Citation/Abstract

In this thesis we construct a version of Morava K-theories k ( n ) in the stable A ¹ homotopy category for any integer n greater than or equal to zero. They come equipped with an [Special characters omitted.] -resolution, which is used to define a characteristic number t satisfying certain multiplicative properties. Let Y and X be smooth, proper, irreducible algebraic varieties and f : Y [arrow right] X be any dominant map. The higher degree formula states that t ( Y ) ≡ deg( f )· t ( X ) mod I _r ( X ), where I _r ( X ) is an ideal in [Special characters omitted.] depending on X and a positive integer r . Finally, we prove that a certain condition on X would imply I _r ( X ) = 0. The so called norm varieties are known to satisfy this condition and play a fundamental role in the proof of the Block-Kato conjecture.