Proof. The coaction on M can be viewed as a map ρ: M −→ A ⊗AΨ M. By
(IdA ⊗ ρ)ρ = (ηL ⊗ IdM )ρ = (IdA ⊗ 1 ⊗ IdM )ρ,
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
A ⊗AΨ A ⊗AΨ M
is exact, so
im ρ ⊆ A ⊗AΨ ker(ρ − 1 ⊗ IdM ) = A ⊗AΨ (A
Since ρ: M −→ Ψ ⊗A M is split by the augmentation
ε ⊗ IdM : Ψ ⊗A M −→ A ⊗A M
ρ is a monomorphism. For each coaction primitive z ∈ A ΨM and a ∈ A, we have
ρ(az) = a ⊗ z ∈ A ⊗AΨ (A
hence im ρ = A ⊗AΨ (A ΨM). So we have shown that
= 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
−→ R is a G-Galois exten-
sion in the sense of . Thus there is an isomorphism of rings
(4.1) R ⊗RG R
where the dual group ring is
The left hand side is visi-
bly a Hopf algebroid and as an
R is ﬁnitely generated projective, so
Lemma 4.1 applies.
Following the outline in , we can identify
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
= R ⊗RG M
this is a module theoretic interpretation of the comodule result of Lemma 4.1.