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This is a more direct version of this question, which was perhaps a bit obtuse. This is a more elementary formulation.

Recall that for a groupoid scheme (or indeed any internal groupoid) $X = (X_1 \rightrightarrows X_0)$ and a map $Y \to X_0$, on can define a new groupoid scheme $X[Y]$ with objects $Y$ and arrows $Y\times_{X_0} X_1 \times_{X_0} Y$. This latter scheme exists (unless I'm horribly mistaken), but we need to check whether the groupoid $X[Y]$ has the same properties as $X$. For example, consider the following properties of $X$ (where I am pulling adjective out of a hat that I have heard of - I am not an algebraic geometer):

  1. $(s,t):X_1 \to X_0 \times X_0$ is quasi-separated, separated, quasi-compact, flat or any other adjective you can think of. More generally we say property 1 holds for $X$ (for a class $P$ of maps).

  2. $s$ and $t$ are etale, smooth, unramified, flat, finite fibres, have local sections over a cover from any of the usual Grothendieck topologies. More generally we say property 2 holds for $X$ (for a given class of maps).

Now clearly if $P$ is a property of a map that is stable under pullback, then if property 1 holds for $X$ then it holds for $X[Y]$ by definition. But what about property 2? I know that in the analogous case of topological groupoids, if $P$ is 'local homeomorphisms', then property 2 holding for $X$ implies it for $X[Y]$, but I don't know enough alg. geom. to claim this for etale maps and groupoid schemes.

In general, for what classes of maps of schemes do we know that property 2 holding for $X$ implies it for $X[Y]$?

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I don't think that 2) is as well behaved, even in the smooth setting. For example, given any orbifold, one can find a Lie groupoid $\G$ presenting it such that the source and target maps are proper, however, this property is not stable under Morita-equivalence. In general, one can only claim that the map in 1) is proper. – David Carchedi Nov 29 '10 at 2:11
It is not as well behaved, as you say, but I'm sure there are non-trivial examples. Given an atlas $p:X \to \mathcal{X}$ of a stack $\mathcal{X}$ (algebraic or topological or otherwise), where $p$ is in some class $P$, then $s,t:X \times_{\mathcal{X}} X \to X$ are in class $P$, and this should have some sort of stability under forming the induced groupoid. – David Roberts Nov 29 '10 at 2:48

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