L-complete Hopf algebroids and their comodules
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
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.
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∗
= 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 . 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,
2000 Mathematics Subject Classiﬁcation. Primary 55N22; Secondary 55T25, 55P60, 16W30,
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)
Volume 504, 2009
c 2009 American Mathematical Society