L-COMPLETE HOPF ALGEBROIDS 3
Proposition 1.1. Let M, N be R-modules with M finitely generated. Then
there is a natural isomorphism
M L0N −→ L0(M N).
In particular,
L0M

= Mm

= R M,
R/mk
L0N

=
R/mk
N =
N/mkN.
Hence, if N is a bounded m-torsion module then it is L-complete.
Proof. See [7, proposition A.4].
A module M is said to be L-complete if η : M −→ L0M is an isomorphism.
The subcategory of L-complete modules M M is a full subcategory and the
functor L0 : M −→ M is left adjoint to the inclusion M −→ M . The category M
has projectives, namely the pro-free modules which have the form
L0F = Fm
for some free R-module F . Thus M has enough projectives and we can do homo-
logical algebra to define derived functors of right exact functors.
By [7, theorem A.6(e)], the category M is abelian and has limits and colimits
which are obtained by passing to M , taking (co)limits there and applying L0. For
the latter there are non-trivial derived functors which by [6] satisfy
colims
M
= Ls colim,
M
so
colims
is trivial for s n. In fact, for a coproduct
α
with M , we
also have
Ln
α
= 0.
The category M has a symmetric monoidal structure coming from the tensor
product in M . For M, N M , let
M⊗N = L0(M N) M .
Note that we also have
M⊗N

=
L0(L0M L0N).
As in [7], we find that (M , ⊗) is a symmetric monoidal category.
For any R-module M, there are natural homomorphisms
R L0M −→ L0(R M) −→ L0M,
so we can view M as a subcategory of MR; since R is a flat R-algebra, for many
purposes it is better to think of M this way. For example, the functor L0 = L0
m
on
M can be expressed as
L0
mM

=
L0
m(R
M),
where L0 m is the derived functor on the category of R-modules M
R
associated to
completion with respect to the induced ideal m R. Finitely generated modules
over R (which always lie in M ) are completions of finitely generated R-modules.
There is an analogue of Nakayama’s Lemma provided by [7, theorem A.6(d)].
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