 Abstract : Beilinson theorem describes the derived category of coherent sheaves on projective space P^n. In particular it gives crucial informations which are helpful to construct a resolution of a sheaf by knowing its cohomology modules. Many classical results can be proved in this way, like Castelnuovo-Mumford criterion, Horrocks splitting criterion and its generalization to sheaves, Hilbert-Burch resolution of arithmetically Cohen-Macaulay codimension two subvarieties, Hilbert syzygy Theorem. This approach is useful in the construction of syzygies of special varieties. In the case of the Veronese variety and other homogeneous varieties, this approach can be combined with representation theory and quiver techniques. If time allows, we will sketch about Ulrich sheaves and supernatural bundles, and their role in Betti tables. |
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