L-COMPLETE HOPF ALGEBROIDS 5
so M is S-divisible.
We will find it useful to know about some basic functors on M and their derived
functors.
Let N be an L-complete R-module. As the functors N (−): M −→ M and
L0 : M −→ M are right exact, so is the endofunctor of M
M N⊗M = L0(N M).
Therefore we can use resolutions by projective objects (i.e., pro-free L-complete
modules) to form the left derived functors, which we will denote Tors
R
(N, −), where
Tor0
R
(N, M) = N⊗M.
If P is pro-free then by definition, Tors
R
(N, P ) = 0 for s 0. On the other hand,
Tors
R
(P, −) need not be the trivial functor (see Appendix B). This shows that
Tors
R
(−, −) is not a balanced bifunctor, i.e., in general
Tors
R
(N, M)

=
Tors
R
(M, N).
By Proposition 1.1, for a finitely generated R-module N0, L0N0 is a finitely
generated R-module which induces the left exact functor
M L0(N0 M)

=
N0 M.
For M M , we can choose a free resolution in M , say
F∗ −→ M 0.
Recalling that L0M

= M and LsM = 0 for s 0, the homology of L0F∗ is
H∗(L0F∗) = L0M

=
M,
hence we have a resolution of M by pro-free modules
L0F∗ −→ M 0.
Then
Tor∗
R
(L0N0,M) = H∗(L0(N0 F∗)).
But now we have
L0(N0 F∗)

= N0 L0F∗ = N0 (F∗)m.
When N is a finitely generated m-torsion module, we have L0N = N and
L0(N F∗)

= N F∗,
therefore
(1.2) Tor∗
R
(N, M) = Tor∗
R
(N, M).
Now take a free resolution
P∗ −→ N 0
with each Ps finitely generated. Then
Tor∗
R
(M, N) = H∗(L0(M P∗))

= H∗(M P∗) = Tor∗
R(M,
N),
hence
(1.3) Tor∗
R
(M, N)

=
Tor∗
R(M,
N).
5
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