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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