Topology IV.b - Algebraic K-theory
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
(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) Elementary matrices and commutators
(b) Definition and elementary properties of K_1
(c) Generalized determinants
(d) K_1 as a Grothendieck group
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 and Resolution Theorems
- Interpret computations of K_0 and K_1
- Explain the extensions of K_0 beyond the context of rings
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.
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)
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
Mo | Tu | We | Th | Fr | |
8-9 | |||||
9-10 | |||||
10-11 | |||||
11-12 | |||||
12-13 | |||||
13-14 | |||||
14-15 | |||||
15-16 | |||||
16-17 | |||||
17-18 | |||||
18-19 | |||||
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20-21 | |||||
21-22 |
Légendes:
Lecture
Exercise, TP
Project, Lab, other