Cohomology operations and algebraic geometry

This manuscript is based on a ten hours series of seminars I delivered in August of 2003 at the Nagoya Institute of Technology as part of the workshop on homotopy theory organized by Norihiko Minami and following the Kinosaki conference in honor of Goro Nishida. One of the most striking applications of homotopy theory in “exotic” contexes is Voevodsky’s proof of the Milnor Conjecture. This conjecture can be reduced to statements about algebraic varieties and “cohomology theories” of algebraic varieties. These contravariant functors are called motivic cohomology with coefficients in abelian groups A. Since they share several properties with singular cohomology in classical homotopy theory, it is reasonable to expect “motivic cohomology operations” acting naturally on these cohomology theories. By assuming the existence of certain motivic Steenrod operations and guessing their right degrees, Voevodsky was able to prove the Milnor Conjecture. This strategy reduced the complete proof of the conjecture to the construction of these operations and to an appropriate category in whcih motivic cohomology is a “representable”. In homotopy theory there are several ways of doing this. We now know two ways of obtaining such operations on motivic cohomology: one is due to P Brosnan [7] and the other to V Voevodsky [22]. The latter approach follows a systematic developement of homotopy categories containing algebraic information of the underlying objects and it is the one we will discuss in this manuscript. It turns out that, if we think of an algebraic variety as something like a topological space with an algebraic structure attached to it, it makes sense to try to construct homotopy categories in which objects are algebraic varieties, as opposed to just topological spaces, and, at the same time motivic cohomology representable. In the classical homotopy