Understanding the definition of the quotient stack $[X/G]$

Question

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.

Answer 1

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.

Answer 2

It follows from the definitions that $[X/G]$ is representable by a scheme $S$ if and only if $X$ is a $G$-torsor over $S$, i.e. the natural map $G\times X\rightarrow X\times _SX$ given by $(g,x)\mapsto (x,gx)$ is an isomorphism. Example 0.4 (Chapter 0, §3, page 11 of 3rd edition) in Mumford's GIT is a variety $X$ with an action of $G=SL(2)$ with trivial stabilizers and geometric quotient $\mathbb{A}^1$, but such that $X$ is not a $G$-torsor over $\mathbb{A}^1$, so $[X/G]$ is not representable.