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coherent cohomology

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Topos Theory

topos theory

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homological algebra

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Contents

Idea

Given a (Noetherian) ringed topos (𝒳,𝒪 X)(\mathcal{X}, \mathcal{O}_X), then a chain complex V V_\bullet of modules over the structure sheaf is said to have (quasi-)coherent cohomology if all its chain homology groups are (quasi-)coherent sheaves (coherent objects).

Properties

Over a (finite-dimensional) Noetherian scheme XX the derived category of quasi-coherent sheaves is canonically equivalent to that of sheaves with quasicoherent cohomology.

The coherent version of the statement is (SGA 6, Exp. II, Corollaire 2.2.2.1) while the quasi-coherent version is (SGA 6, Exp. II, Proposition 3.7, b)). A review appears also as (Orlov 03, prop. 1.3.2).

See also the discussion at triangulated categories of sheaves.

References

Last revised on May 30, 2014 at 09:30:18. See the history of this page for a list of all contributions to it.