Schematic Functorialities of Birational Motivic Homotopy Categories

Abstract

We promote the $n$-birational motivic homotopy category assignment $S\mapsto {\mathcal{H}}^n(S)$ to a $P{r}^L$-valued presheaf on $Corr(\mathrm{Sch}{)}_{uglt,sm}$. As a consequence, the birational motivic homotopy category ${\mathcal{H}}^{b{\mathbb{A}}^1}(X)$ of a scheme $X$ with finitely many generic points decomposes as the cartesian product of the birational motivic homotopy categories of those generic points; in particular, for a variety $V$, ${\mathcal{H}}^{b{\mathbb{A}}^1}(V)\simeq {\mathcal{H}}^{b{\mathbb{A}}^1}(k(V))$. This implies that birational equivalences of schemes in $S{m}_X$ can be detected via the birational contractibility of their generic fibers. Finally, we show that stably birational morphisms and purely transcendental field extensions induce fully faithful embeddings of birational motivic homotopy categories.