5
$\begingroup$

$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Spec{Spec}$I am looking for a comprehensive reference on the theory of the Grothendieck ring of varieties over a field $K$, denoted $K_0(K)$ here, that includes answers to such basic questions as:

(1) If L/K is a Galois extension of fields, then $\Gal(L/K)$ acts on $K_0(L)$. Is "$\Spec(K_0(L))//\Gal(L/K) \simeq \Spec(K_0(L))$" as stacks?

(2) What are the prime ideals of $K_0(K)$?

(3) If $R$ is an integral domain, $K$ its field of fractions, and $k$ the reduction of $R$ modulo a prime ideal, what is the relation between $K_0(K)$ and $K_0(k)$? Is there a nice fibration?

Improve this question
$\endgroup$
2
  • 2
    $\begingroup$ (2) is presumably very hard, because it is equivalent to "what are all additive invariants valued in a domain". For (3), have a look at this paper (which answers the question in characteristic $0$ for a certain widely studied quotient of $K_0$). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jun 4, 2020 at 22:39
  • $\begingroup$ (1) is a bit strange: RHS is a scheme, so why do you ask for isomorphism of stacks? The natural question is whether $K_0(L)^G$ is isomorphic to $K_0(K)$. $\endgroup$
    – Evgeny Shinder
    Commented Jun 5, 2020 at 21:58

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .