There 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.
Exams have to be postponed until May (at least).
Lecture 0: Overview
: 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).
: Perfect modules and regularity
. Perfectness and finite Tor-amplitude. Minimal resolutions over local rings. Regularity of rings.
: 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.
: Algebraic K-theory of perfect complexes
. Perfect complexes. K-theory of perfect complexes. G-theory of coherent complexes.
: 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.
: 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.
: 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.
: 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.
: 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.
: 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.
. The Picard group. Effective and non-effective Cartier divisors. The relationship between Cartier divisors and the Picard group. From Weil divisors to Cartier divisors.
: 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.
: Algebraic geometry
. Affine schemes. Zero loci. Algebraic varieties. Coherent sheaves. K-theory and Chow groups of schemes.
: Comparing K-theory and the Chow groups
. The map from algebraic cycles to K-theory. Compatibility with intersection products, flat inverse image, and rational equivalence, modulo the coniveau filtration. Multiplicity of the coniveau filtration. The comparison theorem.
Sheet 0: pdf
Sheet 1: pdf
Sheet 2: pdf
Sheet 3: pdf
Sheet 4: pdf
Sheet 5: pdf
Sheet 6: pdf
Sheet 7: pdf
Sheet 8: pdf
Sheet 9: pdf
Sheet 10: pdf
Sheet 11: pdf
Sheet 12: pdf
Sheet 13: pdf
Exercise sessions: Fridays 10-12 in M009
by Maria Yakerson
Extra office hour: Fridays 16-17 in M223