Here is a fun example. Take U = Spec(k[x_0, x_1, x_2, …]) and let G_m act by t(x_0, x_1, x_2, …) = (tx_0, t^px_1, t^{p^2}x_2, …) where p is a prime number. Let X = [U/G_m]. This is an algebraic stack. There is a stratification of X by strata

- X_0 is where x_0 is not zero,
- X_1 is where x_0 is zero but x_1 is not zero,
- X_2 is where x_0, x_1 are zero, but x_2 is not zero,
- and so on…
- X_{infty} is where all the x_i are zero

Each stratum is a gerb over a scheme with group \mu_{p^i} for X_i and G_m for X_{infty}. The strata are reduced locally closed substacks. There is no coarser stratification with the same properties.

So clearly, in order to prove a very general result as in the title of this post then we need to allow infinite stratifications…

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