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| In [[abstract algebra]], '''cohomological dimension''' is an invariant of a [[group (mathematics)|group]] which measures the homological complexity of its representations. It has important applications in [[geometric group theory]], [[topology]], and [[algebraic number theory]].
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| == Cohomological dimension of a group ==
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| As most (co)homological invariants, the cohomological dimension involves a choice of a "ring of coefficients" ''R'', with a prominent special case given by ''R'' = '''Z''', the ring of [[integers]]. Let ''G'' be a [[discrete group]], ''R'' a non-zero [[ring (mathematics)|ring]] with a unit, and ''RG'' the [[group ring]]. The group ''G'' has '''cohomological dimension less than or equal to ''n''''', denoted cd<sub>''R''</sub>(''G'') ≤ ''n'', if the trivial ''RG''-module ''R'' has a [[projective resolution]] of length ''n'', i.e. there are [[projective module|projective]] ''RG''-modules ''P''<sub>0</sub>, …, ''P''<sub>''n''</sub> and ''RG''-module homomorphisms ''d''<sub>''k''</sub>: ''P''<sub>''k''</sub><math>\to</math>''P''<sub>''k'' − 1</sub> (''k'' = 1, …, ''n'') and ''d''<sub>0</sub>: ''P''<sub>0</sub><math>\to</math>''R'', such that the image of ''d''<sub>''k''</sub> coincides with the kernel of ''d''<sub>''k'' − 1</sub> for ''k'' = 1, …, ''n'' and the kernel of ''d''<sub>''n''</sub> is trivial.
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| Equivalently, the cohomological dimension is less than or equal to ''n'' if for an arbitrary ''RG''-module ''M'', the [[group cohomology|cohomology]] of ''G'' with coeffients in ''M'' vanishes in degrees ''k'' > ''n'', that is, ''H''<sup>''k''</sup>(''G'',''M'') = 0 whenever ''k'' > ''n''. The ''p''-cohomological dimension for prime ''p'' is similarly defined in terms of the ''p''-torsion groups ''H''<sup>''k''</sup>(''G'',''M''){''p''}.<ref name=GS136>Gille & Szamuely (2006) p.136</ref>
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| The smallest ''n'' such that the cohomological dimension of ''G'' is less than or equal to ''n'' is the '''cohomological dimension''' of ''G'' (with coefficients ''R''), which is denoted ''n'' = cd<sub>''R''</sub>(''G'').
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| A free resolution of '''Z''' can be obtained from a [[free action]] of the group ''G'' on a [[Contractible space|contractible topological space]] ''X''. In particular, if ''X'' is a contractible [[CW complex]] of dimension ''n'' with a free action of a discrete group ''G'' that permutes the cells, then cd<sub>'''Z'''</sub>(''G'') ≤ ''n''.
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| == Examples ==
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| In the first group of examples, let the ring ''R'' of coefficients be '''Z'''.
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| * A [[free group]] has cohomological dimension one. As shown by [[John Stallings]] (for finitely generated group) and Richard Swan (in full generality), this property characterizes free groups.
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| * The [[fundamental group]] of a [[Compact space|compact]], [[Connected space|connected]], [[Orientability|orientable]] [[Riemann surface]] other than the [[sphere]] has cohomological dimension two.
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| * More generally, the fundamental group of a compact, connected, orientable [[Aspherical space|aspherical]] [[manifold]] of [[dimension]] ''n'' has cohomological dimension ''n''. In particular, the fundamental group of a closed orientable hyperbolic ''n''-manifold has cohomological dimension ''n''.
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| * Nontrivial [[finite group]]s have infinite cohomological dimension over '''Z'''. More generally, the same is true for groups with nontrivial [[torsion (algebra)|torsion]].
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| Now let us consider the case of a general ring ''R''.
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| * A group ''G'' has cohomological dimension 0 if and only if its group ring ''RG'' is [[Semisimple algebra|semisimple]]. Thus a finite group has cohomological dimension 0 if and only if its order (or, equivalently, the orders of its elements) is invertible in ''R''.
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| * Generalizing the Stallings–Swan theorem for ''R'' = '''Z''', Dunwoody proved that a group has cohomological dimension at most one over an arbitrary ring ''R'' if and only if it is the fundamental group of a connected [[graph of groups|graph of finite groups]] whose orders are invertible in ''R''.
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| ==Cohomological dimension of a field==
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| The ''p''-cohomological dimension of a field ''K'' is the ''p''-cohomological dimension of the [[Galois group]] of a [[separable closure]] of ''K''.<ref name=Sha94>Shatz (1972) p.94</ref> The cohomological dimension of ''K'' is the supremum of the ''p''-cohomological dimension over all primes ''p''.<ref name=GS138>Gille & Szamuely (2006) p.138</ref>
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| ==Examples==
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| * Every field of non-zero [[characteristic of a field|characteristic]] has cohomological dimension at most 1.<ref name=GS139>Gille & Szamuely (2006) p.139</ref>
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| * Every finite field has absolute Galois group isomorphic to <math>\mathbf{\hat Z}</math> and so has cohomological dimension 1.<ref name=GS140>Gille & Szamuely (2006) p.140</ref>
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| * The field of formal Laurent series ''k''((''t'')) over an [[algebraically closed field]] ''k'' of non-zero characteristic also has absolute Galois group isomorphic to <math>\mathbf{\hat Z}</math> and so cohomological dimension 1.<ref name=GS140/>
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| == See also ==
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| * [[Eilenberg−Ganea conjecture]]
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| * [[Group cohomology]]
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| * [[Global dimension]]
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| == References ==
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| {{reflist}}
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| * {{cite book | first=Kenneth S. | last=Brown |authorlink=Kenneth Brown (mathematician)| title=Cohomology of groups | edition=Corrected reprint of the 1982 original | series=[[Graduate Texts in Mathematics]] | volume=87 | publisher=[[Springer Science+Business Media|Springer-Verlag]] | location=New York | year=1994 | mr=1324339 | isbn=0-387-90688-6 | zbl=0584.20036 }}
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| * {{ cite book | first=Warren | last=Dicks | doi=10.1007/BFb0088140 | title=Groups, Trees, and Projective Modules | series=Lecture Notes in Mathematics | volume=790 | publisher=[[Springer-Verlag]] | location=Berlin | year=1980 | mr=0584790 | isbn=3-540-09974-3 | zbl=0427.20016 }}
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| * {{ cite book | first=Jerzy | last=Dydak | chapter=Cohomological dimension theory | title=Handbook of geometric topology | pages=423–470 | publisher=[[Elsevier|North-Holland]] | location=Amsterdam | year=2002 | editor1-last=Daverman | editor1-first=R. J. | isbn=0-444-82432-4 | mr=1886675 | zbl=0992.55001 }}
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| * {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
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| * {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Galois cohomology | publisher=[[Springer-Verlag]] | year=1997| isbn=3-540-61990-9 | zbl=0902.12004 }}
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| * {{cite book | last=Shatz | first=Stephen S. | title=Profinite groups, arithmetic, and geometry | series=Annals of Mathematics Studies | volume=67 | location=Princeton, NJ | publisher=[[Princeton University Press]] | year=1972 | isbn=0-691-08017-8 | zbl=0236.12002 | mr=0347778 }}
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| * {{ cite journal | first=John R. | last=Stallings | authorlink=John R. Stallings | title=On torsion-free groups with infinitely many ends | journal=[[Annals of Mathematics]] (2) |volume=88 | year=1968 | pages=312–334 | mr=0228573 | zbl=0238.20036 | issn=0003-486X }}
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| * {{cite journal | last=Swan | first=Richard G. | authorlink=Richard G. Swan | title=Groups of cohomological dimension one | journal=[[Journal of Algebra]] | volume=12 | year=1969 | pages=585–610 | mr=0240177 | zbl=0188.07001| issn=0021-8693 }}
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| [[Category:Group theory]]
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| [[Category:Homological algebra]]
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