Proof. For each s 0, the exact sequence of (1.1) and the flatness of M yield
Mm if s = 0,
The spectral sequence of Proposition (1.7) with N = R/m degenerates so that for
each s 0 we obtain
(R/m,L0M) = Tors
(R/m,M) = 0,
hence L0M is pro-free by [7, theorem A.9(3)].
If M is a ﬁnitely generated R-module then it has bounded m-torsion, hence
by [5, theorem 1.9], L0M = Mm and LsM = 0 for s 0. More generally, if F is a
free module, then F ⊗ M has bounded m-torsion, so
Ls(F ⊗ M) =
Fm ⊗ M if s = 0,
0 if s = 0.
If we choose a basis for F , we can write F =
R, and the last observation
amounts to the vanishing of the higher derived functors of the coproduct functor
Mfg −→ M ; M → L0
deﬁned on the subcategory of completions of ﬁnitely generated modules (which
is the same as the subcategory of ﬁnitely generated R -modules). Now for any
resolution P∗ −→ M → 0 of a ﬁnitely generated module M by ﬁnitely generated
projectives, we have
L0(F ⊗ P∗)
= L0F ⊗ P∗.
The left hand side has as its homology the above derived functors, so
H∗L0(F ⊗ P∗) = L0(F ⊗ M) = L0F ⊗ M,
while the right hand side has homology
H∗(L0F ⊗ P∗) = Tor∗
So for s 0, Tors
M) = 0.
For P ∈ M , the functor on M given by M → P ⊗M is right exact. We say
that P is L-flat if the functor P ⊗(−) is exact on M . However, the L-flat modules
are easily identiﬁed, at least when n = dim R = 1, because of the following result.
Proposition 1.9. Let P ∈ M be L-flat. Then P is pro-free.
Proof. The proof is similar to that of Corollary 1.4 and is based on a standard
argument for ﬁnitely presented flat modules over a local ring. Choose a free R-
module F for which F/mF
= P/mP . If f : F −→ P is a homomorphism covering
this isomorphism, there is an extension to a homomorphism f : Fm −→ P . Then
im f + mP = P
m(P/ im f) = P/ im f,
hence im f = P by Nakayama’s Lemma.