The Hopf ring of Morava K-theory. (English) Zbl 0564.55004
Let \({}_*=\{\underline{K(n)}_ i\}\) be the \(\Omega\)- spectrum representing the n-th Morava K-theory for a prime p. Let E be a multiplicative BP module spectrum. The author shows that \(E_*\underline{K(n)}_*\) is a Hopf ring, i.e., each \(E_*\underline{K(n)}_ i\) is a (bi)commutative Hopf algebra with conjugation and product. Using the techniques in [D. C. Ravenel and the author, J. Pure Appl. Algebra 9, 241-280 (1977; Zbl 0373.57020); Am. J. Math. 102, 691-748 (1980; Zbl 0466.55007) and the author, Reg. Conf. Ser. Math. 48 (1982; Zbl 0518.55001)], and the bar spectral sequence with Hopf rings from [R. W. Thomason and the author, Q. J. Math., Oxf. II. Ser. 31, 507-511 (1980; Zbl 0449.55018)], he computes \(H_*\underline{K(n)}_*\). Then he shows that the Atiyah-Hirzebruch spectral sequence collapses, solves all extension problems, and obtains \(E_*\underline{K(n)}_*\).
Reviewer: T.Kobayashi
MSC:
| 55N15 | Topological \(K\)-theory |
| 55S25 | \(K\)-theory operations and generalized cohomology operations in algebraic topology |
| 55T25 | Generalized cohomology and spectral sequences in algebraic topology |
| 55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |
| 55P42 | Stable homotopy theory, spectra |
Keywords:
\(\Omega \) -spectrum; n-th Morava K-theory for a prime p; BP module spectrum; Hopf ring; Hopf algebra; bar spectral sequence; Atiyah- Hirzebruch spectral sequenceReferences:
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