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?

  • 4
    $\begingroup$ It's all ok. Idea: for schemes, $|X' \times_X X"| \rightarrow |X'| \times_ {|X|} |X"|$ is surjective (because $k' \otimes_k k" \ne 0$ for fields), which is why it "works" for etale cover $U$ of a scheme $X$. For general alg. space $X$ define $|X|$ via equiv. rel. on field-valued pts $x$, $x'$ that the scheme $x \times_X x'$ is non-empty (recovers D. Knutson's defn in qcqs case via his "atoms"); of course, have to check this is an 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 $\endgroup$
    – BCnrd
    Apr 18, 2010 at 16:15
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    $\begingroup$ Daniel, just to be clear, I'm saying the assertion of the lemma is ok, not that the proof given is ok. (As you point out, the argument is wrong. You should email deJong to point out the need to revise the argument.) $\endgroup$
    – BCnrd
    Apr 18, 2010 at 16:21
  • $\begingroup$ Sorry, but I don't think I follow your reasoning at all. I agree on your definition of a point, but I don't see how $|U\times_X U| \ to $|U|\times_{|X|}|U|$ becomes injective. I edit my question to clarify. $\endgroup$ Apr 20, 2010 at 13:12
  • 2
    $\begingroup$ No, $|R| \to |U| \times_{|X|}|U|$ is not injective. Take $X = \mathop{\rm Spec} \mathbb R$, $U = \mathop{\rm Spec} \mathbb C$. $\endgroup$
    – Angelo
    Apr 20, 2010 at 14:47


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