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In the Grothendieck ring of varieties, there are ways of distinguishing classes of varieties, for example $\ell$-adic cohomology. The Grothendieck ring of stacks is a localization of the Grothendieck ring of varieties, more precisely it is obtained by inverting $\mathbb{L}$ and the cyclotomic polynomials in $\mathbb{L}$.

If I am not mistaken, to check that $\{X\}\neq\{Y\}$ in $K_0(Var_K)$, where $X$ and $Y$ are varieties (assume smooth, but not proper) it suffices to check that $\sum (-1)^i[H^i(X_{K^{alg}},\mathbb{Q}_l)]$ is not isomorphic to $\sum (-1)^i[H^i(Y_{K^{alg}},\mathbb{Q}_l)]$. Now say that $\{X\}=\{Y\}$ in $K_0(Stacks_K)$. Even when $X$ and $Y$ are actually schemes, then $p(\mathbb{L})(\{X\}-\{Y\})=0$, where $p$ is a product of cyclotomic polynomials and powers of $\mathbb{L}$, and I don't think we can use $\ell$-adic cohomology now. More generally, what if $X$ and $Y$ are actually stacks and not just schemes?

As a more concrete question (I am not necessarily interested in this, if you have a better example), what if $X$ and $Y$ are tori of the same rank, given by an explicit action on lattices?

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I'm not sure why you think there is a problem with applying $\ell$-adic cohomology here. If we want to show that ${X} \neq {Y}$ in $K_0(Stacks_k)$, and we know their associated classes in $K_0$ of Galois representations are different, it suffices to show that cyclotomic polynomials in the Lefschetz motive/cyclotomic character are not zero divisors in the ring of Galois representations. This is clear as the ring of Galois representations is contained in the ring of functions on the Galois group, and zero divisors with the cyclotomic polynomials in the Lefschetz motive are exactly functions supported on elements that act on the cyclotomic character by roots of unity, but these are a density zero subset and so no nontrivial Galois representation has a character supported there.

To do the same thing for stacks, one just has to invert these cyclotomic polynomials in the category of Galois representations - possible by considering infinite formal sums of Galois representations, where all but finitely many have weight below a given constant.

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  • $\begingroup$ Could you recommend some literature to read about such developments? $\endgroup$ – მამუკა ჯიბლაძე May 5 '18 at 20:43
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    $\begingroup$ @მამუკაჯიბლაძე The existence of cohomology with compact supports for stacks is discussed here arxiv.org/abs/math/0603680, and this would be sufficient to define the $\ell$-adic realization motivic measure on the Grothendieck ring of stacks. $\endgroup$ – Will Sawin May 6 '18 at 5:46

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