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In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod ''p'' cohomology. For a given prime number ''p'', the Steenrod algebra ''A''''p'' is the graded Hopf algebra over the field F''p'' of order ''p'', consisting of all stable cohomology operations for mod ''p'' cohomology. It is generated by the Steenrod squares introduced by for ''p''=2, and by the Steenrod reduced ''p''th powers introduced in and the Bockstein homomorphism for ''p''>2. The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory. ==Cohomology operations== A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring, the cup product squaring operation yields a family of cohomology operations: : : Cohomology operations need not be homomorphisms of graded rings, see the Cartan formula below. These operations do not commute with suspension, that is they are unstable. (This is because if ''Y'' is a suspension of a space ''X'', the cup product on the cohomology of ''Y'' is trivial.) Norman Steenrod constructed stable operations : : for all ''i'' greater than zero. The notation ''Sq'' and their name, the Steenrod squares, comes from the fact that ''Sq''''n'' restricted to classes of degree ''n'' is the cup square. There are analogous operations for odd primary coefficients, usually denoted ''P''''i'' and called the reduced ''p''-th power operations. The ''Sq''''i'' generate a connected graded algebra over ''Z/2'', where the multiplication is given by composition of operations. This is the mod 2 Steenrod algebra. In the case ''p'' > 2, the mod ''p'' Steenrod algebra is generated by the ''P''''i'' and the Bockstein operation β associated to the short exact sequence : In the case ''p''=2, the Bockstein element is ''Sq''1 and the reduced ''p''-th power ''P''''i'' is ''Sq''2''i''. 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「Steenrod algebra」の詳細全文を読む スポンサード リンク
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