Graphs, ${\mathbb{F}}_1$-schemes and virtual mixed Tate motives

Abstract

In a number of recent works [6, 7] the authors have introduced and studied a functor ${\mathcal{F}}_k$ which associates to each loose graph $\Gamma$ -which is similar to a graph, but where edges with $0$ or $1$ vertex are allowed - a $k$-scheme, such that ${\mathcal{F}}_k(\Gamma )$ is largely controlled by the combinatorics of $\Gamma$. Here, $k$ is a field, and we allow $k$ to be ${\mathbb{F}}_1$, the field with one element. For each finite prime field ${\mathbb{F}}_p$, it is noted in [6] that any ${\mathcal{F}}_k(\Gamma )$ is polynomial-count, and the polynomial is independent of the choice of the field. In this note, we show that for each $k$, the class of ${\mathcal{F}}_k(\Gamma )$ in the Grothendieck ring $K_0({\text{Sch}}_k)$ is contained in $\mathbb{Z}[\mathbb{L}]$, the integral subring generated by the virtual Lefschetz motive.

8 pages. arXiv admin note: text overlap with arXiv:1607.03814

References in corpus (1)

• The Weyl functor - Introduction to Absolute Arithmetic