Confused with local effectiveness of etale equivalence relations Knutson's Algebraic Spaces book have the following two theorems (we assume that $R\rightarrow U\times U$ is quasi-compact in the following Props):


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*Let $R\rightarrow U\times U$ be an etale equivalence relation where both map $R\rightarrow U\times U\rightarrow U$ are finite (we take the projections here) and $U$ is affine. Then the equivalence relation is effective.

*Let $U_I$ be a Zariski open covering of $U$ and for each $i\in I$ let $R_i\rightarrow U_i\times U_i$ be the induced equivalence relation. Then if each $R_i\rightarrow U_i\times U_i$ is effective, so is $R\rightarrow U\times U$.
This seems to imply that if $R\rightarrow U\times U\rightarrow U$ is finite then it is effective. Is this true?
 A: No, it is not true.
The reason is that you can't always apply (2) to reduce to the situation of (1).
The basic example is as follows: if $G$ is a finite group acting freely on a variety $X$ (over $\mathbb C$, say), and if the orbit of each (closed) point is 
contained in an affine open subvariety, then the quotient variety $X/G$ exists.
But if the condition on orbits doesn't hold, then the quotient variety
won't exist (and there is a famous example of Hironaka illustrating this).

In terms of etale equivalence relations, we are considering the equivalence relation $G \times X \hookrightarrow X \times X$, where $(g,x) \mapsto (x,gx)$.
The argument you have in mind is that you want to reduce to the situation of (1) to get an effective equiv. rel'n, by applying (2).  But to apply (2) in this context, amounts to being able to find an affine open cover $U_i$ of $X$ such that each $U_i$ is stable under the equivalence relation.  (In particular, I think your phrasing of (2) is a little off, and this may be a source of the confusion: in general, if we take a random open cover of $X$, the equivalence relation $R$ won't restrict to an equivalence relation on each $U_i$, because the individual $U_i$ won't contain the full equivalence class of each of their points; so to be able to form $R_i$ as an equivalence relation on $U_i$, you need the additional hypothesis that each $U_i$ is indeed closed under formation of equivalence relations, or equivalently, that the pull-back of $R$ from $U \times U$ to $U_i \times U$ actually lies in $U_i \times U_i$, so that it defines an equivalence relation $R_i$ on $U_i$.)

If $X$ is a quasi-projective variety, then any finite set is contained in an affine open subset, and by a little extra argument, one can show that actually each $G$-orbit is contained in a $G$-invariant affine open subset of $X$.  Thus (2) applies, and we may form $X/G$.  (I've discussed this here, for example). 
But there are non-quasi-projective varieties containing finite sets that don't lie in any single affine open, and then the above argument does not apply.  Indeed, as I mentioned, there is an example due to Hironaka of a group $G$ of order $2$ acting on a proper (non-projective variety) $X$ such that $X/G$ does not exist as a variety.
Indeed, this was one of Artin's motivations for introducing algebraic spaces: Hironaka's example shows that the Hilbert "scheme" of $X$ need not exist as a scheme when $X$ is not projective, but Artin showed that it does exist as an algebraic space.
