The class of a torus in the Grothendieck ring of varieties

Abstract

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.

The Binomial Theorem and motivic classes of universal quasi-split tori

“…Given a separable k-algebra L of degree n, we can form the configuration space Conf n (Spec L), which is a Σ n -torsor over k. Descent along this torsor gives a functor Σ n -Var k → Var k , and L × is the image of G [n] m under this functor. The functor induces a lambda-ring homomorphism K 0 (Σ n -Var k ) → K 0 (Var k ), and we obtain the following result from [Rök11] as a corollary to our main theorem.…”

Virtual motives for synthetic geometries, A. Definition and properties of $K_0(\mathcal{Q}_\ell)$

“…One extra motivation is the following: in incidence geometry, an ovoid of a generalized quadrangle is not intrinsically defined, but defined relative to a generalized quadrangle; so actually, one should see an ovoid O of a generalized quadrangle Γ as a pair (O, Γ). The upshot is that in some sense, Aut(Γ) O sees more structure than Aut(O): it keeps track of the embedding (25) O ֒→ Γ.…”

Motivic classes of some classifying stacks

“…It is possible to compute the class of a quasi-split torus in the Grothendieck ring of algebraic spaces explicitly. This has been done by in [23] for tori over fields and in [4] in the relative setting. The class of a quasi-split torus may be expressed in terms of classes in the Burnside ring.…”