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 descent over the Nisnevich site – as is true for the Zariski site but not in full generality for the etale site, see at algebraic K-theory – Descent for more;
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)
Pushforward along a finite morphism? is an exact functor on Nisnevich sheaves of abelian groups (like étale)
Nisnevich cohomology can be computed using ?ech 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 .
Let be the category of étale schemes of finite presentation over a quasi-compact quasi-separated scheme . An (∞,1)-presheaf on is said to satisfy Nisnevich excision if the following conditions hold:
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.
If is a Noetherian scheme of finite Krull dimension, then the (∞,1)-topos of (∞,1)-sheaves on the small Nisnevich site of has homotopy dimension .
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.
More generally, if is a pro-algebraic space limit of a cofiltered diagram of qcqs algebraic spaces of Krull dimension , then the (∞,1)-topos of (∞,1)-sheaves on the small Nisnevich site of has homotopy dimension .
This is Clausen and A. Mathew, Hyperdescent and etale K-theory, 2019, arXiv:1905.06611, Cor.3.11, Thm.3.12, and Thm.3.17.
fpqc-site fppf-site syntomic site étale site Nisnevich site Zariski site
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
Last revised on February 18, 2021 at 09:22:23. See the history of this page for a list of all contributions to it.