**Algebraic Geometry I**

課程：MATH5251

作者：hchencs

創建於：2020-11-03 19:21:27

更新於：2020-11-03 19:34:25

Term: 2020 fall

Instructor: Weiping Li

Rate of professor: Good

Grades: depends

This is the first course of the one-year sequence in algebraic geometry. It covers the first chapter and 2.1-2.5 of the Hartshorne's book, depending on the speed of lecture every year. Usually the first half ends around divisors and the second ends around Serre duality. The materials are essential for research areas involving algebraic geometry.

General comments

I would say the lecture is very fast paced, comparable to the current MIT course 18.725. The homework is arguably just enough or a bit less than enough for understanding the materials well. But I always found myself running out of time revising what has covered in the lecture while doing homework. An estimate for me would be at least 10 hours of concentration time (=2.5 working days for me) in a week. An advantage of following a book faithfully is that students always got rigorous backups after class.

About background

The first course in commutative algebra covering prime and radical ideals, noetherian rings, localization and some dimension theory such as Krull's dimension and transcendence dimension is essential and would be very involved starting from 1.1. Ideally one knows a little valuation theory but these will be reviewed quickly at section 1.6.

Field extensions like algebraic extension, degree of extension, transcendental extension was used and roughly revised in 10 min. But no Galois theory involved.

Usually the first commutative algebra course is not enough but once we get some maturity we can accept the results without proofs. It would not be practical to expect making up the algebra background systematically while doing this course. For me I can just accept the results from algebra and use them.

Also the basic concepts and equivalent definitions of point set topology like dense, closure, continuous maps, irreducible sets were used without being mentioned.

Basic category theory is helpful but not strictly necessary. Concepts like category, functor, natural transformation, adjoints, and limits, especially direct limit, inverse limit, kernel and cokernel would be helpful and can be looked up when needed.

(Basic homological algebra will be reviewed at the start of chapter 3, but not in the syllabus of 5251 in the first term.)

Assessment

The grade is 100% determined by the final oral examination on one of the homework he randomly selected. So during the term the students are highly autonomous.

Learning outcome

Given all the hardwork in 3 months it is definitely very rewarding. Hartshorne's book is known to be dense, and after one year you will understand most of the junior seminars involving algebraic geometry.

About professor

Personally I like the professor though what he does is just repeating the book. Apart from that he sometimes makes cultural remarks in the AG community and on his own understanding of the subject. He also added some details in the proofs if you don't have enough algebra background. He was in general willing to answer any questions in or after class, replied emails quickly, though no office hours. The professor speaks fluent English and there is no problem in communication.

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