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Images directes en cohomologie cohérente. (Direct images in coherent cohomology). (French) Zbl 0699.14023
Let S be an affine scheme, f: \(X\to S\) a morphism with \({\mathcal F}^ a \)coherent \({\mathcal O}_ X\)-module, flat over S. Suppose that for \(p\geq 0\) the \({\mathcal O}_ S\)-modules \(R^ if_ X{\mathcal F}\) are coherent for \(i\leq p\). In this memoir the author shows that there is a perfect complex \({\mathcal L}\) (representable by a finite complex of locally free \({\mathcal O}_ S\)-modules) and a morphism \(\alpha: {\mathcal L}\to R^ if_ X{\mathcal F}\) such that \(H^ i(\alpha)\) is an isomorphism for \(i\leq p\) provided that X is a “large” open subset of a projective space and \({\mathcal F}\) enjoys certain properties on depth.
To prove this result the author proves many results related to the categories Ind-\({\mathcal A}\) or Pro-\({\mathcal A}\) where \({\mathcal A}\) might be abelian or a suitable category of modules or a derived category of one of these.
Reviewer: P.Cherenack
MSC:
14F20 Étale and other Grothendieck topologies and (co)homologies
18E30 Derived categories, triangulated categories (MSC2010)
14F25 Classical real and complex (co)homology in algebraic geometry
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