The Stacks project

Remark 79.12.12. The condition that $f$ be separated cannot be dropped from Proposition 79.12.11. An example is to take $X$ the affine line with zero doubled, see Schemes, Example 26.14.3, $Y = \mathbf{A}^1_ k$ the affine line, and $X \to Y$ the obvious map. Recall that over $0 \in Y$ there are two points $0_1$ and $0_2$ in $X$. Thus $(X/Y)_{fin}$ has four points over $0$, namely $\emptyset , \{ 0_1\} , \{ 0_2\} , \{ 0_1, 0_2\} $. Of these four points only three can be lifted to an open subscheme of $U \times _ Y X$ finite over $U$ for $U \to Y$ étale, namely $\emptyset , \{ 0_1\} , \{ 0_2\} $. This shows that $(X/Y)_{fin}$ if representable by an algebraic space is not étale over $Y$. Similar arguments show that $(X/Y)_{fin}$ is really not an algebraic space. Details omitted.