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I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=\operatorname{GL}(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcoming" of categorical quotients?

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  • $\begingroup$ What source denotes the categorical quotient by X//G? Normally, it is denoted by X/G. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 28, 2020 at 3:03
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    $\begingroup$ Wikipedia for example, en.wikipedia.org/wiki/Categorical_quotient For me a categorical quotient is a GIT quotient. Then I thought stacky quotient was [X/G]. Did the notation change recently or do I miss something here? $\endgroup$
    – Adam
    Commented Nov 30, 2020 at 19:23
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    $\begingroup$ In modern literature, X/G denotes the colimit of a G-action and X//G denotes the homotopy colimit of the same action. The former is known as the strict, or categorical, quotient, whereas the latter is known as the homotopy, or stacky, quotient. This is how Qiaochu used this terminology in his answer. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 30, 2020 at 22:18
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    $\begingroup$ The GIT quotient is not a quotient of X, but rather a quotient of the locus of semistable points in X, which is a different object. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 30, 2020 at 22:23
  • $\begingroup$ @DmitriPavlov: Thanks, I can change the notation in my question. To respond to your last point though: I thought that GIT quotient = categorical quotient, correct? Also the GIT quotient is both (a) a quotient of X and (b) a quotient of the locus of semistable points in X. (It identifies every non-semistable point with a semistable point.) $\endgroup$
    – Adam
    Commented Dec 1, 2020 at 1:32

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$\DeclareMathOperator\Spec{Spec}$ If $G$ is an affine group scheme acting on an affine scheme $X$ over an algebraically closed field $K$ you can ask what the $K$-points $\Spec K \to X//G$ of the stacky quotient are (my conventions are that $X/G$ is the categorical quotient and $X//G$ is the stacky quotient). By definition they consist of pairs of a $G$-torsor $P \to \Spec K$ over $\Spec K$ and a $G$-equivariant map $P \to X$. Under mild hypotheses on $G$ (I think it suffices that $\mathcal{O}(G)$ be countably generated, if I'm reading Deninger - A remark on the structure of torsors under an affine group scheme correctly), $G$ is the only $G$-torsor over $\Spec K$, so the $K$-points of $X//G$ are $G$-equivariant maps $G \to X$.

These correspond exactly to the $K$-points $X(K)$ of $X$, together with the action of $G(K)$ on them (acting on $G$ from the right); in other words, $(X//G)(K)$ is the stacky quotient / action groupoid / homotopy quotient $X(K)//G(K)$, and in particular its $\pi_0$ (set of isomorphism classes) is exactly the set of orbits of $G(K)$ acting on $X(K)$.

(Actually I don't know if I need to assume $X$ affine here but I'm playing it safe.)

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