where all generators are in degree 0 except u which has degree 2. Thus (apart from
the odd-even grading) the coefficient ring for the covariant functor (En)∗ ∨(−) is a
commutative Noetherian regular complete local ring of dimension n and the theory
of L-complete modules works well. For details of these applications see [7], also [6]
and [2, section 7].
In this paper we consider analogues of Hopf algebroids in the category of L-
complete modules over a commutative Noetherian regular complete local ring, and
relate this to our earlier work of [1]. Since the latter appeared there has been
a considerable amount of work by Hovey and Strickland [8, 9] on localisations of
categories of comodules over MU∗MU and BP∗BP , but that seems to be unrelated
to the present theory. One of our main motivations was to try to understand the
precise sense in which the L-complete theory differs from these other theories and
we intend to return to this in future work.
We introduce a notion of unipotent Hopf algebroid over a field and then consider
the relationship between modules over a Hopf algebroid (k, Γ) and over its asso-
ciated Hopf algebra (k, Γ ) and unicursal Hopf algebroid. We show that if (k, Γ )
is unipotent then so is (k, Γ). As a consequence a large class of Hopf algebroids
over local rings have composition series for finitely generated comodules which are
discrete in the sense that they are annihilated by some power of the maximal ideal.
We end by discussing the important case E∗
for a Lubin-Tate spectrum E.
In particular we verify that finitely generated comodules over this L-complete Hopf
algebroid have Landweber filtrations.
For completeness, in two appendices we continue the discussion of the connec-
tions with twisted group rings begun in [1], and expand on a result of [6] on the
non-exactness of coproducts of L-complete modules.
1. L-complete modules
Let (R, m) be a commutative Noetherian regular local ring, and let n = dim R.
We denote the category of (left) R-modules by M = MR. Undecorated tensor
products will be taken over R, i.e., = ⊗R. We will often write R for the m-adic
completion Rm, and m for mm.
The m-adic completion functor
M Mm
on M is neither left nor right exact. Following [5], we consider its left derived
functors Ls = Ls
(s 0). We recall that there are natural transformations
−−→ L0 −→ (−)m −→ R/m ⊗R (−).
The two right hand natural transformations are epimorphic for each module, and
L0 is idempotent, i.e., L0

= L0. It is also true that Ls is trivial for s n. For
computing the derived functor for an R-module M and s 0 there is a natural
exact sequence of [5, proposition 1.1]:
(1.1) 0
Tors+1(R/mk,M) R
−→ LsM −→ lim
It is an important fact that tensoring with finitely generated modules interacts
well with the functor L0. A module is said to have bounded m-torsion module if it
is annihilated by some power of m.
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