Nisnevich covers and distinguished squares

PDF p. 4: Definition 2.2; PDF p. 5: Nisnevich refinement

For a point $x$ of a scheme $X$, write $\kappa (x)$ for the residue field of the local ring ${\mathcal{O}}_{X,x}$. A family of scheme morphisms $\{f_i:U_i\to X{\}}_{i\in I}$ is a Nisnevich covering family if every $f_i$ is étale in the sense of Étale morphism of schemes and, for every point $x\in X$, there are an index $i$ and a point $u\in U_i$ with $f_i(u)=x$ such that the induced field homomorphism $\kappa (x)\to \kappa (u)$ is an isomorphism. In this situation $u$ is called a residue-field-preserving lift of $x$.

On a specified small category of schemes in which these families are available, the Nisnevich topology is the Grothendieck topology generated by the sieves containing such families. A sieve and a Grothendieck topology have the meanings in Scheme sites and sheafification. For the small Nisnevich site of a scheme $X$, take a suitably small category of étale $X$-schemes, closed under the relevant fibre products, with this topology; the big site requires a separately specified category of schemes over the base.

A distinguished Nisnevich square is a cartesian square of schemes

$\begin{array}{ccc}W& ⟶& V\\ \downarrow & & \downarrow p\\ U& \stackrel{j}{\to }& X\end{array}$

where $j$ is an open immersion, $p$ is étale, and, putting $Z=(X\setminus j(U){)}_{\mathrm{r}\mathrm{e}\mathrm{d}}$ for the closed complement with its reduced induced scheme structure, the base change $V{\times }_{X}Z\to Z$ is an isomorphism. Cartesian means that the displayed map $W\to U{\times }_{X}V$ is an isomorphism. The pair $\{U\to X,V\to X\}$ is its associated covering family.

For a set-valued presheaf $F$, the distinguished-square excision conditions are that $F(\mathrm{\varnothing })$ is a singleton and that, for every such square in the chosen category, the natural map

$F(X)⟶F(U){\times }_{F(W)}F(V)$

is a bijection. Here the fibre product is the set of pairs whose restrictions to $W$ agree. These conditions describe the excision interface; no equivalence between them and the full sheaf condition is asserted here.

The residue-field test is independent of a choice of representative of a point: a scheme morphism gives the indicated homomorphism of residue fields. The generated topology exists because the maximal topology is among the Grothendieck topologies containing the specified sieves, and their intersection is again a Grothendieck topology; the smallness convention makes this intersection legitimate.

For a distinguished square, the open immersion $U\to X$ is étale, as recorded in Étale morphism of schemes, and $V\to X$ is étale by hypothesis. If $x$ lies in $U$, its unique preimage under the open immersion has residue field $\kappa (x)$. Otherwise $x$ is a point of $Z$; the isomorphism $V{\times }_{X}Z\cong Z$ gives a point over $x$ with residue field $\kappa (x)$, whose image in $V$ is a residue-field-preserving lift. Thus the associated pair is a Nisnevich covering family. The empty family covers the empty scheme by the same pointwise definition. The restriction maps of the presheaf commute around the square, so they define the displayed natural map to the fibre product.