8 ANDREW BAKER

Let K = ker f. Tensoring the exact sequence

0 → K −→ Fm −→ P → 0

with R/m, by flatness of P we obtain the exact sequence

0 → K/mK −→ F/mF

∼

=

−→ P/mP → 0

so K/mK = 0. Hence K = 0 by Nakayama’s Lemma.

In general, tensoring with a pro-free module need not be left exact on M as is

shown by an example in Appendix B. In particular, when n 1, inﬁnitely generated

pro-free modules may not always be flat. Instead we can restrict attention to L-

flatness on restricted classes of L-complete modules. We say that P is weakly L-flat

if the functor

Mbt −→ M ; M → P ⊗M

is exact on the subcategory Mbt of bounded m-torsion modules. Then if Q is a flat

module, L0Q is weakly L-flat since for any N ∈ Mbt,

L0Q⊗N

∼

=

L0(Q ⊗ N)

∼

=

Q⊗N.

2. L-complete Hopf algebroids

To ease notation, from now on we assume that (R, m) is a commutative Noe-

therian regular local ring which is m-adically complete, i.e., R = R = Rm. We

assume that R is an algebra over some chosen local subring (k0, m0) so that the

inclusion map is local, i.e., m0 = k0 ∩ m. We write k = R/m for the residue ﬁeld.

Let Γ ∈ Mk0 . We need to assume extra structure on Γ to deﬁne the notion of

an L-complete Hopf algebroid. Unfortunately this is quite complicated to describe.

A (non-unital) ring object A ∈ Mk0 is equipped with a product morphism

ϕ: A ⊗k0 A −→ A which is associative, i.e., the following diagram commutes.

A ⊗k0 A ⊗k0 A

id⊗ϕ

id⊗ϕ

A ⊗k0 A

ϕ

A ⊗k0 A

ϕ

A

It is commutative if

A ⊗k0 A

switch

ϕ

A ⊗k0 A

ϕ

A

also commutes. An R-unit for ϕ is a k0-algebra homomorphism η : R −→ A.

Definition 2.1. A ring object is R-biunital if it has two units ηL,ηR : R −→ A

which extend to give a morphism ηL ⊗ ηR : R ⊗k0 R −→ A.

To distinguish between the two R-module structures on A, we will write RA and

AR. When discussing AR we will emphasise the use of the right module structure

whenever it occurs. In particular, from now on tensor products over R are to be

interpreted as bimodule tensor products R⊗R, even though we often write ⊗.

Definition 2.2. An R-biunital ring object A is L-complete if A is L-complete

as both a left and a right R-module.

8

Let K = ker f. Tensoring the exact sequence

0 → K −→ Fm −→ P → 0

with R/m, by flatness of P we obtain the exact sequence

0 → K/mK −→ F/mF

∼

=

−→ P/mP → 0

so K/mK = 0. Hence K = 0 by Nakayama’s Lemma.

In general, tensoring with a pro-free module need not be left exact on M as is

shown by an example in Appendix B. In particular, when n 1, inﬁnitely generated

pro-free modules may not always be flat. Instead we can restrict attention to L-

flatness on restricted classes of L-complete modules. We say that P is weakly L-flat

if the functor

Mbt −→ M ; M → P ⊗M

is exact on the subcategory Mbt of bounded m-torsion modules. Then if Q is a flat

module, L0Q is weakly L-flat since for any N ∈ Mbt,

L0Q⊗N

∼

=

L0(Q ⊗ N)

∼

=

Q⊗N.

2. L-complete Hopf algebroids

To ease notation, from now on we assume that (R, m) is a commutative Noe-

therian regular local ring which is m-adically complete, i.e., R = R = Rm. We

assume that R is an algebra over some chosen local subring (k0, m0) so that the

inclusion map is local, i.e., m0 = k0 ∩ m. We write k = R/m for the residue ﬁeld.

Let Γ ∈ Mk0 . We need to assume extra structure on Γ to deﬁne the notion of

an L-complete Hopf algebroid. Unfortunately this is quite complicated to describe.

A (non-unital) ring object A ∈ Mk0 is equipped with a product morphism

ϕ: A ⊗k0 A −→ A which is associative, i.e., the following diagram commutes.

A ⊗k0 A ⊗k0 A

id⊗ϕ

id⊗ϕ

A ⊗k0 A

ϕ

A ⊗k0 A

ϕ

A

It is commutative if

A ⊗k0 A

switch

ϕ

A ⊗k0 A

ϕ

A

also commutes. An R-unit for ϕ is a k0-algebra homomorphism η : R −→ A.

Definition 2.1. A ring object is R-biunital if it has two units ηL,ηR : R −→ A

which extend to give a morphism ηL ⊗ ηR : R ⊗k0 R −→ A.

To distinguish between the two R-module structures on A, we will write RA and

AR. When discussing AR we will emphasise the use of the right module structure

whenever it occurs. In particular, from now on tensor products over R are to be

interpreted as bimodule tensor products R⊗R, even though we often write ⊗.

Definition 2.2. An R-biunital ring object A is L-complete if A is L-complete

as both a left and a right R-module.

8