Cohomology of Schemes

Definitions

Homological algebra:
- What is an abelian category?
- What is a delta functor?
- What is the module of relative differential forms?
Functor formalism:
- What is the higher direct image?
- What is the higher pushforward?
Acyclic resolutions:
- What is a flasque sheaf?
- What is a flabby sheaf?
- What is a soft sheaf?
- What is a fine sheaf?
- What is the Godemont resolution?
Chow:
- What is the Chow ring?
- What is the cycle class map?
Special sheaves, bundles, classes:
- What is the tangent sheaf?
- What is the canonical sheaf?
- What is the canonical class?
- What is the cotangent complex?
- What is the conormal bundle?
- What is the anticanonical sheaf?
- What is the dualizing sheaf?
- What is the sheaf of differentials?
- What is a constructible sheaf?
What is the geometric genus?
What is an lax embedding?
What is Cech cohomology?

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?
Why are higher direct images with proper support of l-adic sheaves again l-adic?
Exact sequences:
- What is the exponential exact sequence?
- What is the conormal exact sequence?
- What is the Euler exact sequence?
Why is the moduli space of curves ${\mathcal{M}}_g$ smooth and proper?

Show that the differential in the de Rham complex is not ${\mathcal{O}}_X{\hbox{-}}$linear.
Compute the Cech cohomology of ${\mathbb{P}}^n_{/ {k}} $.
Compute $H^*(X, {\mathcal{O}}_X)$ for $X = {\mathbb{P}}^n_{/ {k}} $.
Slogans to make rigorous:
- For $X\to B$ smooth, why does $\operatorname{Ext} ^0(\Omega_{X/B}, {\mathcal{O}}_X) \cong H^0({\mathbf{T}}_{X/B} )$ hold, and why does it measure infinitesimal automorpisms of $X$?
- What does $\operatorname{Ext} ^1(\Omega_{X/B}, {\mathcal{O}}_X)$ measure 1st order deformations of $X\to B$?
- In general,
Why doesn’t dualizing the cotangent complex yield the tangent complex? What is the correct dualization to take?