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In homological algebra, the Bockstein homomorphism, introduced by , is a connecting homomorphism associated with a short exact sequence :0 → ''P'' → ''Q'' → ''R'' → 0 of abelian groups, when they are introduced as coefficients into a chain complex ''C'', and which appears in the homology groups as a homomorphism reducing degree by one, :β: ''H''''i''(''C'', ''R'') → ''H''''i'' − 1(''C'', ''P''). To be more precise, ''C'' should be a complex of free, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product with ''C'' (some flat module condition should enter). The construction of β is by the usual argument (snake lemma). A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have :β: ''H''''i''(''C'', ''R'') → ''H''''i'' + 1(''C'', ''P''). The Bockstein homomorphism β of the coefficient sequence :0 → Z/''p''Z → Z/''p''2Z → Z/''p''Z → 0 is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the two properties :ββ = 0 if ''p''>2 :β(a∪b) = β(a)∪b + (-1)dim a a∪β(b) in other words it is a superderivation acting on the cohomology mod ''p'' of a space. ==References== * * * * . * 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「Bockstein homomorphism」の詳細全文を読む スポンサード リンク
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