The Grothendieck ring of varieties is defined to be the free abelian group spanned by isomorphism classes of varieties modulo the cut & paste (or scissor) relations, which say that $[X] = [U] + [Y]$, for any closed $Y \subset X$ and where $U$ is the open complement.

Now, we can equally speak of a Grothendieck ring of schemes, or algebraic spaces, and they all turn out to be isomorphic via the obvious inclusions. (for the scheme case one sees this by noticing that $[X] = [X_\text{red}] - 0$) There is also a Grothendieck ring of Artin stacks (everything here is of finite type over a field of characteristic zero, possibly algebraically closed, and with affine diagonal). This ring turns out to be a localisation of the previous one.

My question is: what happens if we include higher stacks?

I think there is work by Toën on the Grothendieck ring for derived stacks, but here I'm only asking about higher (underived) stacks. Do we have a similar phenomenon to the case of varieties VS schemes? In the sense that $[X] = [\pi_0(X)] - 0$, where $\pi_0$ of a derived stack is its underived truncation?

  • $\begingroup$ are there any interesting phenomena you expect to happen? or just for fun? $\endgroup$
    – user138661
    May 20, 2019 at 12:26


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