Productive Elements in Group Cohomology
نویسنده
چکیده
LetG be a finite group and k be a field of characteristic p > 0. A cohomology class ζ ∈ H(G, k) is called productive if it annihilates Ext∗kG(Lζ , Lζ). We consider the chain complexP(ζ) of projective kG-modules which has the homology of an (n−1)-sphere and whose k-invariant is ζ under a certain polarization. We show that ζ is productive if and only if there is a chain map ∆ : P(ζ) → P(ζ) ⊗ P(ζ) such that (id ⊗ ǫ)∆ ≃ id and (ǫ ⊗ id)∆ ≃ id. Using the Postnikov decomposition of P(ζ) ⊗ P(ζ), we prove that there is a unique obstruction for constructing a chain map ∆ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson [6] and Langer [13] on productive elements.
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