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
• 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:
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.
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
ker θ M
Γ⊗ ker θ Γ⊗M
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.