Sheaf cohomology consists of the right derived functors of global sections. The zeroth group is , and higher groups measure obstructions to gluing local sections.
A sheaf is flasque when every restriction map is surjective. Flasque sheaves are acyclic for global sections, so for .
For an open cover , the Čech cochain group iswith the alternating sum of restrictions as differential. Its cohomology is the Čech cohomology of with respect to .
The Čech cochain complex places sections on -fold intersections in degree and uses the alternating sum of restriction maps as its differential.
A Čech cochain is a cocycle when its alternating coboundary vanishes. For a multiplicative one-cochain , this says on triple intersections.
A Čech coboundary is the image of a cochain in the preceding degree. Multiplicatively, a zero-cochain changes a one-cocycle by .
If every nonempty finite intersection of members of an open cover has vanishing higher sheaf cohomology for , then the cover's Čech cohomology computes . An affine open cover of a separated scheme satisfies this condition for a quasi-coherent sheaf because its finite intersections are affine.
For , the short exact sequence of sheaves obtained by restricting to , , and induces a long exact sequence
Articles by others on the same topic
Sheaf cohomology is a fundamental concept in algebraic geometry and topology that provides a way to study the properties of sheaves on topological spaces or schemes. It serves as a powerful tool for capturing global sections of sheaves and understanding their finer structures. ### Key Concepts 1.