By Lindstrum A.
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We see also from (1) that 50 1 Preliminary Notions N = C[∂] ⊗C N −1 , From this we easily obtain Mod qc (DX ) i N = i −1 N −1 . Mod X qc (DY ). 0 It remains to show that i sends Mod c (DX ) to Mod X c (DY ) and that i sends X Mod c (DY ) to Mod c (DX ). We may shrink Y if necessary. If M ∈ Mod qc (DX ) is 0 finitely generated over DX , then i M = C[∂] ⊗C i∗ M is clearly finitely generated over DY . Assume that N ∈ Mod X qc (DY ) is finitely generated over DY . By N = C[∂] ⊗C N −1 the DY -module N is generated by finitely many sections s1 , .
Assume that A = DX (U ) for some affine open subset U of X or A = DX,x for some x ∈ X. (i) A is a left (and right) noetherian ring. (ii) The left and right global dimensions of A are smaller than or equal to 2 dim X. 7. 11). We recall the notion of coherent sheaves. 8. Let R be a sheaf of rings on a topological space X. (i) An R-module M is called coherent if M is locally finitely generated and if for any open subset U of X any locally finitely generated submodule of M|U is locally finitely presented.
7. The natural functors b D b (Mod qc (DX )) −→ Dqc (DX ), D b (Mod c (DX )) −→ Dcb (DX ) give equivalences of categories. Inverse images Let f : X → Y be a morphism of smooth algebraic varieties. We can define a functor Lf ∗ : D b (DY ) → D b (DX ) by using a flat resolution of M ·. 5 Inverse images and direct images II 33 b (D ) to D b (D ). 8. Lf ∗ sends Dqc Y X qc Proof. 9 below. For an algebraic variety Z and = +, −, b, ∅ we denote by Dqc (OZ ) (resp. Dc (OZ )) the full subcategory of D (OZ ) consisting of F · ∈ D (OZ ) such that H i (F · ) ∈ Mod qc (OZ ) (resp.