Fall 2023
Commutative algebra and homological algebra 交换代数和同调代数
- Instructor: Chenglong Yu 余成龙
- E-mail:
yuchenglong {at} tsinghua.edu.cn,
- Office:ShuangQing B523 双清综合楼A座B523, 地址：北京市海淀区逸清南路西延6号院1号，双清公寓马路对面、清华附小（双清校区）西侧。
- Office hours: TBA
- TA: Baiting Xie
- TA's email:
xbt23 {at} mails.tsinghua.edu.cn .
- TA's office hour:
Course structure:
- Lectures:24, 42,
- Labs:
- Textbook:
- Allen Altman and Steven Kleiman, A term of commutative algebra (https://dspace.mit.edu/handle/1721.1/116075.2)
- Atiyah and MacDonald, Introduction to commutative algebra
- Weibel, An introduction to homological algebra
- Additional handouts to be posted on the course website.
- Homework: Homework is very important for this class. Homework will be posted on this website once a week and due the following Thursday in class.
You are encouraged to work in groups on your homework or discuss it with the TA and me. But you should write up and hand in your homework individually.
- Exams: There are two exams in class.
- Grading: The grades are based on 30% Homework + 30% midterm + 40% final exam.
Syllabus:
第一章：环与理想：素理想，极⼤理想 ，局部环，环的幂零根，雅各布森根，扎利斯基拓扑。
第二章：模：模的定义，同态，子模，商模，升链条件，诺特环，诺特模，希尔伯特基定理，张量积.
第三章：同调代数：范畴，Yoneda引理，正合列，分裂, 蛇形引理与五引理, 极限，对偶函子，投射模, 内射模, 平坦模, 复形以及其(上)同调
第四章： 凯莱-哈密顿定理，行列式技巧，Nakayama引理
第五章： 局部化： 环和模的局部化， 理想与局部化，局部化的正合性，支撑
第六章：整性：环的整扩张， Going-up, 诺特正规化，希尔伯特零点定理, 雅各布森环。
第七章：Assciated primes and Primary decomposition
第八章：戴德金环：离散赋值环，一般赋值，分式理想，完备化和hensel引理
第九章：维数理论： 维数，阿廷环，Krull-Chevalley-Samuel 定理，参数理想, 深度，分次环，分次模，阿廷-里斯引理，希尔伯特多项式, 正规的塞尔判定法则，Cohen-Macaulay
Homework
Homework 1
Lecture notes
(Introduction)