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By N. Saavedra Rivano

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Since our sheaf DX is locally free over OX , it is quasicoherent over OX . We mainly deal with DX -modules which are quasi-coherent over OX . 1. For an algebraic variety X we denote the category of quasi-coherent OX -modules by Mod qc (OX ). For a smooth algebraic variety X we denote by Mod qc (DX ) the category of OX -quasi-coherent DX -modules. The category Mod qc (DX ) is an abelian category. It is well known that for affine algebraic varieties X, (a) the global section functor (X, •) : Mod qc (OX ) → Mod( (X, OX )) is exact, (b) if (X, M) = 0 for M ∈ Mod qc (OX ), then M = 0.

15. Assume that X is D-affine. Then for any M ∈ Mod qc (DX ) and i > 0 we have H i (X, M) = 0. Proof. 14 we can take a resolution 0 → M → I0 → I1 → · · · , where Ij are injective objects of Mod qc (DX ) which are flabby. Since Ij are flabby, H i (X, M) is the ith cohomology group of the complex (X, I · ). On the other hand since X is D-affine, the functor (X, •) is exact on Mod qc (DX ), and hence H i (X, M) = 0 for i > 0. , [Ha2, p. 126]). 16. (i) Let F be a quasi-coherent OX -module. For an open subset U ⊂ X consider a coherent OU -submodule G ⊂ F |U of the restriction F |U of F to U .

Namely, a candidate for the direct image Mod(DX ) → Mod(DY ) is obtained by the commutativity of Mod(DX ) −−−−→ Mod(DY ) ⏐ ⏐ ⏐ ⏐ ⊗ (•) X ⊗OX Y op OY (•) op Mod(DX ) −−−−→ Mod(DY ) where the lower horizontal arrow is given by f∗ ((•) ⊗DX DX→Y ). Thus, to a a left DX -module M we can associate a left DY -module ⊗−1 Y ⊗OY f∗ (( ⊗OX M) ⊗DX DX→Y ). 11 we have an isomorphism ( of right f −1 D X ⊗OX M) ⊗DX DX→Y Y -modules, where f −1 D ( Y ((ω ⊗ R) ⊗ s)P = (ω ⊗ RP ) ⊗ s X ⊗OX DX→Y ) ⊗DX M acts on ( (ω ∈ X X, R ⊗OX DX→Y ) ⊗DX M by ∈ DX→Y , s ∈ M, P ∈ DY ).

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Categories Tannakiennes by N. Saavedra Rivano


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