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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


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 {K(n)K(,q),q = 2,3,} 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.

Hopf ring, Dieudonné module, Morava $K$–theory, $p$–divisible group, Serre duality
Mathematical Subject Classification 2000
Primary: 55N22
Secondary: 14L05
Received: 4 December 2006
Revised: 20 February 2007
Accepted: 8 March 2007
Published: 10 May 2007
Victor Buchstaber
Steklov Mathematical Institute
Russian Academy of Sciences
Gubkina 8 Moscow 119991
Andrey Lazarev
Mathematics Department
University of Bristol Bristol BS8 1TW