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

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