- The class of a torus in the Grothendieck ring of varieties
- American Journal of Mathematics
- Volume | Issue number
- 133 | 4
- Pages (from-to)
- Document type
- Faculty of Science (FNWI)
- Korteweg-de Vries Institute for Mathematics (KdVI)
We establish a formula for the classes of certain tori in the Grothendieck ring of varieties, expressing them in terms of the natural lambda-structure on the Grothendieck ring. 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 perform 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.
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