The classifying stack $B\mathbb{G}_m$ parametrizes line bundles: $B\mathbb{G}_m(T) = \lbrace \text{line bundles on } T \rbrace / \cong$ This is because $\mathbb{G}_m$-torsors correspond exactly to line bundles: given a line bundle $\mathcal{L}$, the associated frame bundle $\underline{\mathrm{Isom}}(\mathcal{O}_T, \mathcal{L})$ is a $\mathbb{G}_m$-torsor.

The Picard group $\mathrm{Pic}(T) = H^1(T, \mathbb{G}_m)$ classifies $\mathbb{G}_m$-torsors on $T$. Thus $[T, B\mathbb{G}_m] \cong \mathrm{Pic}(T)$ (where $[T, B\mathbb{G}_m]$ denotes isomorphism classes of maps).

The stack $B\mathrm{GL}_n$ classifies rank-$n$ vector bundles: $B\mathrm{GL}_n(T) = \lbrace \text{rank-}n \text{ vector bundles on } T \rbrace / \cong$ A $\mathrm{GL}_n$-torsor $P \to T$ gives a vector bundle via the associated bundle construction: $\mathcal{E} = P \times^{\mathrm{GL}_n} \mathbb{A}^n$. Conversely, a vector bundle $\mathcal{E}$ gives the frame bundle $\underline{\mathrm{Isom}}(\mathcal{O}_T^n, \mathcal{E})$.

We have $\dim B\mathrm{GL}_n = -n^2$.

Over a base where $n$ is invertible, the stack $B(\mathbb{Z}/n\mathbb{Z})$ classifies etale $\mathbb{Z}/n\mathbb{Z}$-torsors, which correspond to degree-$n$ cyclic etale covers. For $T$ connected, these are classified by $H^1_{\text{et}}(T, \mathbb{Z}/n\mathbb{Z}) \cong \mathrm{Hom}(\pi_1^{\text{et}}(T), \mathbb{Z}/n\mathbb{Z})$.

Consider $\mathbb{G}_m$ acting on $\mathbb{A}^1$ by multiplication. The quotient stack $[\mathbb{A}^1/\mathbb{G}_m]$ has two points:

• The open point $[(\mathbb{A}^1 \setminus \lbrace 0 \rbrace)/\mathbb{G}_m] \cong \mathrm{Spec}\, k$ (the orbit of any nonzero point, with trivial stabilizer).
• The closed point $[\lbrace 0 \rbrace/\mathbb{G}_m] \cong B\mathbb{G}_m$ (the origin, with full $\mathbb{G}_m$ stabilizer).

This stack is "the stacky point with a thickening direction" and provides a local model for understanding stacky structure near points with $\mathbb{G}_m$ stabilizer.

The projective space $\mathbb{P}^n$ can be realized as the quotient stack $\mathbb{P}^n = [(\mathbb{A}^{n+1} \setminus \lbrace 0 \rbrace) / \mathbb{G}_m]$ Since $\mathbb{G}_m$ acts freely on $\mathbb{A}^{n+1} \setminus \lbrace 0 \rbrace$, the quotient stack is actually a scheme. The tautological line bundle on $\mathbb{P}^n$ corresponds to the $\mathbb{G}_m$-torsor $\mathbb{A}^{n+1} \setminus \lbrace 0 \rbrace \to \mathbb{P}^n$.

Since $\mathbb{G}_m = \mathrm{GL}_1$, we have $A^*(B\mathbb{G}_m) \cong \mathbb{Z}[c_1]$ where $c_1$ is the first Chern class. This is the algebraic analogue of $H^*(\mathbb{CP}^{\infty}; \mathbb{Z}) = \mathbb{Z}[c_1]$ in topology.

Concretely, $A^*(B\mathbb{G}_m)$ can be computed as $\varprojlim A^*(\mathbb{P}^n) = \varprojlim \mathbb{Z}[h]/(h^{n+1}) = \mathbb{Z}[[h]]$, but properly interpreted as a polynomial ring.

The Chow ring of $B\mathrm{PGL}_2$ is more subtle. We have $A^*(B\mathrm{PGL}_2) \otimes \mathbb{Q} \cong \mathbb{Q}[c_2, c_3]$ where $c_2$ and $c_3$ have degrees 2 and 3 respectively. The integral Chow ring has 2-torsion phenomena reflecting the non-trivial center $\mu_2 \subset \mathrm{SL}_2 \to \mathrm{PGL}_2$.

A morphism $\mathrm{Spec}\, k \to B\mathbb{G}_m$ is a line bundle on $\mathrm{Spec}\, k$. Since all line bundles on $\mathrm{Spec}\, k$ are trivial, there is (up to isomorphism) a unique morphism $\mathrm{Spec}\, k \to B\mathbb{G}_m$, but its automorphism group is $\mathbb{G}_m(k) = k^{\times}$.

The set of isomorphism classes of maps $\mathbb{P}^1 \to B\mathbb{G}_m$ is $\mathrm{Pic}(\mathbb{P}^1) \cong \mathbb{Z}$, with the map of degree $n$ corresponding to $\mathcal{O}(n)$. The automorphism group of $\mathcal{O}(n)$ is $\mathbb{G}_m = \mathrm{Aut}(\mathcal{O}(n))$.

$\mathbb{G}_m$-gerbes over $S$ are classified by $H^2(S, \mathbb{G}_m)$, which is the cohomological Brauer group $\mathrm{Br}(S)$. For $S = \mathrm{Spec}\, k$ with $k$ a field, $\mathrm{Br}(k)$ classifies central simple algebras over $k$ up to Morita equivalence.

The classifying stack $B\mathbb{G}_m$ is itself a $\mathbb{G}_m$-gerbe over the point, corresponding to the trivial element of $\mathrm{Br}(k) = 0$ (the Brauer group of a point is trivial).

A Severi-Brauer variety of dimension $n$ over $k$ is a variety $X$ such that $X_{\bar{k}} \cong \mathbb{P}^n_{\bar{k}}$. These are classified by $H^1(k, \mathrm{PGL}_{n+1})$, which corresponds to maps $\mathrm{Spec}\, k \to B\mathrm{PGL}_{n+1}$. The Brauer group element associated to a Severi-Brauer variety via $H^1(k, \mathrm{PGL}_{n+1}) \to H^2(k, \mathbb{G}_m)$ is its Brauer class.

The exact sequence of groups gives: $B\mathbb{G}_m \to B\mathrm{GL}_n \to B\mathrm{PGL}_n$ At the level of cohomology over $\mathrm{Spec}\, k$: $k^{\times} \to H^1(k, \mathbb{G}_m) \to H^1(k, \mathrm{GL}_n) \to H^1(k, \mathrm{PGL}_n) \xrightarrow{\delta} H^2(k, \mathbb{G}_m)$ Since $H^1(k, \mathrm{GL}_n) = 0$ (Hilbert 90), this gives an injection $H^1(k, \mathrm{PGL}_n) \hookrightarrow \mathrm{Br}(k)$, identifying $\mathrm{PGL}_n$-torsors with Brauer classes of degree $n$.

The Kummer sequence gives a fiber sequence $B\mu_n \to B\mathbb{G}_m \xrightarrow{(\cdot)^n} B\mathbb{G}_m$ Over a scheme $S$ (with $n$ invertible), this yields the Kummer exact sequence: $\mathcal{O}(S)^{\times} \xrightarrow{(\cdot)^n} \mathcal{O}(S)^{\times} \to H^1(S, \mu_n) \to \mathrm{Pic}(S) \xrightarrow{(\cdot)^n} \mathrm{Pic}(S) \to H^2(S, \mu_n) \to \mathrm{Br}(S)$

Via the Weierstrass equation $y^2 = x^3 + ax + b$ (in characteristic $\neq 2, 3$), the moduli stack of elliptic curves can be presented as $\mathcal{M}_{1,1} \cong [U / \mathbb{G}_m]$ where $U = \mathrm{Spec}\, \mathbb{Z}[1/6][a, b, \Delta^{-1}]$ with $\Delta = -16(4a^3 + 27b^2)$, and $\mathbb{G}_m$ acts by $\lambda \cdot (a, b) = (\lambda^4 a, \lambda^6 b)$.

The map $U \to \mathcal{M}_{1,1}$ is a $\mathbb{G}_m$-torsor, giving the atlas. The $j$-invariant $j = -1728 \cdot \frac{(4a)^3}{\Delta}$ is the $\mathbb{G}_m$-invariant function defining the coarse moduli $\mathbb{A}^1_j$.

For a smooth projective curve $C$ and a reductive group $G$, the moduli stack $\mathrm{Bun}_G(C)$ of principal $G$-bundles on $C$ is a smooth algebraic stack. It can be described via the uniformization theorem: $\mathrm{Bun}_G(C) \cong G(\mathcal{K}) \backslash G(\mathbb{A}) / G(\mathcal{O})$ in the function field / adelic picture. Here $\mathcal{K}$ is the function field, $\mathbb{A}$ is the ring of adeles, and $\mathcal{O}$ is the ring of integral adeles.

For $G = \mathrm{GL}_n$, $\mathrm{Bun}_G(C)$ is the stack of vector bundles of rank $n$ on $C$. This stack plays a central role in the geometric Langlands program.