Synopsis
Varieties.- Some algebra.- Irreducible algebraic sets.- Definition of a morphism: I.- Sheaves and affine varieties.- Definition of prevarieties and morphism.- Products and the Hausdorff Axiom.- Dimension.- The fibres of a morphism.- Complete varieties.- Complex varieties.- Preschemes.- Spec (R).- The category of preschemes.- Varieties are preschemes.- Fields of definition.- Closed subpreschemes.- The functor of points of a prescheme.- Proper morphisms and finite morphisms.- Specialization.- Local Properties of Schemes.- Quasi-coherent modules.- Coherent modules.- Tangent cones.- Non-singularity and differentials.- Étale morphisms.- Uniformizing parameters.- Non-singularity and the UFD property.- Normal varieties and normalization.- Zariski's Main Theorem.- Flat and smooth morphisms.
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