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. A morphism of schemes 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 f−1(x) 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 f−1(x) such that k(x) → k(y) is an isomorphism. A family of morphisms {uα : Xα → 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 ∈ Xα s.t. uα(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 from 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.