Tags: #study-guides

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?

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?
• 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?

Problems

• 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?