Algebraic K-theory and intersection theory

 
 

Time and place

Winter semester 2019/20
Wednesdays 10-12, M009

Description

Intersection theory, a topic at the heart of algebraic geometry, is concerned with the question of describing the intersection of two subvarieties in an ambient smooth algebraic variety. The modern flavour of the subject is highly influenced by Alexander Grothendieck’s revolutionary introduction of algebraic K-theory. In this course, we will introduce algebraic K-theory K0, as well as the closely related theory of Chow rings, of smooth algebraic varieties. We will mostly be occupied with understanding the basic properties of these theories and the precise relationship between the two.

Prerequisites: A good grasp of commutative algebra is necessary. We will not assume any prior knowledge of algebraic geometry, and instead introduce what we need during the course.

 

Latest updates

  • [Nov. 9] Notes for lecture 4 uploaded. Minor correction to notes for lecture 2 made (in the exercise about quasi-isomorphisms of minimal complexes).
  • [Nov. 8] Sheet 4 and solutions to Sheet 3 uploaded.

Lecture notes

Lecture 0: Overview
Lecture 1: Homological algebra crash course. Finiteness conditions on modules (finite generation/presentation, projectivity). Functoriality (restriction/extension of scalars, preservation of finiteness properties, modules over quotient rings). Structure of finitely generated modules. Projective resolutions (Koszul complexes, regular sequences).
Lecture 2: Perfect modules and regularity. Perfectness and finite Tor-amplitude. Minimal resolutions over local rings. Regularity of rings.
Lecture 3: Algebraic K-theory and G-theory. Group completion of monoids and the construction of K-theory of a ring. Construction of G-theory and statement of comparison for regular rings.
Lecture 4: Algebraic K-theory of perfect complexes. Perfect complexes. K-theory of perfect complexes. G-theory of coherent complexes.

Exercises

Sheet 0: pdf, ungraded
Sheet 1: pdf, solutions
Sheet 2: pdf, solutions
Sheet 3: pdf, solutions
Sheet 4: pdf, due Nov. 15

Exercise sessions: Fridays 10-12 in M009 by Maria Yakerson
Extra office hour: Fridays 16-17 in M223

References

  • H. Gillet, K-theory and intersection theory, pdf
  • A. Grothendieck, Classes de faisceaux et théorème de Riemann–Roch, pdf
  • M. Levine, A short course in K-theory, pdf

Contact

adenullel.khanull[email protected]nullr.dnulle
marnullia.yaknull[email protected]nullr.dnulle

← Return to Adeel Khan’s home page.