Ramification Theoretic Methods in Algebraic Geometry

Preface Introduction I. General Ramification Theory 1. Elementary lemmas from ideal theory 2. Primary decomposition in non-noetherian rings 3. Integral dependence 4. Linear algebra 5. Discriminant of an algebra 6. The discriminant ideal 7. Galois theory of local rings II. Valuation Theory 8. Ordered abelian groups 9. Valuations 10. Specialization and composition of valuations 11. Ramification theory of valuations 12. Valuations of algebraic function fields III. Noetherian Local Rings 13. The dimension of a noetherian local ring 14. Quadratic transforms IV. Two-Dimensional Local Domains 15. Limits of quadratic sequences 16. Uniformization in a two-dimensional regular local domain 17. Uniformization for a two-dimensional algebraic function field V. Varieties and Transformations 18. Projective varieties 19. Resolution of singularities of algebraic surfaces Bibliography