Cohomology of quasi-coherent sheaves

This course develops sheaf and Čech cohomology, affine and projective cohomology, proper direct images and base change, formal functions and algebraization, duality, and the comparison between algebraic and analytic coherent sheaves. Its eighteen lessons include worked examples and 108 exercises with solutions.

Begin with sheaves of modules, injective resolutions and bounded-below derived functors in Derived categories of sheaves. The prerequisite and proof guide identifies the precise statements and hypotheses used throughout the course.

Lessons

  1. Cohomology of sheaves on ringed spaces
  2. Čech cohomology
  3. Cohomology of affine schemes and Serre's criterion
  4. Cohomology of projective space
  5. Coherent sheaves on projective schemes: Serre's theorems
  6. Coherence of higher direct images under proper morphisms
  7. Euler characteristics and Hilbert polynomials
  8. Base change and the Grothendieck complex
  9. Semicontinuity and Grauert's theorem
  10. The theorem on formal functions
  11. Zariski's connectedness theorem and Stein factorization
  12. Grothendieck's existence theorem
  13. Algebraization of formal schemes
  14. Ext sheaves and Serre duality on projective space
  15. Dualizing sheaves and Serre duality for projective schemes
  16. The right adjoint of derived pushforward
  17. Complex analytic spaces and analytification
  18. Serre's comparison theorems and Chow's theorem

Editable sources

Complete reader and source archive · Complete LaTeX · Exact proof providers