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Transactions of the American Mathematical Society

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On the structure of $P(n)_\ast P((n))$ for $p=2$


Author: Christian Nassau
Journal: Trans. Amer. Math. Soc. 354 (2002), 1749-1757
MSC (1991): Primary 55N22; Secondary 55P43
Published electronically: January 7, 2002
MathSciNet review: 1881014
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Abstract: We show that $P(n)_\ast(P(n))$ for $p=2$ with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation $\epsilon$ nor the coproduct $\Delta$are multiplicative. As a consequence the algebra structure of $P(n)_\ast(P(n))$ is slightly different from what was supposed to be the case. We give formulas for $\epsilon(xy)$ and $\Delta(xy)$ and show that the inversion of the formal group of $P(n)$is induced by an antimultiplicative involution $\Xi:P(n)\rightarrow P(n)$. Some consequences for multiplicative and antimultiplicative automorphisms of $K(n)$ for $p=2$ are also discussed.


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Additional Information

Christian Nassau
Affiliation: Johann Wolfgang Goethe-Universität Frankfurt, Fachbereich Mathematik, Robert Mayer Strasse 6-8, 60054 Frankfurt, Germany
Email: nassau@math.uni-frankfurt.de

DOI: http://dx.doi.org/10.1090/S0002-9947-02-02920-3
Keywords: Hopf algebroids, Morava $K$-theory, bordism theory, noncommutative ring spectra
Received by editor(s): July 3, 2000
Published electronically: January 7, 2002
Article copyright: © Copyright 2002 American Mathematical Society