(SISSA, Trieste, Italy)

- The group of algebraic cycles associated to a scheme. Fundamental cycle associated to a pure-dimensional scheme.
- Definition of proper push-forward and flat pullback. Functoriality and compatibility among them.
- Rational equivalence, alternative definitions of rational equivalence.
- The Chow group of algebraic cycles modulo rational equivalence.
- Proper pushforward and flat pullnack preserve rational equivalence. Flat pullback from a scheme to a vector bundle over it defines a (degree-shifting) isomorphism of Chow groups.
- Intersection with the zero section of a bundle
- Cones, abelian cones, normal cone.
- Fundamental exact sequences and relationship with the cotangent complex; regular embedding, lci morphism.
- Degeneration to the normal cone.
- Segre classes of a cone. Chern classes, splitting principle. Compatibility between Chern classes, pushforward and pullback.
- Gysin pullback for an lci morphism of schemes. Definition of bivariant classes, Gysin pullback as bivariant class.
- Excess intersection formula, principle of the conservation of number
- Statement of Grothendieck Riemann Roch theorem; outline of proof

(Please note that the lectures at IMSc. start half an hour earlier than announced before)

Four lectures at CMI on 20, 21, 23 and 24 January, 2014 at 14:00 to 15:30

Professor Fantechi will give a colloquium on 22, January, 2014 at 15:30 at CMI.