L-COMPLETE HOPF ALGEBROIDS 3

Proposition 1.1. Let M, N be R-modules with M ﬁnitely generated. Then

there is a natural isomorphism

M ⊗ L0N −→ L0(M ⊗ N).

In particular,

L0M

∼

= Mm

∼

= R ⊗ M,

R/mk

⊗ L0N

∼

=

R/mk

⊗ N =

N/mkN.

Hence, if N is a bounded m-torsion module then it is L-complete.

Proof. See [7, proposition A.4].

A module M is said to be L-complete if η : M −→ L0M is an isomorphism.

The subcategory of L-complete modules M ⊆ M is a full subcategory and the

functor L0 : M −→ M is left adjoint to the inclusion M −→ M . The category M

has projectives, namely the pro-free modules which have the form

L0F = Fm

for some free R-module F . Thus M has enough projectives and we can do homo-

logical algebra to deﬁne derived functors of right exact functors.

By [7, theorem A.6(e)], the category M is abelian and has limits and colimits

which are obtained by passing to M , taking (co)limits there and applying L0. For

the latter there are non-trivial derived functors which by [6] satisfy

colims

M

= Ls colim,

M

so

colims

is trivial for s n. In fact, for a coproduct

α

Mα with Mα ∈ M , we

also have

Ln

α

Mα = 0.

The category M has a symmetric monoidal structure coming from the tensor

product in M . For M, N ∈ M , let

M⊗N = L0(M ⊗ N) ∈ M .

Note that we also have

M⊗N

∼

=

L0(L0M ⊗ L0N).

As in [7], we ﬁnd that (M , ⊗) is a symmetric monoidal category.

For any R-module M, there are natural homomorphisms

R ⊗ L0M −→ L0(R ⊗ M) −→ L0M,

so we can view M as a subcategory of MR; since R is a flat R-algebra, for many

purposes it is better to think of M this way. For example, the functor L0 = L0

m

on

M can be expressed as

L0

mM

∼

=

L0

m(R

⊗ M),

where L0 m is the derived functor on the category of R-modules M

R

associated to

completion with respect to the induced ideal m R. Finitely generated modules

over R (which always lie in M ) are completions of ﬁnitely generated R-modules.

There is an analogue of Nakayama’s Lemma provided by [7, theorem A.6(d)].

3

Proposition 1.1. Let M, N be R-modules with M ﬁnitely generated. Then

there is a natural isomorphism

M ⊗ L0N −→ L0(M ⊗ N).

In particular,

L0M

∼

= Mm

∼

= R ⊗ M,

R/mk

⊗ L0N

∼

=

R/mk

⊗ N =

N/mkN.

Hence, if N is a bounded m-torsion module then it is L-complete.

Proof. See [7, proposition A.4].

A module M is said to be L-complete if η : M −→ L0M is an isomorphism.

The subcategory of L-complete modules M ⊆ M is a full subcategory and the

functor L0 : M −→ M is left adjoint to the inclusion M −→ M . The category M

has projectives, namely the pro-free modules which have the form

L0F = Fm

for some free R-module F . Thus M has enough projectives and we can do homo-

logical algebra to deﬁne derived functors of right exact functors.

By [7, theorem A.6(e)], the category M is abelian and has limits and colimits

which are obtained by passing to M , taking (co)limits there and applying L0. For

the latter there are non-trivial derived functors which by [6] satisfy

colims

M

= Ls colim,

M

so

colims

is trivial for s n. In fact, for a coproduct

α

Mα with Mα ∈ M , we

also have

Ln

α

Mα = 0.

The category M has a symmetric monoidal structure coming from the tensor

product in M . For M, N ∈ M , let

M⊗N = L0(M ⊗ N) ∈ M .

Note that we also have

M⊗N

∼

=

L0(L0M ⊗ L0N).

As in [7], we ﬁnd that (M , ⊗) is a symmetric monoidal category.

For any R-module M, there are natural homomorphisms

R ⊗ L0M −→ L0(R ⊗ M) −→ L0M,

so we can view M as a subcategory of MR; since R is a flat R-algebra, for many

purposes it is better to think of M this way. For example, the functor L0 = L0

m

on

M can be expressed as

L0

mM

∼

=

L0

m(R

⊗ M),

where L0 m is the derived functor on the category of R-modules M

R

associated to

completion with respect to the induced ideal m R. Finitely generated modules

over R (which always lie in M ) are completions of ﬁnitely generated R-modules.

There is an analogue of Nakayama’s Lemma provided by [7, theorem A.6(d)].

3