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Example text

P I ) form a vector space basis for A. Proof. It is a consequence of the Adem relations that the operations Sq I (resp. the operations P I ), I admissible, span the graded vector space A. In fact, let I = (i1 , . . , in ) (resp. I = ( 0 , i1 , 1 , . . , in , n )) be an admissible sequence. Its moment is defined to be i1 + 2i2 + · · · + nin (resp. i1 + 1 + 2(i2 + 2 ) + · · · ). If I is not admissible there exists h, 1 ≤ h ≤ n − 1 such that ih < 2ih+1 . Using the Adem relations one gets: [ih /2] Sq I = t Sq I Sq ih +ih+1 −t Sq t Sq I , 0 where t ∈ F2 , 0 ≤ t ≤ [ih /2], I = (i1 , .

In these formulas Sq 0 (resp. P 0 ) for p = 2 (resp. p > 2) is understood to be the unit. The mod p cohomology H ∗ (X; Z/p) of a space X will be denoted by H ∗ X ˜ ∗ X. 1 Theorem (Steenrod, Adem). For any space X, H ∗ X is in a natural way a graded A-module. Classically, β (Sq 1 if p = 2) acts as the Bockstein homomorphism associated to the sequence 0 → Z/p → Z/p2 → Z/p → 0. E. Steenrod constructed the operations Sq i and the operation P i , and J. Adem showed that the Adem relations above act trivially on the mod p cohomology of any space.

Power algebras can be used to define the action of the Steenrod algebra on H ∗ (X). In this section we describe precisely the algebraic connection between power algebras and unstable algebras over the Steenrod algebra. This clarifies the role of the power maps γ. Again let F = Z/p be the field of p elements where p ≥ 2 is a prime. If V is a F-vector space with basis x1 , . . 1) V = Fx1 ⊕ · · · ⊕ Fxn Let Vec be the category of F-vector spaces and F-linear maps. The zero-vector space is denoted by V = 0.

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Aus dem Nachlass von R. Brauer by Jørn Børling Olsson


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