By Noyes A.A., Beckman A.O.
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Extra info for A Periodic Table of the Structure of Atoms and Its Relation to Ion-Formation and Valence
Then we have gr N · π∗ L· with L· = OT ∗ X ⊗π −1 OX n 0 π −1 X → · · · →OT ∗ X ⊗π −1 OX π −1 X →i∗ OX , where OT ∗ X ⊗π −1 OX 0 π −1 X (= OT ∗ X ) → i∗ OX is given by ϕ → i∗ (ϕ ◦ i) and d : OT ∗ X ⊗π −1 OX k π −1 X → OT ∗ X ⊗π −1 OX k−1 π −1 X is given by d(ϕ ⊗ θ1 ∧ · · · ∧ θk ) = (−1)i+1 ϕσ1 (θi ) ⊗ θ1 ∧ · · · ∧ θi · · · ∧ θk . , [Matm, Theorem 43]). Since π is an affine morphism, π∗ L· is also acyclic. Let Y and Z be smooth algebraic varieties and set X = Y × Z. Let f : X → Y and g : X → Z be the projections.
By the relation [∂, y] = 1 we get yN j ⊂ N j +1 , ∂N j ⊂ N j −1 and ∼ θ induces an isomorphism j × : N j → N j for ∀ j = 0. Therefore, ∂y = θ + 1 : N j → N j is an isomorphism for ∀ j = −1. In particular, if j < −1, both morphisms y ∂ N j → N j +1 → N j are isomorphisms. Let us show that ∞ N −i . N= (1) i=1 Since N is a quasi-coherent OY -module supported in X, any s ∈ N is annihilated by y k for a sufficiently large k. Hence it suffices to prove the following assertion: k Ker(y k : N → N ) ⊂ N −j (k ≥ 1).
Sr contained in N −1 . Then i N = i −1 N −1 is generated as a DX -module by the sections s1 , . . , sr . The proof is complete. b,X b (D ) (resp. (DY ) (resp. Dcb,X (DY )) the subcategory of Dqc Denote by Dqc Y Dcb (DY )) consisting of complexes N˙ whose cohomology sheaves H ∗ (N˙) are supported by X. 2. For = qc or c the functor : D b (DX ) → D b,X (DY ) i gives an equivalence of triangulated categories. Its quasi-inverse is given by Ri = i † : D b,X (DY ) → D b (DX ). Proof. It is easily seen that i sends D b (DX ) to D b,X (DY ) and Ri sends D b,X (DY ) to D b (DX ).