Nisnevich Topology - Definition

Definition

A morphism of schemes f : YX is called a Nisnevich morphism if it is an étale morphism such that for every (possibly non-closed) point xX, there exists a point yY 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 xX, 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 : XαX} is a Nisnevich cover if each morphism in the family is étale and for every (possibly non-closed) point xX, there exists α and a point yXα 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 UX 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 topology allows proper birational morphisms as coverings.
  • The h topology allows De Jong's alterations as coverings.
  • The l′ topology allows morphisms as in the conclusion of Gabber's local uniformization theorem.

The cdh and l′ topologies are incomparable with the étale topology, and the h topology is finer than the étale topology.

Read more about this topic:  Nisnevich Topology

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