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\(A_ \infty\) structures on some spectra related to Morava \(K\)-theories. (English) Zbl 0772.55003
The author studies the fine structure of several families of spectra constructed starting from Brown-Peterson homology at a fixed prime \(p\). He begins with the homology theories \(E(n)_ *(\cdot)\), which have coefficient rings \[ E(n)_ *=\mathbb{Z}_{(p)} [v_ 1,\dots,v_ n,v_ n^{-1}] \qquad | v_ i|=2(p^ i -1); \] let \(\ell_ n\) denote the ideal \((p,v_ 1,\dots,v_{n-1})\) in \(E(n)_ *\). The author then forms the \(\ell_ n\)-adic completion \(\widehat {E(n)}\) of \(E(n)\), and goes on to show that this spectrum admits a unique topological \(A_ \infty\) structure compatible with its canonical ring spectrum structure. He also studies the canonical morphism of ring spectra \(\widehat {E(n)}\to K(n)\) mapping to Morava \(K\)-theory, and makes the points that \(K(n)^*(\cdot)\) is best regarded as “\(\widehat{E(n)}\) modulo \(\ell_ n\)”, and that in many ways \(\widehat{E(n)}\) is more fundamental.
This study rests on recent work of Alan Robinson dealing with \(A_ \infty\) structures on ring spectra and their module spectra. For example, see his paper [Obstruction theory and the strict associativity of Morava \(K\)-theories, in “Advances in homotopy theory”, Proc. Conf. in Honor of I. M. James, Cortona/Italy 1988, Lond. Math. Soc. Lect. Note Ser. 139, 143-152 (1989; Zbl 0688.55008)].

MSC:
55N22 Bordism and cobordism theories and formal group laws in algebraic topology
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