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.)
| 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) |
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.
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.
Here is a list of references for the course: