Algebraic Cycles and Algebraic 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

  •   Definition and first properties of 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 theorems (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.