## Weekly outline

### NEWS

• Extra lecture on Jan 29, Thursday 10:15-12:00 at KTH, room 3418. We will discuss the homework assignments and solutions to them.
• Please fill in the Course Evaluation. It is open throughout Jan 2015.
• Written assignment #4 is out (Deadline: January 15). Send the solutions by email (dary@kth.se) or leave them at the mathematics department at KTH.
• Written assignment #3 is out (Deadline: December 11). There was a small misprint in problem 3: the condition is that (a,b,c)!=(0,0,0) (not abc!=0).
• The tutorial and the lecture on Friday Oct 24 are cancelled. Next tutorial and lecture take place on Thursday Oct 30.
• Written assignment #2 is out (Deadline: November 6 extended until November 13)
• D'oh! There was a stupid mistake in problem 2 on assignment #1 - check out the corrected version.
• Careful: A finite A-module' is not necessarily finite, it's finitely generated'!
• Written assignment #1 is out! (Deadline: October 9)

### Teachers

Lectures: Benjamin Nill (SU) and David Rydh (KTH).
Tutorials: Olof Bergvall (SU).

### Schedule

The lectures will start on September 4, the tutorials on September 11.

Last lecture on Dec 11.

Extra lecture on Jan 29, Thursday 10:15-12:00 at KTH, room 3418.

### Textbook

Supplemental material: 1) Local Properties 2) Sheaves

### Course description

Grading criteria see below. Homework questions for self-study and tutorials see below.

• ### 9 oktober - 15 oktober

#6 Finite ring extensions and Noether normalization (Reid 4.1-4.6). [Olof replaced Benjamin]

• ### 16 oktober - 22 oktober

#7 What is geometry? Weak nullstellensatz. Maximal ideals of polynomial rings over an (alg. closed) field. Max-spectra and morphisms. On the geometry of finite extensions and Noether Normalization. (Reid 4.9-4.10 and 5.1-5.2) [David replaced Benjamin]

• ### 23 oktober - 29 oktober

No tutorial/lecture. Neither on Thursday Oct 23 nor on Friday Oct 24.

• ### 30 oktober - 5 november

#8 Varieties. V and I. Nullstellensatz. Irreducible varieties. Zariski topology on varieties. (Reid 5.3-5.7, 5.9)

Some on noetherian topological spaces and irreducible components. (Reid 5.10-5.11)

• ### 6 november - 12 november

#9 Zariski topology on Spec A. Some pictures of spectra. Polynomial maps between varieties. (Reid 5.12-5.14) [Benjamin replaced David]

• ### 13 november - 19 november

#10 Rings of fractions S-1A. Universal property of S-1A. Description Af=A[X]/Xf-1. Contraction and extension of ideals along A->S-1A and A->>A/K. Description of ideals and prime ideals of S-1A. Localization at a prime ideal AP. (Reid 6.1-6.4)

The universal property is treated briefly in Reid (6.2 (c) and exercise 6.11) but very important.

• ### 20 november - 26 november

#11 Modules of fractions. Exactness of localization. Local properties. Residue fields (Reid 6.5-6.8). Open subsets and principal open subsets Df of varieties (basis of topology, see Sheaves sections 1–2).

• ### 27 november - 3 december

#12 Presheaves and sheaves. (see Sheaves S3–6)

• ### 4 december - 10 december

#13 Support of modules. Associated points. Disassembly of modules. (Reid 7.1-7.5)

• ### 11 december - 17 december

#14 Primary decomposition (7.6-7.13).