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 Ψ M and a ∈ A, we have ρ(az) = a ⊗ z ∈ A ⊗AΨ (A Ψ M), hence im ρ = A ⊗AΨ (A Ψ M). So we have shown that M ∼ A ⊗AΨ (A Ψ M). 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.

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