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:
- The Nisnevich cohomological dimension (and even the homotopy dimension) of a scheme is bounded by its Krull dimension? (like Zariski)
- Fields have trivial shape for the Nisnevich topology (like Zariski)
- Algebraic K-theory satisfies Nisnevich descent (like Zariski)
- Any smooth closed pair (i.e., is a closed subscheme of and and are both smooth over a common base scheme) is Nisnevich-locally isomorphic to (like étale)
- Pushforward along a finite morphism? is an exact functor on Nisnevich sheaves of abelian groups (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 non-Noetherian 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 .
As an excision property
Let be the category of étale schemes of finite type over a Noetherian scheme . An (∞,1)-presheaf on is said to satisfy Nisnevich excision if the following conditions hold:
- is contractible.
- If is a closed subscheme of and if is a morphism in which is an isomorphism over , then the square
is an (∞,1)-pullback square. Intuitively, this says that the space of sections of over with support in (i.e., the homotopy fiber of ) does not depend on . This is Definition 2.5 in DAG XI.
An (∞,1)-presheaf on is an (∞,1)-sheaf for the Nisnevich topology if and only if it satisifes Nisnevich excision.
This is Morel-Voevosky, Prop. 1.16 or DAG XI, Thm. 2.9.
This is DAG XI, Theorem 2.24. As a consequence, Postnikov towers are convergent in the (∞,1)-topos of (∞,1)-sheaves on the Nisnevich site over , and in particular that (∞,1)-topos is hypercomplete.
fpqc-site fppf-site syntomic site étale site Nisnevich site Zariski site
A quick overview is at the beginning of the talk slides
- Jardine, Motivic spaces and the motivic stable category (pdf) .
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