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.
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