I want to explain a particularly easy example of a motivic cellular decomposition: That of -dimensional projective space. We will have a look at the cohomology, the Chow groups and the algebraic K-theory of projective space -- a discussion probably interesting to non-motivic people as well. After these invariants, I will look at the motive and the A¹-homotopy type. Then I want to describe the decomposition of the motive (and the homotopy type) homotopy-theoretically, by means of cofiber sequences. (We will see that projective space is not isomorphic to a coproduct of motivic spheres with the same motive). Of course, nothing is new, I'm just working out exercises here.
In this part 1, I discuss only the cohomology of . Part 2 contains a discussion of the intersection theory and bundles and part 3 contains the motivic stuff. I intentionally left out usage of projective bundle formulas, as I will discuss them separately.
Continue reading «Invariants of projective space I: Cohomology»