L-COMPLETE HOPF ALGEBROIDS 9

Definition 2.3. Suppose that Γ is an L-complete commutative R-biunital ring

object with left and right units ηL,ηR : R −→ Γ, and has the following additional

structure:

• a counit: a k0-algebra homomorphism ε: Γ −→ R;

• a coproduct: a k0-algebra homomorphism ψ : Γ −→ Γ⊗Γ = ΓR⊗RΓ;

• an antipode: a k0-algebra homomorphism χ: Γ −→ Γ.

Then Γ is an L-complete Hopf algebroid if

• with this structure, Γ becomes a cogroupoid object,

• if Γ is pro-free as a left (or equivalently as a right) R-module,

• the ideal m R is invariant, i.e., mΓ = Γm.

We often denote such a pair by (R, Γ) when the structure maps are clear.

The cogroupoid condition is essentially the same as that spelt out in [13, deﬁni-

tion A1.1.1] but interpreted in the context of L-complete bimodules. In particular

we have a relationship between the two notions of L-completeness for Γ since the

antipode χ induces an isomorphism of R-modules χ:

R

Γ

∼

=

ΓR. The pro-freeness

condition is a disguised version of flatness required to do homological algebra.

Definition 2.4. Let (R, Γ) be an L-complete Hopf algebroid and let M ∈ M .

Then an R-module homomorphism ρ: M −→ Γ⊗M makes M into a left (R, Γ)-

comodule or Γ-comodule if the following diagrams commute.

M

ρ

ρ

Γ⊗M

ψ⊗id

Γ⊗M

ρ⊗id

Γ⊗Γ⊗M

M

ρ

∼

=

Γ⊗M

ε⊗id

R⊗M

There is a similar deﬁnition of a right Γ-comodule.

Let (R, Γ) be an L-complete Hopf Algebroid. Then given a morphism of Γ-

comodules θ : M −→ N, there is a commutative diagram of solid arrows

0

ker θ M

θ

ψ

N

ψ

Γ⊗ ker θ Γ⊗M

id⊗θ

Γ⊗N

but if id ⊗ θ is not a monomorphism then the dotted arrow may not exist or be

unique. If Γ⊗(−) always preserved exactness then this would not present a problem,

but this is not so easily ensured in great generality.

If Γ is pro-free, then as already noted, Γ⊗(−) is exact on the categories Mbt

and Mfg, so in each of these contexts the above diagram always has a completion

by a unique dotted arrow. Therefore the categories of Γ-comodules in Mbt and

Mfg are abelian since they have kernels and all the other axioms are satisﬁed.

Example 2.5. Let (R, Γ) be a flat Hopf algebroid over the commutative Noe-

therian regular local ring R, and assume that mΓ = Γm. By Lemma 1.8,

L0(RΓ) = Γm = L0(ΓR),

where Γm denotes the completion with respect to the ideal mΓ which equals Γm.

9

Definition 2.3. Suppose that Γ is an L-complete commutative R-biunital ring

object with left and right units ηL,ηR : R −→ Γ, and has the following additional

structure:

• a counit: a k0-algebra homomorphism ε: Γ −→ R;

• a coproduct: a k0-algebra homomorphism ψ : Γ −→ Γ⊗Γ = ΓR⊗RΓ;

• an antipode: a k0-algebra homomorphism χ: Γ −→ Γ.

Then Γ is an L-complete Hopf algebroid if

• with this structure, Γ becomes a cogroupoid object,

• if Γ is pro-free as a left (or equivalently as a right) R-module,

• the ideal m R is invariant, i.e., mΓ = Γm.

We often denote such a pair by (R, Γ) when the structure maps are clear.

The cogroupoid condition is essentially the same as that spelt out in [13, deﬁni-

tion A1.1.1] but interpreted in the context of L-complete bimodules. In particular

we have a relationship between the two notions of L-completeness for Γ since the

antipode χ induces an isomorphism of R-modules χ:

R

Γ

∼

=

ΓR. The pro-freeness

condition is a disguised version of flatness required to do homological algebra.

Definition 2.4. Let (R, Γ) be an L-complete Hopf algebroid and let M ∈ M .

Then an R-module homomorphism ρ: M −→ Γ⊗M makes M into a left (R, Γ)-

comodule or Γ-comodule if the following diagrams commute.

M

ρ

ρ

Γ⊗M

ψ⊗id

Γ⊗M

ρ⊗id

Γ⊗Γ⊗M

M

ρ

∼

=

Γ⊗M

ε⊗id

R⊗M

There is a similar deﬁnition of a right Γ-comodule.

Let (R, Γ) be an L-complete Hopf Algebroid. Then given a morphism of Γ-

comodules θ : M −→ N, there is a commutative diagram of solid arrows

0

ker θ M

θ

ψ

N

ψ

Γ⊗ ker θ Γ⊗M

id⊗θ

Γ⊗N

but if id ⊗ θ is not a monomorphism then the dotted arrow may not exist or be

unique. If Γ⊗(−) always preserved exactness then this would not present a problem,

but this is not so easily ensured in great generality.

If Γ is pro-free, then as already noted, Γ⊗(−) is exact on the categories Mbt

and Mfg, so in each of these contexts the above diagram always has a completion

by a unique dotted arrow. Therefore the categories of Γ-comodules in Mbt and

Mfg are abelian since they have kernels and all the other axioms are satisﬁed.

Example 2.5. Let (R, Γ) be a flat Hopf algebroid over the commutative Noe-

therian regular local ring R, and assume that mΓ = Γm. By Lemma 1.8,

L0(RΓ) = Γm = L0(ΓR),

where Γm denotes the completion with respect to the ideal mΓ which equals Γm.

9