Derived categories of coherent sheaves and integral functors

07/28/2021

We provide an introduction to the theory of derived categories and derived functors. To achieve this, we begin by studying the triangulated structure on the homotopy category of complexes over an abelian category $\ mathscr{A}$, and define its derived category $D(\mathscr{A})$ by formally inverting quasi-isomorphisms. In this way, the derived category, although not abelian, inherits a canonical structure of a triangulated category, and derived functors are defined as initial objects in the category of extensions that preserve the distinguished triangles. We apply these constructions to the abelian category $\mathrm{Coh}_X$ of coherent sheaves on a smooth projective variety $X$, with the help of tools such as spectral sequences and $\delta$-functors. Finally, we introduce integral functors. Given two such varieties $X$ and $Y$, these are geometrically motivated functors $D^b(\mathrm{Coh}_X)\to D^b(\mathrm{Coh}_Y)$ between the derived categories, which are extensively used in present-day Algebraic Geometry and Mathematical Physics.

Authors in the community:

• 

  J

  João Gabriel Santos Ruano

  ist425693

Supervisors of this institution:

• 

  E

  Emílio Franco Gómez

  ist428517

mathematics - Mathematics

por - Portuguese

Instituto Superior Técnico