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Picard groups of higher real \(K\)-theory spectra at height \(p-1\). (English) Zbl 1374.14006
We consider the formal group law \(F_n(X, Y) \in \mathbb{F}_{p^n}[[X, Y]]\) with \(p\)-series \([p]_{F_n}(X)=X^{p^n}\) where we suppose that \(p\) is an odd prime and \(n=p-1\). Denote by \(\mathbb{S}_n\) the group of automorphisms of \(F_n\) given by power series \(f \in \mathbb{F}_{p^n}[[X]]\) such that \(f(0)=0\), \(f'(0)\neq 0\) and \(F_n(f(X), f(Y))=f(F(X, Y))\). Then the assignment \(f \mapsto f'(0)\) induces a homomorphism \(\mathbb{S}_n \to \mathbb{F}_{p^n}^\times\) which is surjective and whose kernel is a pro-\(p\)-group. This allows us to define a semidirect product of \(\mathbb{S}_n\) and \(\text{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\), which is written \(\mathbb{G}_n=\mathbb{S}_n \rtimes \text{Gal}(\mathbb{F}_{p^n}/\mathbb{F}_p)\). Let \(E_n\) be the \(n\)th Morava \(E\)-theory, i.e. an \({\mathbf E}_\infty\)-ring spectrum with \(\pi_*E_n=W(\mathbb{F}_{p^n})[[u_1, \ldots, u_{n-1}]][u^{\pm{1}}]\) where \(W(\mathbb{F}_{p^n})\) denotes the ring of Witt vectors over \(\mathbb{F}_{p^n}\). Then \(\mathbb{G}_n\) acts on \(E_n\) through \({\mathbf E}_\infty\)-ring maps. If we denote by \(E_n^{hG}\) the homotopy fixed points of \(E_n\) for a finite subgroup \(G \subset \mathbb{G}_n\), then the main theorem of this paper (Theorem 4.1) states that the Picard group \(\text{Pic}(E_n^{hG})\) is a cyclic group generated by the suspension \(\Sigma E_n^{hG}\). In order to prove this, the authors first check that the extension \(E_n^{hG} \to E_n\) is a faithful \(G\)-Galois extension of ring spectra, thereby obtaining the descent spectral sequence for Picard groups. The proof is done by analyzing this spectral sequence, using the technique developed in the second part of the paper. Incidentally, this paper consists of two parts I and II, both having three sections. We then refer to a remark on the example of the \(C_2\)-Galois extension \(KO \to KU\). It states that the theorem above is also true when \(p=2\), \(n=1\); as a result we have \(\text{Pic}(KO)=\mathbb{Z}/8\), generated by \(\Sigma KO\).

MSC:
14C22 Picard groups
55P43 Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.)
55Q51 \(v_n\)-periodicity
55S99 Operations and obstructions in algebraic topology
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