Let $S$ be a scheme and let $X, Y$ be schemes over $S$. Consider the stack $\mathop{\mathscr{Mor}}_S(X, Y)$ fibered in groupoids over $(\mathsf{Sch}/S)_{\mathrm{fppf}}$ defined as follows: The objects of $\mathop{\mathscr{Mor}}_S(X, Y)$ consist of morphisms from $X \times_S T$ to $Y \times_S T$ over $T$ for some $T \in \mathop{\mathrm{Ob}}\left((\mathsf{Sch}/S)_{\mathrm{fppf}}\right)$, and morphisms of $\mathop{\mathscr{Mor}}_S(X, Y)$ are given by cartesian diagrams. Obviously $\mathop{\mathscr{Mor}}_S(X, Y)$ is not equivalent to a category fibered in sets due to the existence of nontrivial automorphisms. 0D1C shows that the functor associated to $\mathop{\mathscr{Mor}}_S(X, Y)$ is an algebraic space over $S$ under several assumptions on $X$ and $Y$. I wonder is $\mathop{\mathscr{Mor}}_S(X, Y)$ an algebraic stack over $S$?
$\begingroup$
$\endgroup$
3

3$\begingroup$ This is called the "Hom stack" and it is not necessarily an algebraic stack, see Tag 0AF8 (in this example, $X \to S$ is proper and flat and $Y \to S$ is proper and smooth). $\endgroup$– Minseon ShinCommented Dec 26, 2020 at 8:38

4$\begingroup$ @MinseonShin The example in The Stacks Project gives nonalgebraicity in the case that $Y$ is a stack (not a scheme). This example contradicts a theorem (including the "corrected" version in the erratum) for a published article about stacks. Another incorrect theorem in that article is corrected in Proposition 2.3.4 of Lieblich's article, "Remarks on the stack of coherent algebras": arxiv.org/pdf/math/0603034.pdf $\endgroup$– Jason StarrCommented Dec 26, 2020 at 12:54

1$\begingroup$ @JasonStarr Oops, I'd missed that, thank you. $\endgroup$– Minseon ShinCommented Dec 26, 2020 at 18:20
Add a comment
