Hilbert's Theorem 90 and the Segal Conjecture for Elementary Abelian p-Groups
Copyright policy )Hilbert's Theorem 90 and the Segal Conjecture for Elementary Abelian p-Groups
Let V be an n-dimensional vector space over \({\mathbb{F}}_ p\) and set \(S=H^*(V; {\mathbb{F}}_ p)\). Part of the homological algebra that goes into the proof of the Segal conjecture is an isomorphism \[ Tor_*^{A_ p}({\mathbb{F}}_ p,S[L^{-1}])\cong \oplus^{N}_{1}Tor_*^{A_ p}(F_ p,\Sigma^{-n}{\mathbb{F}}_ p) \] where \(N=p^{\left( \begin{matrix} n\\ 2\end{matrix} \right)}\). Here L is the product of all two-dimensional classes. The authors give a proof of this isomorphism, using the Steinberg module of V, St(V). They prove that there exists an \(A_ p[GL(V)]\)-module map f: S[L\({}^{-1}]\to \Sigma^{-n}St(V)\) which induces an isomorphism on \(Tor^{A_ p}({\mathbb{F}}_ p,-)\).
abelian p-groups, stable homotopy, Segal conjecture, Adams spectral sequence, Stable homotopy groups, classifying space of a finite group, Steinberg module, elementary, Steenrod algebra, Adams spectral sequences
abelian p-groups, stable homotopy, Segal conjecture, Adams spectral sequence, Stable homotopy groups, classifying space of a finite group, Steinberg module, elementary, Steenrod algebra, Adams spectral sequences
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