# Coursebooks

## Algebraic K-theory

#### Lecturer(s) :

Hess Bellwald Kathryn

English

#### Summary

Algebraic K-theory, which to any ring R associates a sequence of groups, can be viewed as a theory of linear algebra over an arbitrary ring. We will study in detail the first two of these groups and applications of algebraic K-theory to number theory, algebraic topology, and representation theory.

#### Content

1. K_0 : Grothendieck groups, stability, tensor products, change of rings, the Dévissage, Resolution and Localization theorems and their applications
2. K_1 : elementary matrices, commutators and determinants, long exact sequences relating K_0 and K_1

#### Keywords

Rings and modules, Grothendiek group

#### Learning Prerequisites

##### Required courses

Second-year algebra and topology courses

##### Recommended courses

Rings and modules (Anneaux et modules)

##### Important concepts to start the course

Elementary ring and field theory

#### Learning Outcomes

By the end of the course, the student must be able to:
• Compute group completions of various semi-groups
• Interpret the universal properties of group completions, Grothendieck groups, and universal determinants
• Compute the Grothendieck group of important subcategories of modules
• Apply the Dévissage, Resolution and Localization theorems
• Sketch the proofs of the Dévissage, Resolution, and Localization theorems
• Explain the functoriality of K_0
• Compare the Grothendieck-type and matrix-based approaches to definining K_1
• Prove elementary properties of K_1

#### Transversal skills

• Assess one's own level of skill acquisition, and plan their on-going learning goals.
• Continue to work through difficulties or initial failure to find optimal solutions.
• Demonstrate a capacity for creativity.

#### Assessment methods

Each student must hand in one exercise each week for correction, which will determine 30% of the final grade.

The student's performance on the final written exam will determine the other 70% of the grade.

Dans le cas de l¿art. 3 al. 5 du Règlement de section, l¿enseignant décide de la forme de l¿examen qu¿il communique aux étudiants concernés.

### In the programs

• Mathematics - master program, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic K-theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 2
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic K-theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks
• Applied Mathematics, 2019-2020, Master semester 4
• Semester
Spring
• Exam form
Written
• Credits
5
• Subject examined
Algebraic K-theory
• Lecture
2 Hour(s) per week x 14 weeks
• Exercises
2 Hour(s) per week x 14 weeks

MoTuWeThFr
8-9
9-10
10-11
11-12
12-13
13-14MAA330
14-15
15-16MAA330
16-17
17-18
18-19
19-20
20-21
21-22
Lecture
Exercise, TP
Project, other

### legend

• Autumn semester
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• Lecture in French
• Lecture in English
• Lecture in German