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 = (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
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