The Nisnevich topology is a certain Grothendieck topology on the category of schemes which is finer than the Zariski topology but coarser than the étale topology. It retains many desirable properties from both topologies:
For a closed immersion between affine schemes that are smooth over a base , the Nisnevich sheaf is isomorphic to , where is the normal bundle of in (like étale)
Nisnevich cohomology can be computed using Čech cohomology (like étale)
The Nisnevich topology plays a central rôle in motivic homotopy theory.
An family of morphisms of Noetherian schemes is a Nisnevich cover if each is an étale map and if every field-valued point lifts to one of the . This is a pretopology on the category of Noetherian schemes, and the associated topology is the Nisnevich topology.
The Nisnevich site over a Noetherian scheme usually refers to the site given by the category of smooth schemes of finite type over equipped with the Nisnevich topology. The small Nisnevich site of is the subsite consisting of étale -schemes.
For a general affine scheme , one defines a sieve on to be a covering sieve for the Nisnevich topology if there exist a Noetherian affine scheme , a morphism , and a Nisnevich covering sieve on such that . On an arbitrary scheme , a sieve is a Nisnevich covering sieve if there exists an open cover by affine schemes such that is a Nisnevich covering sieve on for all .
A quick overview is at the beginning of the talk slides
A detailed discussion is in section 3.1.1 of
or in the lecture notes
A self-contained account of the Nisnevich -topos including the non-Noetherian case is in