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In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. == Cohomological dimension of a group == 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 with a unit, and ''RG'' the group ring. The group ''G'' has cohomological dimension less than or equal to ''n'', denoted cd''R''(''G'') ≤ ''n'', if the trivial ''RG''-module ''R'' has a projective resolution of length ''n'', i.e. there are projective ''RG''-modules ''P''0, …, ''P''''n'' and ''RG''-module homomorphisms ''d''''k'': ''P''''k''''P''''k'' − 1 (''k'' = 1, …, ''n'') and ''d''0: ''P''0''R'', such that the image of ''d''''k'' coincides with the kernel of ''d''''k'' − 1 for ''k'' = 1, …, ''n'' and the kernel of ''d''''n'' is trivial. Equivalently, the cohomological dimension is less than or equal to ''n'' if for an arbitrary ''RG''-module ''M'', the cohomology of ''G'' with coeffients in ''M'' vanishes in degrees ''k'' > ''n'', that is, ''H''''k''(''G'',''M'') = 0 whenever ''k'' > ''n''. The ''p''-cohomological dimension for prime ''p'' is similarly defined in terms of the ''p''-torsion groups ''H''''k''(''G'',''M'').〔Gille & Szamuely (2006) p.136〕 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''R''(''G''). A free resolution of Z can be obtained from a free action of the group ''G'' on a 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 cdZ(''G'') ≤ ''n''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cohomological dimension」の詳細全文を読む スポンサード リンク
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