*K*-theory

Outline of course and guide to the literature

**Lecturer** Abhijnan Rej (Bonn and Durham)

**Time/Place** Tuesday 3:15 pm/EH 101

Lecture 0: November 9th *Overview* Special time (5:15pm) and place (CM105)

Lecture 1: November 13th *Review of preliminaries *

- Sheaves
- Preschemes and schemes; smoothness, projectivity and quasiprojectivity
- Vector bundles, locally free sheaves and Grothendieck group

Guide to reading: Kunz- *Introduction to commutative algebra and algebraic geometry* (first few chapters), Hartshorne chapter 2 (first couple of sections) and Hatcher- *Vector bundles and K-theory*.

Lecture 2: November 20th *Introduction to cycles*

- Equivalence relations on cycles; adequate relations (time permitting)
- Chow group and the Grothendieck group

Guide to reading: Hartshorne (appendices on tanscendental methods). Murre's paper on pure motives (beginning sections).

Lecture 3: November 27th *Basic intersection theory and characteristic classes*

- Characteristic classes of smooth schemes
- Moving lemma (special cases) and intersection number.- focus on surfaces.
- Riemann-Roch theorem
*s*(emphasis on the plural!) - Intersections of cycles of a complex manifold

Guide to reading: I will distribute some notes; also Hartshorne's appendix on intersection theory. A couple of chapters of Griffiths-Harris (for the complex manifold story). The bible of intersection theory is Fulton's book of the same name though we wont need so much machinery for our purpose.