Motives of Projective Bundles

Friday, December 14th, 2012 | Author:

Given a vector bundle E-->X of rank r+1 one can take the projective space of lines in each fiber, which results in a projective bundle P(E)-->X. A projective bundle formula for a functor F from spaces to rings tells us that F(P(E)) is a free F(X)-module of rank r.

In this post I look at some computations around projective bundle formulae for the Chow ring, the algebraic K-Theory and the (Chow) motive of some spaces, in particular flag varieties. We recover some results from the previous posts on cohomology, cycles & bundles and motive of projective space.

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Divisorial Jungle

Thursday, November 29th, 2012 | Author:

I'd like to compile a short list of definitions of Weil and Cartier Divisors, Line Bundles and Invertible Sheaves, Class Groups and Picard Groups, Cohomology, (higher) Chow Groups and K-theory for algebraic schemes and their relations. I intentionally omit proofs, but there are some ideas. I couldn't resist to jot down some properties of the objects which are important to me (homotopy invariance, existence of pullbacks and pushforwards).

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