Cohomology of Projective Space and Ext Groups and Sheaves


張茗遠 ( National Taiwan University )
2022-07-14  14:00 - 16:00
Room 201, Astronomy and Mathematics Building

With the help of Čech cohomology, we can compute the cohomology of sheaves more explicitly. One important application is describing the cohomology of the sheaves O(n) on projective spaces over noetherian ring explicitly.

Then, we will see the vanishing of cohomology of coherent sheaves after enough twisting, and give a criterion of ampleness by cohomology.

Next, we will introduce the fuctor Ext and its sheafified version, and show some basic properties using the trick of universal delta-functor. It turns out that we can calculate the sheafified Ext functor using a locally free resolution in the first variable. Together with the previous results, we will show that under certain assumptions Ext is actually the global section of sheafified Ext after enough twisting.