Students should achieve mastery of the topics listed on the Math 664i Wiki page. This means they 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.

As this is a terminal course, the instructor has freedom to choose the topic. One possibility is to study local properties of schemes.

• Quasi-coherent sheaves
• Nonsingularity and differentials
• Etale morphisms
• Uniformizing parameters
• Normal varieties and normalization
• Zariski's main theorem
• Flat and smooth morphisms