# Nisnevich covers of algebraic spaces

Does every algebraic space have a Nisnevich cover by a scheme?

(Assume that the algebraic space is quasi-separated, quasi-compact and over a scheme $S$.)

Background:

Every algebraic space has an etale cover by a scheme by definition: an algebraic space is a sheaf $$F : (Sch/S)_{fppf} \to (Set)$$ with representable diagonal $\Delta : F \to F \times F$ and an etale, surjective morphism $U \to F$ from a sheaf associated to an $S$-scheme $U$.

A Nisnevich cover in this context is a surjective, etale morphism $f : U \to X$ such that for all $x \in X$, there exists a $u \in U$ with $f(u) = x$ such that the induced morphism $k(x) \to k(u)$ is an isomorphism.

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Yes, if you restrict to quasi-separated algebraic spaces (to avoid pathologies). Donald Knutson proved that for each $x \in X$ there is an etale scheme neighborhood $(U,u) \rightarrow (X,x)$ with $k(u) = k(x)$. (This underlies how $k(x)$ is defined.) So with a gigantic disjoint union you get a huge Nisnevich scheme cover. If $X$ is also quasi-compact then it has a dense open subscheme $V$ (also proved by Knutson) and by noetherian induction you get a quasi-compact Nisnevich scheme cover applying the previous Knutson theorem at the finitely many generic points of $X-V$. – user30180 Jun 8 '13 at 3:29
In the preceding comment, to carry out noetherian induction it is of course necessary to assume $X$ is noetherian. But the general qcqs case can be reduced to the noetherian case because any qcqs algebraic space is affine over a noetherian algebraic space (by a variant of the result of Thomason & Trobaugh according to which any qcqs scheme is affine over a noetherian scheme). – user30180 Jun 8 '13 at 5:14
A reference for what ayanta explains above is [Knutson, Algebraic spaces, Chapter 2, Theorem 6.4]. – Olivier Benoist Jun 8 '13 at 13:30
Ayanta, thanks for the thorough explanation. If you post it as an answer, I can accept it. I should have specified quasi-separated in the definition, and quasi-compact is a reasonable hypothesis. I should have also clarified what you assumed about etale meaning locally of finite presentation. A definition of $k(x)$ in terms of a universal factorization of $Spec K \to X$ (for $X$ locally noetherian) can be found in Laumon,Moret-Bailly, Definition 11.2. Maybe it uses the result you cite...I need Knuston's book. Olivier, I don't have the book in front of me, but thanks for the reference. – expz Jun 10 '13 at 10:55