MATH 8310: Schemes, sheaves, and cohomology
Plan of the course
Scheaves
- Topological spaces.
- Sheaves of abelian groups.
- Ringed spaces.
Schemes
- Algebraic varieties over an algebraically closed field.
- Affine schemes, Spec R.
- (Quasi)coherent sheaves on Spec R.
- Projective schemes.
- Proj R.
- General schemes.
Cohomology
- Categories and universal constructions. Additive and abelian categories.
- Homological algebra.
- Derived functors.
- Examples from commutative algebra.
- Cohomologies of sheaves.
- Vanishing theorems.
- Riemann-Roch theorem.
- Applications to classification problems in algebraic geometry.