Grothendieck Rings and Motivic Integration
This thesis consists of three parts:
In Part I we study the Burnside ring of the finite group G. This ring has a natural structure of a lambda-ring. However, a priori the images of the G-set S under the lambda-operations can only be computed recursively. We establish an explicit formula, expressing these images as linear combination of classes of G-sets. This formula is derived in two ways: First we give a proof that uses the theory of representation rings in an essential way. We then give an alternative, more intrinsic, proof. This second proof is joint work with Serge Bouc.
In Part II we establish a formula for the classes of certain tori in the Grothendieck ring of varieties, in terms of its lambda-structure. More explicitly, we will see that if L* is the torus of invertible elements in the n-dimensional separable k-algebra L, then the class of L* can be expressed as an alternating sum of the images of the spectrum of L under the lambda-operations, multiplied by powers of the Lefschetz class. This formula is suggested from the cohomology of the torus, illustrating a heuristic method that can be used in other situations. To prove the formula will require some rather explicit calculations in the Grothendieck ring. To be able to make these we introduce a homomorphism from the Burnside ring of the absolute Galois group of k, to the Grothendieck ring of varieties over k. In the process we obtain some information about the structure of the subring generated by zero-dimensional varieties.
In Part III we give a version of geometric motivic integration that specializes to p-adic integration via point counting. This has been done before for stable sets; we extend this to more general sets. The main problem in doing this is that it requires to take limits, hence the measure will have to take values in a completion of the localized Grothendieck ring of varieties. The standard choice is to complete with respect to the dimension filtration. However, since the point counting homomorphism is not continuous with respect to this topology we have to use a stronger one. We thus begin by defining this stronger topology; we will then see that many of the standard constructions of geometric motivic integration work also in this setting. Using this theory, we are then able to give a geometric explanation of the behavior of certain p-adic integrals, by computing the corresponding motivic integrals.
Source Type:Doctoral Dissertation
Keywords:MATHEMATICS; Mathematics; matematik
Date of Publication:01/01/2009