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Algebraic Geometry and Number Theory

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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