Zariski-local framed $\mathbb{A}^1$-homotopy theory

Abstract

For any (not necessarily perfect) field $k$ we obtain equivalences of $\infty$-categories \[\mathbf{H}^{\mathrm{fr},\mathrm{gp}}(k)\simeq \mathbf{H}^{\mathrm{fr},\mathrm{gp}}_{\mathrm{zf}}(k) \text{ and } \mathbf{DM}(k)\simeq\mathbf{DM}_{\mathrm{zar}}(k).\] We also construct an equivalence of $\infty$-categories \[ \mathbf{H}^{\mathrm{fr},\mathrm{gp}}(S) \simeq \mathbf{H}^{\mathrm{fr},\mathrm{gp}}_{\mathrm{zf}}(S) \] of group-like framed motivic spaces over a separated noetherian scheme $S$ of finite Krull dimension with respect to the Nisnevich topology at one side and the Zariski fibre topology $\mathrm{zf}$ generated by the Zariski one and the trivial fibre topology (introduced by Druzhinin, Kolderup and Østvær) on the other side. Over a field, the Zariski fibre topology equals the Zariski topology and the result follows from the previous one. To prove it in the case of a general base scheme, we prove a localisation theorem for $\mathbf{H}^{\mathrm{fr},\mathrm{gp}}_{\mathrm{zf}}(-)$ employing the ideas from the proof of the {\it affine localisation theorem} for the trivial fibre topology by the first author, Kolderup and Østvær.