# Topology IV.a -Algebraic K-theory

MATH-488 / **5 credits**

**Teacher: **

**Language:** English

**Remark:** pas donné en 2023-24

## 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

- K_0 : Grothendieck groups, stability, tensor products, change of rings, the Dévissage, Resolution and Localization theorems and their applications
- 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

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Topology IV.a -Algebraic K-theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Topology IV.a -Algebraic K-theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Topology IV.a -Algebraic K-theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

**Semester:**Fall**Exam form:**Written (winter session)**Subject examined:**Topology IV.a -Algebraic K-theory**Lecture:**2 Hour(s) per week x 14 weeks**Exercises:**2 Hour(s) per week x 14 weeks**Type:**optional

## Reference week

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