On connective \(K\)-theory of elementary abelian 2-groups and local duality (Q740176)

The connective complex \(K\)-theory of elementary abelian 2-groups \(V\) was first computed by \textit{E. Ossa} [Lect. Notes Math. 1375, 269--275 (1989; Zbl 0683.55001)], who proved that it splits into copies of \(\mathbb{Z}/2\) and of the connective \(K\)-theory of \(B\mathbb{Z}/2\). In this paper the author attempts to study this in a more conceptual approach. More precisely, the author shows that \(ku^*(BV_+)\) and \(ku_*(BV_+)\) can be determined as functors of \(V\) to graded \(\mathbb{Z}[v]\)-modules where \(ku\) denotes the connective complex \(K\)-theory with coefficient ring \(ku^*=\mathbb{Z}[v]\), \(v\) being the Bott element of degree 2. If for example we consider the case of \(ku\)-cohomology, then this can be roughly described as follows: Let \(P_{\mathbb{Z}_2}\) denote the functor \(V \mapsto \mathbb{Z}_2[V]\) and \(\overline{P}_{\mathbb{Z}_2}\) the augmentation ideal where \(\mathbb{Z}_2\) denotes the 2-adic integers. Let \(Q_0\) and \(Q_1\) denote the derivations of degrees 1 and 3 on the symmetric algebra \(SV\) of \(V\) induced by the iterated Frobenius maps \(x \mapsto x^2\) and \(x \mapsto x^4\). We write \(L_n(V)\) for the image of \(Q_1 : \text{Ker} \, Q_0 \subset S^{n-3}V \to S^nV\) where \(S^iV \subset SV\) is the \(i\)th symmetric power of \(V\). Then if \(n=2d > 0\) there exists an exact sequence \[ 0 \to L_n(V^\sharp) \to ku^n(BV_+) \to \mathbb{Z}\oplus\overline{P}^d_{\mathbb{Z}_2}(V^\sharp) \to 0; \] otherwise we have an isomorphism \[ L_n(V^\sharp) \cong ku^n(BV_+) \] where \(V^\sharp\) is the dual of \(V\). The proof heavily depends on an analysis of the action of a morphism \(\mathfrak{Q} : H\mathbb{Z} \to \Sigma^3H\mathbb{Z}\) whose mod 2 reduction coincides with the Milnor derivation \(Q_1\) and which is given by the composite \(H\mathbb{Z} \to \Sigma^3ku \to \Sigma^3H\mathbb{Z}\) of morphisms induced from the cofibre sequence \(\Sigma^2ku \overset{v}{\to} {ku} H\mathbb{Z}\). The second part of this paper is devoted to looking at results of \textit{R. R. Bruner} and \textit{J. P. C. Greenless} [The connective \(K\)-theory of finite groups. Mem. Am. Math. Soc. 165, No. 785, viii+127 (2003; Zbl 1032.19003)] from the viewpoint outlined above.