | first1=Jean-Pierre | author1-link = Jean-Pierre Serre | title=Local fields | url={{google books|3LAJCAAAQBAJ|plainurl=yes}} | publisher=Springer-Verlag | location=Berlin, New York | series=[[Graduate Texts in Mathematics]] | isbn=978-0-387-90424-5 | zbl=0423.12016 | mr=554237 | year=1979 | volume=67 }}
* {{Citation | last1=Serre | first1=Jean-Pierre | title=Cohomologie galoisienne | edition=Fifth | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-58002-7 | mr = 1324577 | year=1994 | volume=5}}
* {{Citation | last1=Shatz | first1=Stephen S. | title=Profinite groups, arithmetic, and geometry | publisher=[[Princeton University Press]] | location=Princeton, NJ | isbn=978-0-691-08017-8 | mr = 0347778 | year=1972}}
* Chapter 6 of {{Weibel IHA}}
* {{Citation | last1=Weibel | first1=Charles A. | contribution=History of homological algebra | title=History of Topology |publisher=[[Cambridge University Press]] | isbn=0-444-82375-1 | mr = 1721123 | year=1999 | pages=797–836}}
[[Category:Algebraic number theory]]
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