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

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