Students' Algebraic Geometry Seminar : Exceptional Objects and Semi Orthogonal Decomposition

Now that we have seen the basics of derived and triangulated categories in the previous lectures of the seminar, we shall move ahead with using these to understand the geometry of our schemes in context. One of the most important tools for this is what is called an exceptional object. Exceptional objects and semiorthogonal decomposition help us find smaller pieces in our triangulated categories which act as building blocks. They can also detect various properties of a scheme such as whether it is Fano or what its genus is if it is a curve etc. 
Time permitting I will show the explicit semi orthogonal decomposition of the bounded derived category of projective space.