# Algebra 2, Sommer 2017

If you're taking this course, please enroll on Jogustine. This course is the follow-up of the course Algebra 1 given the preceding semester by Manuel Blickle.

Practical information

Teacher: Ariyan Javanpeykar

Exercise session: Robert Wilms

Course: Tuesday 16h-18h and Wednesday 14h-16h in Room 04-426.

Ubungsgruppe: Wednesday 12h-14h in Room 04-512.

Prerequisites: Algebra 1

Course material and description

In this course, we continue our study of commutative rings. We will follow Atiyah-Macdonald's book for a large part. We will also use Looijenga's notes.

Here are some typed up notes for Vorlesung 7 and 8 by Robert Wilms: Nullstellensatz.
Here are some typed up notes for Vorlesung 23-26 by Robert Wilms: Krull

Homework

We recommend you write your solutions in LateX. This will increase the quality of your homework, and it's a great way to practice writing mathematics with LateX. To use LateX, download MikTeX and an editor (like TeXmaker or TeXworks). A good tutorial for using LateX can be found under this link.

1. April 28th. Blatt 1
2. May 5th. Blatt 2
3. May 15th. Blatt 3
4. May 19th.Blatt 4
5. May 26th. Blatt 5
6. June 2nd. Blatt 6
7. June 9th. Blatt 7
8. June 16th. Blatt 8
9. June 23rd.Blatt 9
10. June 30th. Blatt 10
11. July 7th. Blatt 11
12. July 14th. Blatt 12

ORAL EXAMS: Tuesday, July 25th 2017. (You need at least 72 points to get the Prufungszulassung.)

Schedule

1. Tuesday April 18th. Rings, division algebra's, Hamilton quaternions, polynomial rings, matrixrings, endomorphism rings of abelian groups, zero divisors, integral domains, homomorphism theorem, isomorphism theorem, Chinese remainder theorem.
2. Wednesday April 19th. Proof of Chinese remainder theorem. Division with remainder. Principal ideal domains (PIDs). Cyclicity of finite subgroups of R* (after Gauss). The unit group of a finite field is cyclic. The polynomial ring K[x] over a field K is a PID. Irreducible elements. Nonzero elements in a PID factor.
3. Tuesday April 25th. Prime elements. Irreducible elements in a PID are prime. Unique factorization in PIDs (hence K[x]).  Prime ideals and unique factorization of prime ideals in PIDs. Example: Z[i] is a PID. Decomposition of prime numbers in Z[i].
4. Wednesday April 26th. Solving the Diophantine equation y^2=x^3+1. Euclidean rings. Examples: Z[rho] and Z[sqrt{-2}] and Z[sqrt{-5}]. Factorizing polynomials. Unique factorization domains.
5. Tuesday May 2nd. Algebraically closed fields. Maximal ideals and Zorn's lemma. The Zariski topology on \mathbb{A}^n; from ideals to algebraic sets. Regular functions on \mathbb{A}^n. The coordinate ring of a closed subset of \mathbb{A}^n. Hilbert's Nullstellensatz (proof later). Consequence of HNS: correspondence between closed subsets and radical ideals, points and maximal ideals.
6. Wednesday May 3rd. No Lecture.
7. Tuesday May 9th. (Robert Wilms) Section 2 in Looijenga's notes.
8. Wednesday May 10th. (Robert Wilms) Section 3 in Looijenga's notes.
9. Tuesday May 16th. Irreducibility,  irreducible components of noetherian spaces, irreducible components and prime ideals.
10. Wednesday May 17th. Cayley-Hamilton. Krull dimension of a ring. Dimension of a noetherian irreducible topological space. Main theorem: dim A^n = n. (Proof later.) Definition of projective space.
11. Tuesday May 23rd. Projective Hilbert's Nullstellensatz. Examples of projective varieties.
12. Wednesday May 24th. Morphisms of quasi-projective varieties. For affine varieties: Coordinate ring = ring of regular functions. Function field of a variety. Local ring of a variety a point.
13. Tuesday May 30th. Equivalence of categories: Affine varieties and affine k-algebra's. D(f) is isomorphic to an affine variety. Examples. Frobenius.
14. Wednesday May 31st. The cuspidal curve y^2=x^3. Products of quasi-affine varieties.
15. Tuesday June 6th.Statement of main theorem of elimination theory. projective varieties are complete.  Properties of complete varieties (e.g, O(X) =k). If f:X->Y is surjective, and X and Y are projective, then dim X >= dim Y.
16. Wednesday June 7th. dim P^n = n by using projections from a point (and assuming main theorem of elimination theory).
17. Tuesday June 13th. Proof of main theorem of elimination theory. Veronese embedding.
18. Wednesday June 14th. No Lecture
19. Tuesday June 20th. Dimension of A^n =n.
20. Wednesday June 21st. Dimension and families of varieties. Local noetherian rings and tangent spaces.
21. Tuesday June 27th. Cotangent space. Examples. Basic properties of local regular rings.
22. Wednesday June 28th.
23. Tuesday July 4th. (Robert Wilms)
24. Wednesday July 5th. (Robert Wilms)
25. Tuesday July 11th. (Robert Wilms)
26. Wednesday July 12th. (Robert Wilms)