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ON THE HOMOLOGY THEORY OF MODULES

Journal of the Faculty of Science. Section I 01/1954;
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    ABSTRACT: Let Γ be the bigraded algebra with Γ s,t =H t-s (B(ℤ/2) ×s ;ℤ/2) and γ k ∈Γ 1,k+1 the canonical generators. Let A be the usual mod 2 Steenrod algebra and ℋ the bigraded Steenrod algebra with generators {Sq i ∣i≥0} subject to the Adem relations (bigraded by degree and by length of monomials and with Sq 0 ≠1) as introduced by A. Liulevicius [“The factorization of cyclic reduced powers by secondary cohomology operations.” Mem. Am. Math. Soc. 42 (1962; Zbl 0131.38101)]. Γ s,* has an A-action dual to the A-action on H * (B(ℤ/2) ×s ;ℤ/2) and Γ A denotes the A annihilated elements with respect to this action, which form a subalgebra of Γ. ℤ/2⊗ Σ Γ and (ℤ/2⊗ Σ Γ) A are the algebra and subalgebra defined by (ℤ/2⊗ Σ Γ) s,* =ℤ/2⊗ Σ s Γ s,* and (ℤ/2⊗ Σ Γ) s,* A =((ℤ/2⊗ Σ Γ) s,* ) A respectively. The main results of this paper are that there is a left action of ℋ on ℤ/2⊗ Σ Γ making it an ℋ-algebra uniquely determined by the condition Sq 0 (γ k )=γ 2k+1 ∀k and that (ℤ/2⊗ Σ Γ) A is stable under this action and so is also an ℋ-algebra.
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