# The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

Consider the affine space $\mathbb{A}^n$ (over some base scheme) with the usual $\mathrm{GL}_n$-action. What does the quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$ classify? If $n=1$, then we get $[\mathbb{A}^1 / \mathbb{G}_m]$, which classifies line bundles together with a global section, right? In general, $[\mathbb{A}^n / \mathrm{GL}_n]$ classifies vector bundles of rank $n$ together with some additional data - is it again just a global section?

I would also be happy if someone can add a geometric picture of $[\mathbb{A}^n / \mathrm{GL}_n]$, at least over some algebraically closed field. What are the points of this stack, and what are their stabilizers? What's your geometric intuition for $[\mathbb{A}^n / \mathrm{GL}_n]$?

PS: I am quite new to stack-land and hope that this question is not too trivial.

PPS: In the comments it is suggested to do the same as for $n=1$, so let me better explain what I've done for $n=1$. A $T$-point of $[\mathbb{A}^1/\mathbb{G}_m]$ is a $\mathbb{G}_m$-bundle $P \to T$ with a $\mathbb{G}_m$-map $P \to \mathbb{A}^1$. Then $P$ corresponds to an invertible sheaf $\mathcal{L}$ on $T$ via $P = \mathrm{Spec}_T(\bigoplus_{z \in \mathbb{Z}} \mathcal{L}^{\otimes z})$. Assuming that $T$ is affine, the $\mathbb{G}_m$-map therefore corresponds to a section of $\bigoplus_{z \in \mathbb{Z}} \mathcal{L}^{\otimes z}$, say $\sum_z a_z$, which is compatible with the $\mathcal{O}(\mathbb{G}_m)$-coaction, which comes down to $\sum_z a_z u^z = \sum_z a_z u$, i.e. $a_z=0$ for $z \neq 1$. We end up with a section $a_1$ of $\mathcal{L}$. For $n >1$, I have tried the same, but the quasi-coherent algebra induced by some locally free sheaf of rank $n$, whose spectrum is the corresponding $\mathrm{GL}_n$-bundle, is quite complicated, at least globally. Locally, we just have $P = \mathrm{GL}_n \times T$, and the $\mathrm{GL}_n$-map $P \to \mathbb{A}^n$ corresponds to a map $T \to \mathbb{A}^n$, i.e. $n$ global sections of $T$, which is one section of the free sheaf $\mathcal{O}_T^n$ of rank $n$. But a) I am not sure if this glues properly, and b) I would prefer a global argument as for $n=1$.

• This stack classifies vector bundles of rank $n$ together with $n$ global sections. You see it just like you see that $A^1/G_m$ classifies line bundles with a global section. Oct 12 '14 at 21:03
• This stack classifies vector bundles of rank $n$ together with $1$ global section. You see it just like you see that $A^1/G_m$ classifies line bundles with a global section. Oct 12 '14 at 21:49
• @Matthieu. Hmmm... I don't think so. The stack that classifies vector bundles of rank $n$ together with $n$ global sections is $[Mat_{n\times n}/GL_n]$. Oct 12 '14 at 21:50
• The comment with the most upvotes will be the correct one? ;) Anyway, this kind of disagreement shows that my question is perhaps not so silly, and that it requires a detailed answer ... Oct 12 '14 at 22:29
• Their comments were tied, so I upvoted yours to make it a 3-way tie. Oct 13 '14 at 0:13

## 3 Answers

The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-equivariant map. Morphisms $(P,f) \to (P',f')$ are torsor isomorphisms $g: P \to P'$ satisfying $f = f' g$. Here, $X$ is the vector representation $\mathbf{O}^n$, and $G = GL_n$.

However, the vector representation of $GL_n$ is faithful, so you may replace the $GL_n$-torsor with the associated bundle $P \times^{GL_n} \mathbf{O}^n$ to get an equivalent stack. Here, objects are pairs $(V, f)$, where $V$ is a rank $n$ vector bundle over $T$, and $f: V \to X$ is a $GL_n$-equivariant map to the vector representation, or more usefully, an equivariant sheaf map to the trivial rank $n$ bundle. Morphisms $(V,f) \to (V',f')$ are given by vector bundle isomorphisms $g$ satisfying $f = f' g$.

By taking dual vector bundles, we get an equivalent stack, whose objects are pairs $(V,h)$, where $V$ is a vector bundle, and $h$ is a $GL_n$-equivariant map from the trivial rank $n$ bundle to $V$. Since $GL_n$ acts transitively on the nonzero vectors in the vector representation (and its dual), any map $h$ is uniquely determined by the image of a fixed nonzero vector in the dual vector representation, i.e., a distinguished section of the vector bundle. Then we have a stack whose objects are vector bundles with sections, and morphisms are vector bundle isomorphisms that take sections to sections.

Edit: Here is a picture of the stack: $- \cdot$

The dash is an open dense copy of $B(Aff_{n-1})$ (since the stabilizer of a nonzero vector is the affine group $Aff_{n-1}$) and the dot is a closed copy of $BGL_n$.

I'll use the definition of stack as a (weak) functor from the category of schemes to that of groupoids (as opposed to the definition as a fibered category over the category of schemes).

The prestack associated to the action of $GL_n$ on $\mathbb A_n$ is, by definition, given by $$X \mapsto \left\{\begin{matrix}\text{Objects: maps s:X \to \mathbb A^n}\\ Hom(s_1,s_2): \text{maps f:X\to GL_n such that f\cdot s_1 = s_2} \end{matrix}\right\}$$ That prestack is easily seen to be equivalent to $$X \mapsto \left\{\begin{matrix}\text{Objects: Sections s:X\to \mathbb A^n\times X of the trivial vector bundle \mathbb A^n\times X\to X}\\ \text{Hom(s_1,s_2): Vector bundle isos f:\mathbb A^n\times X\to \mathbb A^n\times X s.t. s_2=f\circ s_1}\end{matrix}\right\}$$ The associated stack is then given by $$X \mapsto \left\{\begin{matrix}\text{Objects: Vector bundles V \to X, together with a section s:X\to V}\\ \text{Hom((V_1,s_1),(V_2,s_2)): Vector bundle iso f:V_1 \to V_2 such that s_2=f\circ s_1}\end{matrix}\right\}$$

Indeed, any vector bundle with section is locally of the form trivial vector bundle with section. Moreover, any vector bundle with section $(V,s)$ over $X$ can be described by an open cover of $X$, on each open of the cover the data of a trivial vector bundle with section, and gluing isomorphisms subject to the obvious cocycle condition. That's exactly what does the associated stack to the prestack of tivial vector bundles with section.

• Thank you, this is a nice way of seeing it. I was using the definition of $[X/G]$ as $[X/G](T) =$ groupoid of $G$-bundles on $T$ with a $G$-map to $X$. You use the atlas of this algebraic stack. Oct 12 '14 at 22:33

Let $$\text{G}$$ be a group. Then maps $$S\ \longrightarrow \ \text{BG}$$ from a test scheme $$S$$ is the same thing as a $$\text{G}$$-bundle $$P\to S$$ (or rather the groupoid of such).

Now let $$X$$ be another scheme with an action of $$\text{G}$$. Then to any $$\text{G}$$ bundle we can build its associated $$X$$ bundle: $$(P\to S)\ \rightsquigarrow \ (P\times_\text{G}X\to S).$$ This $$X$$ bundle has another description, as the pullback $$\require{AMScd}$$ $$\begin{CD} P\times_\text{G}X @>>> X/\text{G}\\ @V V V @VV V\\ S @> P> > \text{BG} \end{CD}$$ Answer: Choosing a lift of the map $$P$$ to $$S\ \longrightarrow \ X/\text{G}$$ is thus the same as choosing a section of the $$X$$-bundle $$P\times_\text{G}X$$ associated to $$P$$.

Examples:

1. $$\mathbf{A}^1/\mathbf{G}_m$$ classifies $$\mathbf{G}_m$$ bundles (a.k.a. line bundles) along with a section of the associated line bundle.
2. $$\mathbf{A}^n/\text{GL}_n$$ classifies $$\text{GL}_n$$ bundles with a section of its associated vector bundle.
3. $$\mathbf{P}^1=(\mathbf{A}^2\setminus 0)/\mathbf{G}_m$$ classifies line bundles together with two sections which do not vanish simultaneously.
4. Similarly for $$\mathbf{P}^n$$.
5. A variation on 1. If instead $$\mathbf{G}_m$$ acts on $$\mathbf{A}^1$$ with weight $$n$$, then $$\mathbf{A}^1/\mathbf{G}_m$$ classifies line bundles with a section of its $$n$$th tensor power. Similarly for e.g. weighted projective spaces.
6. More generally, if $$\text{P}$$ is a parabolic subgroup of the reductive group $$\text{G}$$, then the generalised flag variety $$\text{G}/\text{P}$$ classifies $$\text{P}$$ bundles with a trivialisation of the associated $$\text{G}$$ bundle. This gives the functor of points for e.g. Grassmannians $$\text{Gr}(k,n)$$.