MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 To├źn 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?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.