#### Volume 7, issue 2 (2007)

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Dieudonné modules and $p$–divisible groups associated with Morava $K$–theory of Eilenberg–Mac Lane spaces

### Victor Buchstaber and Andrey Lazarev

Algebraic & Geometric Topology 7 (2007) 529–564
 arXiv: math.AT/0507036
##### Abstract

We study the structure of the formal groups associated to the Morava $K$–theories of integral Eilenberg–Mac Lane spaces. The main result is that every formal group in the collection $\left\{K{\left(n\right)}^{\ast }K\left(ℤ,q\right),q=2,3,\dots \right\}$ for a fixed $n$ enters in it together with its Serre dual, an analogue of a principal polarization on an abelian variety. We also identify the isogeny class of each of these formal groups over an algebraically closed field. These results are obtained with the help of the Dieudonné correspondence between bicommutative Hopf algebras and Dieudonné modules. We extend P Goerss’ results on the bilinear products of such Hopf algebras and corresponding Dieudonné modules.

##### Keywords
Hopf ring, Dieudonné module, Morava $K$–theory, $p$–divisible group, Serre duality
Primary: 55N22
Secondary: 14L05
##### Publication
Received: 4 December 2006
Revised: 20 February 2007
Accepted: 8 March 2007
Published: 10 May 2007
##### Authors
 Victor Buchstaber Steklov Mathematical Institute Russian Academy of Sciences Gubkina 8 Moscow 119991 Russia http://www.mi.ras.ru/~buchstab/ Andrey Lazarev Mathematics Department University of Bristol Bristol BS8 1TW UK http://www.maths.bris.ac.uk/~maxal/