A 3-Fold Vector Product in R ^8 by Zvengrowski P.

By Zvengrowski P.

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Locally free OY -modules of finite ranks). Hence the assertion is obvious. 10. Note that Lf ∗ DY = DX→Y ⊗L f −1 DY = DX→Y . 2). We see from this that the functor Lf ∗ for f : X → Y does not necessarily send Dcb (DY ) to Dcb (DX ). We call Lf ∗ the inverse image functor on derived categories of D-modules. We will also use the shifted inverse image functor f † = Lf ∗ [dim X − dim Y ] : D b (DY ) → D b (DX ). defined by f † M · = Lf ∗ M · [dim X − dim Y ]. The shifted one will be more practical in considering the Riemann–Hilbert correspondence.

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 ).

Considering (X, π ∗ M) as a C× -module, we get its weight space decomposition (X, π ∗ M) = γ (M)(l) , l∈Z where z ∈ C× acts on γ (M)(l) by zl . In particular, (X, M) = γ (M)(0) . Now let us consider the Euler vector field θ = ni=0 xi ∂i (here {xi } is a linear coordinate system of V and ∂i = ∂/∂xi ) on V . If we define the action of θ on π ∗ M by θ ⊗ Id, we have γ (M)(l) = {u ∈ (X, π ∗ M) | θu = lu }. Moreover, we can easily check xi (γ (M)(l) ) ⊂ γ (M)(l+1) , ∂i (γ (M)(l) ) ⊂ γ (M)(l−1) . Set Z = {0} × Y ⊂ V × Y , and let j : X → V × Y and k : Z → V × Y be the embeddings.

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