Some properties of LubinTate cohomology for classifying spaces of finite groups
Abstract
We consider brave new cochain extensions $F(BG_+,R)\to F(EG_+,R)$, where $R$ is either a LubinTate spectrum $E_n$ or the related 2periodic Morava Ktheory $K_n$, and $G$ is a finite group. When $R$ is an EilenbergMac Lane spectrum, in some good cases such an extension is a $G$Galois extension in the sense of John Rognes, but not always faithful. We prove that for $E_n$ and $K_n$ these extensions are always faithful in the $K_n$ local category. However, for a cyclic $p$group $C_{p^r}$, the cochain extension $F({BC_{p^r}}_+,E_n) \to F({EC_{p^r}}_+,E_n)$ is not a Galois extensions because it ramifies. As a consequence, it follows that the $E_n$theory EilenbergMoore spectral sequence for $G$ and $BG$ does not always converge to its expected target.
 Publication:

arXiv eprints
 Pub Date:
 May 2010
 arXiv:
 arXiv:1005.1662
 Bibcode:
 2010arXiv1005.1662B
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Rings and Algebras;
 55P43 (Primary);
 13B05 (Secondary);
 55N22;
 55P60
 EPrint:
 Minor changes, section on Frobenius algebra structure removed. Final version: to appear in Central European Journal of Mathematics under title `Galois theory and LubinTate cochains on classifying spaces'