Algebraic geometry II - schemes and sheaves
MATH-510
Lecture 25 (11.12.24) summary
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1) Acyclic resolution computes derived functors.
2) Injective sheaves are flasque, flasque sheaves are acyclic with respect to the global section functor.
3) Consequences: Sheaf cohomology computed in the category of abelian sheaves and in the category of modules over the structure sheaf are the same.
4) On a noetherian affine scheme, the quasi-coherent sheaf given by an injective module is flasque.
5) On a noetherian affine scheme, quasi-coherent sheaves have no higher cohomologies.
6) Converse of 5, characterization of noetherian affine schemes as those where quasi-coherent sheaves have no higher cohomologies. No proof was given.
2) Injective sheaves are flasque, flasque sheaves are acyclic with respect to the global section functor.
3) Consequences: Sheaf cohomology computed in the category of abelian sheaves and in the category of modules over the structure sheaf are the same.
4) On a noetherian affine scheme, the quasi-coherent sheaf given by an injective module is flasque.
5) On a noetherian affine scheme, quasi-coherent sheaves have no higher cohomologies.
6) Converse of 5, characterization of noetherian affine schemes as those where quasi-coherent sheaves have no higher cohomologies. No proof was given.