Nisnevich equivalences of local essentially smooth open pairs

Abstract

In this note, we prove a Nisnevich local equivalence \begin{equation*} X/(X-Z)\simeq X^\prime/(X^\prime-Z^\prime) \end{equation*} for essentially smooth local schemes $X$ and $X^\prime$ of the same dimension over a base scheme $B$, and arbitrary isomorphic closed subschemes $Z$ and $Z^\prime$, with immediate applications in motivic homotopy theory.