10
$\begingroup$

Consider the category of reduced schemes of finite type over $\mathbb{Z}$. Take the Grothendieck group of this category, i.e. the free abelian group on isomorphism classes, modulo the usual "syzygy" relations given by closed-open complements. This is a ring under the multiplication induced by taking fiber product of schemes.

Consider also the ring of "point-counting functions", i.e. the ring whose elements are functions from the set of prime powers to the set of integers.

There is an obvious ring homomorphism from the Grothendieck ring to the ring of point-counting functions. Question: What is known about the kernel of this map? Are there any conjectures describing this kernel precisely?

My guess is that if you restrict to mixed Tate motives and to polynomial functions, the map is known to be injective. Is there a precise statement of this guess somewhere in the literature?

Any references or related comments are welcome.

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ Question: Let $X$ be a scheme with non-trivial Brauer group, so there are maps $E \to X$ which are $\mathbb{P}^r$ bundles in the etale topology but not in the Zariski topology. Then I think $\# E(\mathbb{F}_q) = \# \left( X \times \mathbb{P}^r \right)(\mathbb{F}_q)$. Are the two sides also equal in the Grothendieck ring? $\endgroup$
    – David E Speyer
    Commented Feb 16, 2015 at 23:00
  • 4
    $\begingroup$ That said, the kernel is certainly non-empty. Take two isogenous elliptic curves over $\mathcal{O}_K[1/N]$, where $K$ is some number field. Then their difference will map to zero. But $[E_1]\neq [E_2]$ because their generic fibers are not stably birational (here I am using a result of Larsen and Lunts that the Grothendieck ring distinguishes non-stably birational varieties in characteristic zero). $\endgroup$
    – Daniel Litt
    Commented Feb 17, 2015 at 2:20

0

Reset to default

You must log in to answer this question.