2 ANDREW BAKER

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∗

∨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

m

(s 0). We recall that there are natural transformations

Id

η

−−→ L0 −→ (−)m −→ R/m ⊗R (−).

The two right hand natural transformations are epimorphic for each module, and

L0 is idempotent, i.e., L0

2

∼

= 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 →

lim1

k

Tors+1(R/mk,M) R

−→ LsM −→ lim

k

Tors

R(R/mk,M)

→ 0.

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.

2

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∗

∨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

m

(s 0). We recall that there are natural transformations

Id

η

−−→ L0 −→ (−)m −→ R/m ⊗R (−).

The two right hand natural transformations are epimorphic for each module, and

L0 is idempotent, i.e., L0

2

∼

= 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 →

lim1

k

Tors+1(R/mk,M) R

−→ LsM −→ lim

k

Tors

R(R/mk,M)

→ 0.

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.

2