L-COMPLETE HOPF ALGEBROIDS 7

Proof. For each s 0, the exact sequence of (1.1) and the flatness of M yield

LsM =

Mm if s = 0,

0 otherwise.

The spectral sequence of Proposition (1.7) with N = R/m degenerates so that for

each s 0 we obtain

Tors

R

(R/m,L0M) = Tors

R

(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 =

α

R, and the last observation

amounts to the vanishing of the higher derived functors of the coproduct functor

Mfg −→ M ; M → L0

α

Mα

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∗

R(L0F,

M).

So for s 0, Tors

R(L0F,

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

we have

im f + mP = P

and so

m(P/ im f) = P/ im f,

hence im f = P by Nakayama’s Lemma.

7

Proof. For each s 0, the exact sequence of (1.1) and the flatness of M yield

LsM =

Mm if s = 0,

0 otherwise.

The spectral sequence of Proposition (1.7) with N = R/m degenerates so that for

each s 0 we obtain

Tors

R

(R/m,L0M) = Tors

R

(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 =

α

R, and the last observation

amounts to the vanishing of the higher derived functors of the coproduct functor

Mfg −→ M ; M → L0

α

Mα

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∗

R(L0F,

M).

So for s 0, Tors

R(L0F,

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

we have

im f + mP = P

and so

m(P/ im f) = P/ im f,

hence im f = P by Nakayama’s Lemma.

7