Algebra II

Course at HU Berlin, Winter Term 16/17

Kay Rülling



Content

This course is an introduction to commutative algebra, that is we study commutative rings, modules over such and their properties. The topic is not only very interesting in its own, it is also fundamental for all further developments in the fields of, e.g., algebra, algebraic geometry, arithmetic geometry, and number theory. For example, in the integers \(\mathbb{Z}\) or the polynomial ring with coefficients in a field \(k[x]\) we can write every element as a product of prime elements. Is this true in general? (No!) Is there a generalization of this statement for more general rings? (Yes!) As another example, we draw a curve in the plane, i.e. the zero set in the reals of a polynomial in two variables. We have the immediate impulse to say this "thing" should be of dimension 1. But how to make it rigorous? And what happens if we consider more polynomials in higher dimensions? And what is the dimension of \(\mathbb{Z}\)? This and much more is what commutative algebra is about.

In the course we will learn about rings, modules, limits, tensor product, localization, flatness, chain conditions (noetherian, artinian), Noether normalization, dimension theory,....

Prerequisites: Basic knowledge in algebra. (It is good to know the definition of rings and modules but it is not strictly necessary.)

Data

Lecturer Time Room
Lecture Kay Rülling Di 11-13 Johann von Neumann-Haus 1.115 (RUD25)
Do 13-15 Johann von Neumann-Haus 1.013 (RUD25)
Exercise Kay Rülling Di 13-15 Erwin Schrödinger Zentrum 1304 (RUD26)

The first lecture is on October 18, 2016. The first exercise session is on October 25.
If you have any questions concerning the lectures or the exercises feel free to contact me.

Exercises

Every Tuesday there will be a new exercise sheet on this web page, the solutions are discussed in the exercise session of the following week.

  1. Exercise sheet, October 18, 2016
  2. Exercise sheet, October 25, 2016
  3. Exercise sheet, November 1, 2016 ( ver 2, 8.11., see here for an example of a prime ideal in an infinite product of domains which is not the inverse image of a prime ideal on one of the factors under the projection)
  4. Exercise sheet, November 8, 2016
  5. Exercise sheet, November 15, 2016
  6. Exercise sheet, November 22, 2016
  7. Exercise sheet, November 29, 2016
  8. Exercise sheet, December 6, 2016
  9. Exercise sheet, Christmas 2016
  10. Exercise sheet, January 3, 2017
  11. Exercise sheet, January 10, 2017 (ver 2, 17.1., see here for solutions for Exc 11.1, (iii), and Exc 11.2)
  12. Exercise sheet, January 17, 2017
  13. Exercise sheet, January 24, 2017
  14. Exercise sheet, January 31, 2017
  15. There was a gap in the proof of the last Corollary (dimension formula for polynomial rings) in the lecture on February, 16. See here for a corrected proof.

Exam

The exam was be on Tuesday 14.02.2017, 11:00-14:00, in room 1.115, RUD 25.
The second exam will be on Tuesday 14.03.2017, 11:00-14:00, in room 1.303, RUD 26.
There is an alternative second exam (for those who don't have time to come to the second exam) on Friday 24.03.2017, 11:00-14:00, in room 1.012, RUD 25.

You may bring three two-sided (or six one-sided) handwritten pages of DIN A4 paper to each exam. Besides this and a pen you are not allowed to bring any other material.

References

Here is a list of references for the course:

  1. A. Altman, S. Kleiman, A Term of commutative algebra. Available at http://web.mit.edu/18.705/www/13Ed.pdf.
  2. M.F. Atiyah, I.G. MacDonald, Introduction to commutative algebra. Westview Press, 1994.
  3. H. Matsumura, Commutative ring theory. Cambridge University Press, 1989.
  4. D. Eisenbud, Commutative Algebra with a view toward algebraic geometry. Springer, 1995.