Topology IV.b - Algebraic K-theory

MATH-488 / 5 credits

Teacher: Hess Bellwald Kathryn

Language: 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 some of their applications to other areas of mathematics.

Content

1. Introduction: motivations and relations with other fields


2. K_0 and classification of modules
(a) Definition and elementary properties of K_0 for rings
i. Group completion
ii. Grothendieck groups
iii. Dévissage
iv. The Resolution Theorem
v. Stability
vi. Multiplicative structure
(b) Functoriality of K_0
i. Exact functors
ii. Naturality of K_0(R)
iii. Localization and K_0
(c) K_0 beyond rings
i. K_0 of symmetric monoidal categories
ii. K_0 of abelian categories
iii. K_0 of exact categories


3. K_1 and classification of invertible matrices
(a) The Bass-Whitehead group of a ring
(b) A Grothendieck-type approach to K_1
(d) Long exact sequences

 

4. A glimpse of higher algebraic K-theory

 

Keywords

K-theory, Grothendieck group, group completion

Learning Prerequisites

Required courses

Algebra IV - rings and modules

Recommended courses

Representation theory I - finite groups

Topology III - homology

Important concepts to start the course

Categories and functors

Projective modules

Learning Outcomes

By the end of the course, the student must be able to:

  • Apply the Dévissage, Resolution, and Localization Theorems
  • Interpret computations of K_0 and K_1
  • Prove the most important results in the course
  • Contextualise the universal properties of K_0 and K_1

Transversal skills

  • Assess one's own level of skill acquisition, and plan their on-going learning goals.
  • Demonstrate a capacity for creativity.
  • Demonstrate the capacity for critical thinking
  • Take feedback (critique) and respond in an appropriate manner.
  • Make an oral presentation.

Teaching methods

Ex cathedra (interactive)

Written exercises

Expected student activities

Active participation in lectures

Weekly homework to hand in

Assessment methods

Weekly homework:

  • 30% of the final grade

Oral exam:

  • 70% of the final grade
  • 10% of exam grade determined by the quality of the oral presentation at the board

Resources

Bibliography

1. Bruce A. Magurn, An Algebraic Introduction to K-theory, Encyclopedia of
Mathematics and its Applications 87, Cambridge University Press, 2009.

 

2. Joseph J. Rotman, An Introduction to Homological Algebra, Academic
Press, 1979.

 

3. Charles Weibel, The K-Book: An introduction to algebraic K-theory,
Graduate Studies in Mathematics 145, American Mathematical Society,
2013.

Ressources en bibliothèque

Moodle Link

In the programs

  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Topology IV.b - Algebraic K-theory
  • Courses: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Topology IV.b - Algebraic K-theory
  • Courses: 2 Hour(s) per week x 14 weeks
  • Exercises: 2 Hour(s) per week x 14 weeks
  • Type: optional
  • Semester: Spring
  • Exam form: Oral (summer session)
  • Subject examined: Topology IV.b - Algebraic K-theory
  • Courses: 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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