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In [[algebraic geometry]], the '''Nisnevich topology''', sometimes called the '''completely decomposed topology''', is a [[Grothendieck topology]] on the category of [[scheme (mathematics)|scheme]]s which has been used in [[algebraic K-theory]], [[A¹ homotopy theory]], and the theory of [[motive (algebraic geometry)|motive]]s.  It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of [[adele ring|adeles]].


== Definition ==
A morphism of schemes {{nowrap|''f'' : ''Y'' &rarr; ''X''}} is called a '''Nisnevich morphism''' if it is an [[étale morphism]] such that for every (possibly non-closed) point ''x'' ∈ ''X'', there exists a point ''y'' ∈ ''Y'' such that the induced map of [[residue field]]s ''k''(''x'') → ''k''(''y'') is an isomorphism.  Equivalently, ''f'' must be flat, unramified,  locally of finite presentation, and for every point ''x'' ∈ ''X'', there must exist a point ''y'' in the fiber {{nowrap|''f''<sup>&minus;1</sup>(''x'')}} such that ''k''(''x'') → ''k''(''y'') is an isomorphism.


A family of morphisms {''u''<sub>α</sub> : ''X''<sub>α</sub> → ''X''} is a '''Nisnevich cover''' if each morphism in the family is étale and for every (possibly non-closed) point ''x'' ∈ ''X'', there exists ''α'' and a point ''y'' ∈ ''X''<sub>α</sub> s.t. ''u''<sub>α</sub>(''y'') = ''x'' and the induced map of [[residue field]]s ''k''(''x'') → ''k''(''y'') is an isomorphism. If the family is finite, this is equivalent to the morphism <math>\coprod u_\alpha</math> from <math>\coprod X_\alpha</math> to ''X'' being a Nisnevich morphism. The Nisnevich covers are the covering families of a pretopology on the category of schemes and morphisms of schemes. This generates a topology called the '''Nisnevich topology'''. The category of schemes with the Nisnevich topology is notated ''Nis''.
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The '''small Nisnevich site of '''''X''''' ''' has as underlying category the same as the small étale site, that is to say, objects are schemes ''U'' with a fixed étale morphism ''U'' → ''X'' and the morphisms are morphisms of schemes compatible with the fixed maps to ''X''. Admissible coverings are Nisnevich morphisms.
 
The '''big Nisnevich site of '''''X''''' ''' has  as underlying category  schemes with a fixed map to ''X'' and morphisms the morphisms of ''X''-schemes. The topology is the one given by Nisnevich morphisms.
 
The Nisnevich topology has several variants which are adapted to studying singular varieties. Covers in these topologies include [[resolution of singularities|resolutions of singularities]] or weaker forms of resolution.
* The '''cdh topology''' allows proper birational morphisms as coverings.
* The '''h topology''' allows De Jong's alterations as coverings.
* The '''l&prime; topology''' allows morphisms as in the conclusion of Gabber's local uniformization theorem.
The cdh and l&prime; topologies are incomparable with the [[étale topology]], and the h topology is finer than the étale topology.
 
== Local rings in the Nisnevich topology ==
{{seealso|Henselian ring}}
 
If ''x'' is a point of a scheme ''X'', then the local ring of ''x'' in the Nisnevich topology is the [[henselization]] of the local ring of ''x'' in the Zariski topology.
 
== Applications ==
Nisnevich introduced his topology to provide a cohomological interpretation of the class set of an affine group scheme, which was originally defined in adelic terms. He used it to partially prove a conjecture of [[Alexander Grothendieck]] and [[Jean-Pierre Serre]] which states that a rationally trivial [[torsor]] under a reductive group scheme over an integral regular Noetherian base scheme is locally trivial in the [[Zariski topology]]. One of the key properties of the Nisnevich topology is the existence of a descent [[spectral sequence]].  Let ''X'' be a Noetherian scheme of finite Krull dimension. Let ''G''<sub>''n''</sub>(''X'') denote the Quillen K-groups of the category of coherent sheaves on ''X'', and let <math>\tilde G_n^{\,\text{cd}}(X)</math> denote the sheavings of these groups with respect to the Nisnevich topology. Then there is a convergent spectral sequence
:<math>E^{p,q}_2 = H^p(X_\text{cd}, \tilde G_q^{\,\text{cd}}) \Rightarrow G_{q-p}(X)</math>
for {{nowrap|p &ge; 0}}, {{nowrap|q &ge; 0}}, and {{nowrap|p - q &ge; 0}}. If <math>\ell</math> is a prime number not equal to the characteristic of ''X'', then there is an analogous convergent spectral sequence for K-groups with coefficients in <math>\mathbf{Z}/\ell\mathbf{Z}</math>.
 
The Nisnevich topology has also found important applications in [[algebraic K-theory]], [[A¹ homotopy theory]] and the theory of [[motive (algebraic geometry)|motive]]s.
 
== References ==
* {{cite conference
| first = Yevsey A.
| last = Nisnevich
| year = 1989
| title = The completely decomposed topology on schemes and associated descent spectral sequences in algebraic K-theory
| booktitle = Algebraic K-theory: connections with geometry and topology. Proceedings of the NATO Advanced Study Institute held in Lake Louise, Alberta, December 7--11, 1987. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, 279
| editor = J. F. Jardine and V. P. Snaith
| publisher = Kluwer Academic Publishers Group
| location = Dordrecht
| pages = 241–342
}}, available at [http://www.cims.nyu.edu/~nisnevic/ Nisnevich's website]
 
[[Category:Algebraic geometry]]
[[Category:Topos theory]]

Latest revision as of 00:00, 29 September 2014


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