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Nisnevich site

Contents

Idea

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 topology plays a central rôle in motivic homotopy theory.

Definition

An family of morphisms of Noetherian schemes {p i:V iU}\{p_i:V_i\to U\} is a Nisnevich cover if each p ip_i is an étale map and if every field-valued point SpeckUSpec k\to U lifts to one of the V iV_i. 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 SS usually refers to the site given by the category of smooth schemes of finite type over SS equipped with the Nisnevich topology. The small Nisnevich site of SS is the subsite consisting of étale SS-schemes.

For non-Noetherian schemes

For a general affine scheme XX, one defines a sieve SS on XX to be a covering sieve for the Nisnevich topology if there exist a Noetherian affine scheme YY, a morphism f:XYf: X\to Y, and a Nisnevich covering sieve TT on YY such that f *(T)Sf^\ast(T)\subset S. On an arbitrary scheme XX, a sieve SS is a Nisnevich covering sieve if there exists an open cover {U iX}\{U_i\to X\} by affine schemes such that S /U iS_{/U_i} is a Nisnevich covering sieve on U iU_i for all ii.

As an excision property

Let Et/SEt/S be the category of étale schemes of finite type over a Noetherian scheme SS. An (∞,1)-presheaf FF on Et/SEt/S is said to satisfy Nisnevich excision if the following conditions hold:

  • F()F(\emptyset) is contractible.
  • If ZZ is a closed subscheme of XEt/SX\in Et/S and if XXX'\to X is a morphism in Et/SEt/S which is an isomorphism over ZZ, then the square
F(X) F(XZ) F(X) F(XZ) \array{ F(X) &\to& F(X-Z) \\ \downarrow && \downarrow \\ F(X') &\to& F(X'-Z) }

is an (∞,1)-pullback square. Intuitively, this says that the space of sections of FF over XX with support in ZZ (i.e., the homotopy fiber of F(X)F(XZ)F(X) \to F(X-Z)) does not depend on XX. This is Definition 2.5 in DAG XI.

Proposition

An (∞,1)-presheaf on Et/SEt/S 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.

Properties

Homotopy dimension

Proposition

If SS is a Noetherian scheme of finite Krull dimension?, then the (∞,1)-topos of (∞,1)-sheaves on the small Nisnevich site of SS has homotopy dimension dim(S)\leq\dim(S).

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 SS, and in particular that (∞,1)-topos is hypercomplete.

fpqc-site \to fppf-site \to syntomic site \to étale site \to Nisnevich site \to Zariski site

References

A quick overview is at the beginning of the talk slides

  • Jardine, Motivic spaces and the motivic stable category (pdf) .

A detailed discussion is in section 3.1.1 of

or in the lecture notes

A self-contained account of the Nisnevich (,1)(\infty,1)-topos including the non-Noetherian case is in

Revised on September 7, 2014 07:17:42 by Urs Schreiber (141.0.8.158)