# Moduli stacks of relative morphisms

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 non-trivial 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 $$S$$?

• 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). Commented Dec 26, 2020 at 8:38
• @MinseonShin The example in The Stacks Project gives non-algebraicity 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 Commented Dec 26, 2020 at 12:54
• @JasonStarr Oops, I'd missed that, thank you. Commented Dec 26, 2020 at 18:20