6 ANDREW BAKER
Combining (1.2) and (1.3), we obtain the following restricted result on Tor∗
R
as
a balanced bi-functor. As far as we can determine, there is no general analogue of
this for arbitrary L-complete modules N which are finitely generated as R-modules.
Proposition 1.6. Let M, N be L-complete R-modules, where N is a finitely
generated m-torsion module. Then
Tor∗
R
(M, N)

= Tor∗
R
(M, N)

= Tor∗
R
(N, M).
When N is a finitely generated m-torsion module, we may also consider the
composite functor M −→ M for which
M L0(N M).
Since
L0(N M) = N L0M = N M,
this functor has for its left derived functors Tor∗
R(N,
−) and there is an associated
composite functor spectral sequence.
Proposition 1.7. Let N be a finitely generated m-torsion module. Then for
each R-module M, there is a natural first quadrant spectral sequence
Es,t
2
= Tors
R
(N, LtM) = Tors
R
(N, LtM) =⇒ Tors+t(N,
R
M).
Proof. Let P∗ −→ N 0 and Q∗ −→ M 0 be free resolutions. Since R is
Noetherian, we can assume that each Ps is finitely generated, so
L0(P∗ Q∗)

= P∗ L0Q∗.
Taking first homology, then second homology, and using (1.2) together with the
fact that each L0Qt is projective in M , we obtain
H∗
II
H∗
I
(P∗ L0Q∗) = H∗
II
Tor∗
R
(N, L0Q∗)
= H∗
II
Tor∗
R
(N, L0Q∗)
= H∗
II
Tor0
R
(N, L0Q∗)
= H∗
II
(N L0Q∗)
= H∗
II
(N Q∗)
= Tor∗
R
(N, M).
The resulting spectral sequence collapses at its
E2-term.
Taking second homology
then first homology we obtain
H∗
I
H∗
II
(P∗ L0Q∗) = H∗
I
(P∗ L∗M)
= Tor∗
R(N,
L∗M).
This is the E2-term of a spectral sequence converging to Tor∗
R(N,
M) as claimed.
Lemma 1.8. Let M be a flat R-module. Then
LsM =
Mm if s = 0,
0 otherwise,
and L0M is pro-free.
6
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