Spring 2024 Algebra (under construction)



上課時間: 週三67,週五678.
上課地點:
天數101
Office hour: 週三3:30--5:30 at 天數412


The purpose of this course is to introduce basic tools in non-commutative rings and commutative algebra.
The goal is to equip students with standard knowledge to study advanced topics
in representation theory, number theory and algebraic geometry.

References:

Noncommutative algebra, B. Farb and K. Dennis (GTM144)
Algebra Chapter 8, Modules et anneaux semi-simples, N. Bourbaki.
Introduction to commutative algebra, Atiyah-Macdonald.
Commutative algebra with a view toward algebraic geometry,
D. Eisenbud (GTM150)

Syllabus:


Theory of semi-simple rings and applications
Week 1: Semisimple modules.

Week 2:
The Weddburn-Artin Theorem
Week 3: Density theorem, simple rings and
the Jacobson radical
Week 4: Jacobson's theorem. Tensor products of modules.
Week 5: Central simple algebras and the Skolem-Noether Theorem.
Week 6-7: Brauer groups and K_2 groups of fields.
Week 8: Midterm.
Introduction to commutative algebras: theory of affine schemes

Week 9  : The language of affine schemes.
Week 10: Algebras of finite type over a field and Hilbert's Nullstellensatz.
Week 11: Etale morphisms and de Rham cohomology of affine schemes.
Week 12: Basic homological algebras: injective resolutions, derived functors and spectral sequences.
Week 13: Noetherian rings and Hilbert's basis theorem.
Week 14: Heights of primes and principal ideal theorem.
Week 15: ???
Week 16: Final exam

習題課時間:  每週五第八節 
助教: 李俊緯 (d09221001AT ntu.edu.tw)


作業:
繳交每週 發佈於NTU Cool的作業。

評量方式:

作業 (30%) 週五第八節結束前繳交,請利用NTU COOL上傳作業, 逾期不收。

期中考(35%) April 10, 12:30--15:10 (Week 1-6)

期末考(35%) June 7, 13:30-15:10