Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor. For more detailed information visit the Math 663 Wiki page.

• Serre varieties including: Sheaves and affine varieties, Definition of varieties and morphisms, Products and separation, Dimension, Fibres of a morphism, Complete varieties, Relation with complex analytic varities.
• Schemes including: Affine schemes, Schemes, Relationship between varieties and schemes, Closed subschemes, Functor of points, Proper varieties and finite morphisms