Definitions
- What is a flasque sheaf?
- What is a fine sheaf?
- What is the Godemont resolution?
- What is the higher direct image?
- What is the higher pushforward?
- What is an abelian category?
- What is a delta functor?
- What is the dualizing sheaf?
- What is the module of relative differential forms?
- What is the sheaf of differentials?
- What is the geometric genus?
- What is the canonical class?
- What is the Chow ring?
- What is the cycle class map?
Results
- Why does the category of sheaves of \({\mathcal{O}}_X\) modules have enough injectives?
- What is Serre’s vanishing characterization of affine schemes among Noetherian schemes?
- What is the theorem on formal functions?
- What is the semicontinuity theorem?
- How is the sheaf of differentials related to the singularities/smoothness of a scheme?
- What is Bertini’s theorem?
- Show that singular cohomology is isomorphic to Cech cohomology for Noetherian separated schemes.
- What is the cohomological criterion for ampleness?
- Show that the dualizing sheaf is isomorphic to the canonical sheaf for nonsingular projective varieties.
- What is the Lefschetz hyperplane theorem
- What is Hodge duality?
- What is Serre duality?
- What is Grothendieck vanishing?
- What is Serre’s vanishing theorem?
Problems
- Compute the Cech cohomology of ${\mathbb{P}}^n_{/ {k}} $.
- Compute \(H^*(X, {\mathcal{O}}_X)\) for $X = {\mathbb{P}}^n_{/ {k}} $.