Example 100.29.2. Let $k$ be a field. Take $U = \mathop{\mathrm{Spec}}(k[x_0, x_1, x_2, \ldots ])$ and let $\mathbf{G}_ m$ act by $t(x_0, x_1, x_2, \ldots ) = (tx_0, t^ p x_1, t^{p^2} x_2, \ldots )$ where $p$ is a prime number. Let $\mathcal{X} = [U/\mathbf{G}_ m]$. This is an algebraic stack. There is a stratification of $\mathcal{X}$ by strata

$\mathcal{X}_0$ is where $x_0$ is not zero,

$\mathcal{X}_1$ is where $x_0$ is zero but $x_1$ is not zero,

$\mathcal{X}_2$ is where $x_0, x_1$ are zero, but $x_2$ is not zero,

and so on, and

$\mathcal{X}_{\infty }$ is where all the $x_ i$ are zero.

Each stratum is a gerbe over a scheme with group $\mu _{p^ i}$ for $\mathcal{X}_ i$ and $\mathbf{G}_ m$ for $\mathcal{X}_{\infty }$. The strata are reduced locally closed substacks. There is no coarser stratification with the same properties.

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