Proposition 1.1. Let M, N be R-modules with M ﬁnitely generated. Then
there is a natural isomorphism
M ⊗ L0N −→ L0(M ⊗ N).
= R ⊗ M,
⊗ N =
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  satisfy
= Ls colim,
is trivial for s n. In fact, for a coproduct
Mα with Mα ∈ M , we
Mα = 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
L0(L0M ⊗ L0N).
As in , 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 can be expressed as
where L0 m is the derived functor on the category of R-modules M
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)].