## Exam

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 10^{th}, from 9am to 11am.
Otherwise, any time after April 4^{th} should work, preferably the week of the 6^{th} or 20^{th}.

## 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.

Lecture 13:

*Algebraic geometry*. Affine schemes. Zero loci. Algebraic varieties. Coherent sheaves. K-theory and Chow groups of schemes.

Lecture 14:

*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.

## Exercises

Sheet 0:

pdf, ungraded

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,

solutions
Sheet 13:

pdf (bonus sheet)

Exercise sessions: Fridays 10-12 in M009
by

Maria Yakerson
Extra office hour: Fridays 16-17 in M223