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

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