Cohomology of a Noetherian Affine Scheme and Cech Cohomology


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

We will prove the vanishing of i-th cohomology for all qusi-coherent sheaves on a noetherian affine scheme, using that the sheaf associated to an injective A-module is flasque. Then we will give a characterization on affineness: given a noetherian scheme X, the affineness of X is equivalent to the vanishing for all i-th coholomology of quasi-coherent sheaves and all i>0, and to the vanishing of the first cohomology of coherent sheaves of ideals.

We will define Cech cohomology and its sheafifed version, and investigate its relation with the usual cohomology: If X is a Noetherian separated scheme, the sheaf is quasi-coherent, and the covering is open affine, then Cech cohomology with respect to a sheaf and an open cover coincide with the cohomology for a sheaf.