1 The Nisnevich topology
Nisnevich localization is relatively standard. This is obtained by defining the Nisnevich topology on , and then imposing the usual sheaf condition. Consequently, the Nisnevich localization functor is a standard sheafification functor, and automatically enjoys the nice formal properties of sheafification. For example, it is an exact functor.
Let . A Nisnevich cover of is a finite family of étale morphisms such that there is a filtration
of by finitely presented closed subschemes such that for each strata , there is some such that
admits a section.
Any Zariski cover is a Nisnevich cover. Any Nisnevich cover is an étale cover.
Let be a field of characteristic not , and . Consider the covering
This forms a Nisnevich cover with the filtration iff .
It turns out to check that something is a Nisnevich sheaf, it suffices to check it for very particular covers with two opens.
An elementary distinguished square is a pullback diagram
of -schemes in such that is a Zariski open immersion, is étale, and is an isomorphism of schemes, where is equipped with the reduced induced scheme structure.
We define to be the full subcategory of consisting of presheaves that satisfy descent with respect to Nisnevich covers. Such presheaves are also said to be Nisnevich local. This is an accessible subcategory of and admits a localization functor .
Every representable functor satisfies Nisnevich descent, since they in fact satisfy étale descent. Note that these functors are valued in discrete spaces.
Then is Nisnevich local iff and for every elementary distinguished square
the induced diagram
is a pullback diagram.
is an elementary distinguished square, then, when considered a square , this is a pushout diagram.
In particular, this holds when is a Zariski cover.