Math 664: Algebraic Geometry 2.
Algebraic Geometry 2.
Cohomology of schemes. Classification problems. Applications.
Desired Learning Outcomes
Minimal learning outcomes
As this is a terminal course, the instructor has freedom to choose the topic. One possibility is to study local properties of schemes. Here are the outcomes for such a course.
Students should achieve mastery of the topics listed below. 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.
- Quasi-coherent sheaves
- Nonsingularity and differentials
- Etale morphisms
- Uniformizing parameters
- Normal varieties and normalization
- Zariski's main theorem
- Flat and smooth mrophisms
Possible textbooks for this course include (but are not limited to):
- The Red Book of Varieties and Schemes, Second, Expanded Edition