Contemporary Mathematics L-complete Hopf algebroids and their comodules Andrew Baker Abstract. We investigate Hopf algebroids in the category of L-complete mod- ules over a commutative Noetherian regular complete local ring. The main examples of interest in algebraic topology are the Hopf algebroids associated to Lubin-Tate spectra in the K(n)-local stable homotopy category, and we show that these have Landweber filtrations for all finitely generated discrete modules. Along the way we investigate the canonical Hopf algebras associated to Hopf algebroids over fields and introduce a notion of unipotent Hopf algebroid generalising that for Hopf algebras. Introduction In this paper we describe some algebraic machinery that has been found useful when working with the K(n)-local homotopy category, and specifically the cooper- ation structure on covariant functors of the form E∗ (−), where E∨(X) = π∗(LK(n)(E X)) is the homotopy of the Bousfield localisation of E X with respect to Morava K- theory K(n). Our main focus is on algebra, but our principal examples originate in stable homotopy theory. In studying the K(n)-local homotopy category, topologists have found it helpful to use the notion of L-complete module introduced for other purposes by Greenlees and May in [5]. It is particularly fortunate that the Lubin-Tate spectrum En associated with a prime p and n 1 has for its homotopy ring π∗En = W Fpn[[u1,...,un−1]][u, u−1], 2000 Mathematics Subject Classification. Primary 55N22 Secondary 55T25, 55P60, 16W30, 13K05. Key words and phrases. L-complete module, Hopf algebroid, Hopf algebra, Lubin-Tate spec- trum, Morava K-theory. The author was partially funded by a YFF Norwegian Research Council grant whilst a visiting Professor at the University of Oslo, and by an EPSRC Research Grant EP/E023495/1. I would like to thank the Oslo topologists for their support and interest in the early stages of this work, Mark Hovey, Uli Kr¨ ahmer and Geoffrey Powell for helpful conversations, and finally the referee for perceptive comments and suggesting some significant improvements in our exposition. This paper is dedicated to the mountains of the Arolla valley. c 0000 (copyright holder) 1 Volume 504, 2009 c 2009 American Mathematical
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