Let $R \rightrightarrows U \to X$ be a presentation of an algebraic space by schemes. Does this induce an exact sequence $|R| \rightrightarrows |U| \to |X|$ on underlying points?

The reason I ask is that this is stated as a lemma in the Stack Project (Lemma 33.4.4). However, the proof uses:

"Since $j = (s,t):R \to U \times U$ is a monomorphism we see that $|R| \to |U|\times|U|$ is injective."

But this can't be true in general. For instance $|U \times U| \to |U|\times|U|$ is not injective when $U$ is the affine line since all generic points in $|U\times U|$ corresponding to dimension 1 curves map to the same pair of generic points. Of course, this equivalence relation is not étale. The question is: Does étaleness prevent such collapses to happen? In that case, how?

Edit: It is pretty clear that $|R| \rightrightarrows |U|$ becomes a pre-equivalence relation with $|U| \to |X|$ as quotient. In fact, I think this is the only property used later in the text, which makes the error unproblematic.

Then the only question is: Is $|R| \to |U| \times_{|X|}|U|$ injective?

scheme$x \times_X x'$ is non-empty (recovers D. Knutson's defn in qcqs case via his "atoms"); of course, have to check thisisan equiv. rel. You ask if $|R| = |U| \times_ {|X|} |U|$. Crux is $R = U \times_ X U$, so $R$ contains non-empty $u \times_ X u'$. QED – BCnrd Apr 18 '10 at 16:15