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The Grothendieck group of varieties $K_0(\textrm{Var}_k)$ over a field $k$ is the Abelian group generated by isomorphism classes of quasi-projective $k$-varieties, subject to the scissor relation (under which $[Y]=[Z]+[Y\setminus Z]$ for every closed subvariety $Z$ of a variety $Y$). It actually is a ring with product given by $[X]\cdot [Y]=[X\times_k Y]$.

I am trying to better understand this ring, and I have some questions on it:

  1. How can one prove that the generating set (of isomorphism classes of quasi-projective $k$-varieties) is actually a set and not a proper class?
  2. Is $[\emptyset]$ in this ring? It seems to be the only reasonable choice for an (additive) identity element. What is the additive inverse $-[Y]$ of a generator $[Y]\in K_0(\textrm{Var}_k)$?
  3. Not every $A\in K_0(\textrm{Var}_k)$ is represented by a variety. How do we write explicitly such an element?

I accept any suggestion on how to think about this ring. Thank you!

Brenin
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1 Answers1

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  1. They can all be seen as subschemes of some $\mathbb{P}^n$, hence of $\coprod_{n \geq 0} \mathbb{P}^n$. More generally, one can show easily that - up to isomorpism - there is only a set of finite type schemes over $k$.
  2. Sure, $\emptyset$ is projective, and yes, $[\emptyset]$ is the zero in the ring (since $[\emptyset]=[\emptyset]+[\emptyset]$ is a scissor relation). The multiplicative unit is $[\mathrm{Spec}(k)]$. The additive inverses are introduced formally in this construction of the Grothendieck ring. If $Y$ is a variety, then (usually) $-[Y]$ is not of the form $[Z]$ for a variety $Z$.
  3. I think that $-[\mathrm{Spec}(k)]$ is not the class of a variety. Otherwise $[X]+[\mathrm{Spec}(k)]=[\emptyset]$ for some $X$, but this can certainly not be deduced by any scissor relation (perhaps I will add a more convincing proof later).
Martin Brandenburg
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  • Can someone add a proof for 3.? I have tried several attempts to define a homomorphism from $K_0(\mathrm{Var}_k)$ to a simpler ring such that the classes of varieties get mapped to a proper subring, but didn't succeed so far. – Martin Brandenburg Jan 25 '14 at 22:21
  • Dear Martin, thanks for your answer. As for 1-2, I am still wondering how to deal with the convention that $\dim \emptyset=-\infty$... Also, for a variety $Z$, how can $-[Z]$ be the class of a variety? Is there an example in which this happens (and $Z$ is nonempty)? – Brenin Jan 26 '14 at 14:50