14 ANDREW BAKER

Proof. The coaction on M can be viewed as a map ρ: M −→ A ⊗AΨ M. By

coassociativity,

(IdA ⊗ ρ)ρ = (ηL ⊗ IdM )ρ = (IdA ⊗ 1 ⊗ IdM )ρ,

hence

IdA ⊗ ρ − IdA ⊗ 1 ⊗ IdM : im ρ −→ A ⊗AΨ A ⊗AΨ M

must be trivial. By flatness of A,

0 → A ⊗AΨ ker(ρ − 1 ⊗ IdM ) A ⊗AΨ M

IdA⊗ρ

−IdA⊗1⊗IdM

A ⊗AΨ A ⊗AΨ M

is exact, so

im ρ ⊆ A ⊗AΨ ker(ρ − 1 ⊗ IdM ) = A ⊗AΨ (A

Ψ

M).

Since ρ: M −→ Ψ ⊗A M is split by the augmentation

ε ⊗ IdM : Ψ ⊗A M −→ A ⊗A M

∼

= M,

ρ is a monomorphism. For each coaction primitive z ∈ A ΨM and a ∈ A, we have

ρ(az) = a ⊗ z ∈ A ⊗AΨ (A

Ψ

M),

hence im ρ = A ⊗AΨ (A ΨM). So we have shown that

M

∼

= A ⊗AΨ (A ΨM).

Remark 4.2. The above algebra can be interpreted scheme-theoretically as

follows. Given a flat morphism of aﬃne schemes f : X −→ Y , X ×Y X becomes a

groupoid scheme with a unique morphism u → v whenever f(u) = f(v). Comodules

for the representing Hopf algebroid are equivalent to OX -modules with descent data,

and the category of such comodules is equivalent to that of OY -modules. See [15,

section 17.2] for an algebraic version of this when f is faithfully flat.

Example 4.3. Let R be a commutative ring and let G be a ﬁnite group which

acts faithfully on R by ring automorphisms so that

RG

−→ R is a G-Galois exten-

sion in the sense of [3]. Thus there is an isomorphism of rings

(4.1) R ⊗RG R

∼

=

R ⊗RG

RGG∗,

where the dual group ring is

RGG∗

= Map(G,

RG).

The left hand side is visi-

bly a Hopf algebroid and as an

RG-module,

R is ﬁnitely generated projective, so

Lemma 4.1 applies.

Following the outline in [1], we can identify

R ⊗RG

RGG∗

∼

=

RG∗

with the dual of the twisted group ring RG and thus it also carries a natural Hopf

algebroid structure. It is easy to verify that this structure agrees with that on

R ⊗RG R under (4.1).

Interpreting a R ⊗RG R-comodule M as equivalent to a RG-module, we can

use the Galois theoretic isomorphism RG

∼

=

EndRG R to show that there is an

isomorphism of RG-modules

M

∼

= R ⊗RG M

G,

and since

M

G

∼

=

R

R⊗RG R

M,

this is a module theoretic interpretation of the comodule result of Lemma 4.1.

14

Proof. The coaction on M can be viewed as a map ρ: M −→ A ⊗AΨ M. By

coassociativity,

(IdA ⊗ ρ)ρ = (ηL ⊗ IdM )ρ = (IdA ⊗ 1 ⊗ IdM )ρ,

hence

IdA ⊗ ρ − IdA ⊗ 1 ⊗ IdM : im ρ −→ A ⊗AΨ A ⊗AΨ M

must be trivial. By flatness of A,

0 → A ⊗AΨ ker(ρ − 1 ⊗ IdM ) A ⊗AΨ M

IdA⊗ρ

−IdA⊗1⊗IdM

A ⊗AΨ A ⊗AΨ M

is exact, so

im ρ ⊆ A ⊗AΨ ker(ρ − 1 ⊗ IdM ) = A ⊗AΨ (A

Ψ

M).

Since ρ: M −→ Ψ ⊗A M is split by the augmentation

ε ⊗ IdM : Ψ ⊗A M −→ A ⊗A M

∼

= M,

ρ is a monomorphism. For each coaction primitive z ∈ A ΨM and a ∈ A, we have

ρ(az) = a ⊗ z ∈ A ⊗AΨ (A

Ψ

M),

hence im ρ = A ⊗AΨ (A ΨM). So we have shown that

M

∼

= A ⊗AΨ (A ΨM).

Remark 4.2. The above algebra can be interpreted scheme-theoretically as

follows. Given a flat morphism of aﬃne schemes f : X −→ Y , X ×Y X becomes a

groupoid scheme with a unique morphism u → v whenever f(u) = f(v). Comodules

for the representing Hopf algebroid are equivalent to OX -modules with descent data,

and the category of such comodules is equivalent to that of OY -modules. See [15,

section 17.2] for an algebraic version of this when f is faithfully flat.

Example 4.3. Let R be a commutative ring and let G be a ﬁnite group which

acts faithfully on R by ring automorphisms so that

RG

−→ R is a G-Galois exten-

sion in the sense of [3]. Thus there is an isomorphism of rings

(4.1) R ⊗RG R

∼

=

R ⊗RG

RGG∗,

where the dual group ring is

RGG∗

= Map(G,

RG).

The left hand side is visi-

bly a Hopf algebroid and as an

RG-module,

R is ﬁnitely generated projective, so

Lemma 4.1 applies.

Following the outline in [1], we can identify

R ⊗RG

RGG∗

∼

=

RG∗

with the dual of the twisted group ring RG and thus it also carries a natural Hopf

algebroid structure. It is easy to verify that this structure agrees with that on

R ⊗RG R under (4.1).

Interpreting a R ⊗RG R-comodule M as equivalent to a RG-module, we can

use the Galois theoretic isomorphism RG

∼

=

EndRG R to show that there is an

isomorphism of RG-modules

M

∼

= R ⊗RG M

G,

and since

M

G

∼

=

R

R⊗RG R

M,

this is a module theoretic interpretation of the comodule result of Lemma 4.1.

14