# 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.

• [Jan. 25] Solutions to Sheet 11 updated (solution to exercise 4 was missing).
• [Jan. 25] Solutions to Sheet 10 uploaded.
• [Jan. 24] Solutions to Sheet 11 uploaded.
• [Jan. 24] Sheet 12 uploaded.
• [Jan. 22] Lecture 12 uploaded.
• [Jan. 22] Minor revision to Lecture 11 uploaded (Remark 14 expanded, Remark 17 added).

## Exam

The exam will be a 30 minute oral exam. If you want to take it, send me an email with a proposed time. There are a few slots on February 10th, from 9am to 11am. Otherwise, any time after April 4th should work, preferably the week of the 6th or 20th.

## 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.
Lecture 5: G-theory of coherent complexes. Truncations. G-theory of coherent complexes (Proof of Theorem from 4.3). K-theory vs. G-theory (Proof of Theorem from 3.2). Tor-amplitude of complexes.
Lecture 6: Products, functoriality, and nil-invariance. Cap and cup products. Functoriality in K-theory, base change and projection formula. Functoriality in G-theory. Invariance of G-theory under quotients by nilpotent ideals.
Lecture 7: Dévissage, localization, and supports. Dévissage in G-theory. Quotients of abelian categories. Localization sequence in G-theory. The (underlying set of the) spectrum of a ring; points of rings and their residue fields.
Lecture 8: K-theory with supports and intersection numbers. Supports of modules. G-theory and K-theory with supports. Cup products in G-theory and in K-theory with supports. Intersection numbers and Serre's formula. Irreducible subsets of the Zariski spectrum.
Lecture 9: The coniveau filtration and algebraic cycles. The coniveau filtration. Algebraic cycles; the cycle associated to a module. Rational equivalence and the Chow group. Direct images. Inverse images.
Lecture 10: Smooth algebras and their Chow cohomology. Smooth algebras over a field. Chow cohomology groups. Quasi-smooth homomorphisms. Some more dimension theory. Proper intersections. Intersection products and the Chow ring.
Lecture 11: Divisors. The Picard group. Effective and non-effective Cartier divisors. The relationship between Cartier divisors and the Picard group. From Weil divisors to Cartier divisors.
Lecture 12: More on divisors. Regular local rings of dimension 1. Effective vs. non-effective Cartier divisors. Multiplicities of Cartier divisors and examples. From Cartier divisors to Weil divisors. The Cartier divisor class group.

## Exercises

Sheet 1: pdf, solutions
Sheet 2: pdf, solutions
Sheet 3: pdf, solutions
Sheet 4: pdf, solutions
Sheet 5: pdf, solutions
Sheet 6: pdf, solutions
Sheet 7: pdf, solutions
Sheet 8: pdf, solutions
Sheet 9: pdf, solutions
Sheet 10: pdf, solutions
Sheet 11: pdf, solutions
Sheet 12: pdf, due Jan. 31

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