nLab coherent cohomology

Idea

Given a (Noetherian) ringed topos $(\mathcal{X}, \mathcal{O}_X)$, then a chain complex $V_\bullet$ 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 $X$ 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

• Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys, 58 (2003), 3, 89-172 (English transl. pdf, Russian orig. pdf)