Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old homological algebra (of derived functors between abelian categories) was established. This new homological algebra, of derived categories and derive
d functors between them, provides a significantly richer and more flexible machinery than the old homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the old homological algebra. This paper is an edited version of the notes for a two-lecture minicourse given at MSRI in January 2013. Sections 1-5 are about the general theory of derived categories, and the material is taken from my manuscript A Course on Derived Categories (available online). Sections 6-9 are on more specialized topics, leaning towards noncommutative algebraic geometry.
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-li
We introduce the notion of central extension of gerbes on a topological space. We then show that there are obstruction classes to lifting objects and isomorphisms in a central extension. We also discuss pronilpotent gerbes. These results are used in
a subsequent paper to study twisted deformation quantization on algebraic varieties.
