Select Commutative Algebra in the fall semester, and then pick a specialization in the spring; either Algebraic Geometry or Algebraic Number Theory.

### Commutative Algebra

I Basic notions for rings and modules

- Rings, ideals, modules
- Localization
- Primary decomposition
- Integrality
- Noetherian and Artinian Rings
- Noether normalization and Nullstellensatz
- Discrete valuation rings, Dedekind domains and curves
- Graded Modules and Completions
- Dimension theory, Hilbert functions, Regularity
- Sheaves and affine schemes

### Algebraic Geometry

I Varieties

- Projective Varieties
- Morphisms and Rational Maps
- Nonsingular Varieties
- Intersections of Varieties

II Schemes

- Basic properties of schemes
- Separated and proper morphisms
- Quasi-coherent sheaves
- Weil and Cartier divisors, line bundles and ampleness
- Differentials
- Sheaf cohomology

III Curves

- Residues and duality
- Riemann-Roch
- Branched coverings
- Projective embeddings
- Canonical curves and Clifford’s Theorem

### Algebraic Number Theory

- Local fields
- Global fields
- Valuations
- Weak approximation
- Chinese Remainder Theorem
- Ideal class groups
- Minkowski’s theorem and Dirichlet’s unit theorem
- Finiteness of class numbers
- Ramification, different and discriminants
- Quadratic symbols and quadratic reciprocity law
- Zeta functions and L-functions
- Chebotarev’s density theorem
- Preview of class field theory