The purpose of this paper is to lay the foundations of a theory of invariants inétale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces.and central simple algebras of degree n corresponding to P GL n -torsors) this gives a unified approach to the cohomological invariants for various types of structures.Suppose that M is an algebraic stack smooth over k 0 , for example the stack
Let g be an even positive integer, and let p be a prime number. We compute the cohomological invariants with coefficients in Z/pZ of the stacks of hyperelliptic curves H g over an algebraically closed field k 0 .
Using the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves
${\mathcal {H}}_g$
over any field of characteristic
$0$
. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.
We compute the class of the classifying stack of the exceptional algebraic group G 2 and of the spin groups Spin 7 and Spin 8 in the Grothendieck ring of stacks, and show that they are equal to the inverse of the class of the corresponding group. Furthermore, we show that the computation of the motivic classes of the stacks BSpin n can be reduced to the computation of the classes of B∆n, where ∆n ⊂ Pinn is the "extraspecial 2-group", the preimage of the diagonal matrices under the projection Pinn → On to the orthogonal group.
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