Nisnevich topology

In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.
Definition.
A morphism of schemes formula_1 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" in the fiber such that the induced map of residue fields "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 such that "k"("x") → "k"("y") is an isomorphism.
A family of morphisms {"u"α : "X"α → "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"α s.t. "u"α("y") = "x" and the induced map of residue fields "k"("x") → "k"("y") is an isomorphism. If the family is finite, this is equivalent to the morphism formula_2 from formula_3 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".
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 resolutions of singularities or weaker forms of resolution.
The cdh and l′ topologies are incomparable with the étale topology, and the h topology is finer than the étale topology.
Equivalent conditions for a Nisnevich cover.
Another equivalent conditionpg 21 for a family of morphisms formula_4 of schemes to be a Nisnevich covering is if
There is an alternative characterization using finite sequences of finitely presented closed subschemesformula_8where the second condition is replaced with the hypothesis thatformula_9admits a section. Notice that when evaluating these morphisms on formula_10-points, this implies the map is a surjection. Conversely, taking the trivial sequence formula_11 gives the result in the opposite direction.
Motivation.
One of the key motivations for introducing the Nisnevich topology in motivic cohomology is the fact that a Zariski open cover formula_12 does not yield a resolution of Zariski sheavesformula_13whereformula_14is the representable functor over the category of presheaves with transfers. For the Nisnevich topology, the local rings are Henselian, and a finite cover of a Henselian ring is given by a product of Henselian rings, showing exactness.
Local rings in the Nisnevich topology.
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. This differs from the Etale topology where the local rings are "strict" henselizations. One of the important points between the two cases can be seen when looking at a local ring formula_15 with residue field formula_16. In this case, the residue fields of the Henselization and strict Henselization differformula_17so the residue field of the strict Henselization gives the separable closure of the original residue field formula_16.
Examples of Nisnevich Covering.
Consider the étale cover given by
If we look at the associated morphism of residue fields for the generic point of the base, we see that this is a degree 2 extension
This implies that this étale cover is not Nisnevich. We can add the étale morphism formula_21 to get a Nisnevich cover since there is an isomorphism of points for the generic point of formula_22.
Conditional covering.
If we take formula_23 as a scheme over a field formula_7, then a coveringpg 21 given byformula_25where formula_26 is the inclusion and formula_27, then this covering is Nisnevich if and only if formula_28 has a solution over formula_7. Otherwise, the covering cannot be a surjection on formula_7-points. In this case, the covering is only an Etale covering.
Zariski coverings.
Every Zariski coveringpg 21 is Nisnevich but the converse doesn't hold in general. This can be easily proven using any of the definitions since the residue fields will always be an isomorphism regardless of the Zariski cover, and by definition a Zariski cover will give a surjection on points. In addition, Zariski inclusions are always Etale morphisms.
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, and let "G""n"("X") be the Quillen K-groups of the category of coherent sheaves on "X". If formula_31 is the sheafification of these groups with respect to the Nisnevich topology, there is a convergent spectral sequence
for , , and . If formula_33 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 formula_34.
The Nisnevich topology has also found important applications in algebraic K-theory, A¹ homotopy theory and the theory of motives.