I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles.
Explicitly, let $G$ be an affine smooth group $S$-scheme with right action $\rho:X\times G\to X$ on a noetherian $S$-scheme $X$. The quotient stack $[X/G]$ is the pseudofunctor
$$
[X/G]:(\mathsf{Sch}/S)^{\operatorname{op}}\to\mathsf{Grpds}
$$
defined as follows.
For an $S$-scheme $U$ let $[X/G](U)$ be the category whose objects are diagrams
$$
U\xleftarrow{\pi}E\xrightarrow{\alpha}X
$$
where $\pi$ is a principal $G$-bundle and $\alpha$ is a $G$-equivariant morphism. The morphisms in $[X/G](U)$ are the isomorphisms of principal $G$-bundles commuting with the $G$-equivariant morphisms.
For a morphism of $S$-schemes $f:U^\prime\to U$ let
$$
[X/G](f):[X/G](U)\to[X/G](U^\prime)
$$
be the functor induced by pullbacks of principal $G$-bundles.
It is not clear to me how to determine when $[X/G]$ is representable in general. My questions are:
Question 1. Is there a sufficient condition we can impose on $\rho$ to ensure that $[X/G]$ is representable? The Wikipedia article on quotient stacks says something about when the categorical quotient $X/G$ exists the canonical map $[X/G]\to\operatorname{Hom}(-,X/G)$ need not necessarily be an isomorhpism. This is somewhat opaque to me as I'm not even sure how $[X/G]\to\operatorname{Hom}(-,X/G)$ is defined.
Question 2. What is an example where $[X/G]$ is not representable?
If these questions are too broad, I'd be very grateful if someone could point me in the direction of a good reference.
Best Answer
The functor $[X/G]$ is not representable whenever there is non-trivial isotropy of the action of $G$ on $X$.
Let us consider the most extreme case: when $X = \bullet$ is a point (the terminal object) and $G$ is any non-trivial group. In such a case, $Hom(-,\bullet/G)$ is a singleton, as $\bullet / G = \bullet$, which is still terminal.
However, $[\bullet / G]$ is much more interesting than that! If you trace through the definition provided above (which is worth doing at least once in your life), we find that $[\bullet / G](U)$ is the collection of principal $G$-bundles over $U$, which is non-trivial most of the time. Thus it follows that, as a stack, $[\bullet / G]$ is the classifying space of $G$.
Particular examples are $[\bullet / \mathbb{G}_m]$ being the collection of line bundles, etc.
As for the question about how the morphism $[X/G] \to Hom(-,X/G)$ is defined:
In your diagram, the morphism $\alpha : E \to X$ is equivariant. You can thus complete the diagram to
$$ \begin{array}{c c c} E & \to & X \\ \downarrow & & \downarrow\\ U & \to & X/G \end{array} $$
which yields the desired map.