Making $H^*(\mathbf{P}^\infty)=\lim H^*(\mathbf{P}^n)=k[t]$ precise using stacks

Question

The stack $B\mathbf{G}_m$, i.e. morally $\mathbf{P}^\infty$, has (etale) cohomology $\mathbf{Q}_\ell[t]$.

The scheme $\mathbf{P}^n$ has cohomology $\mathbf{Q}_\ell[t]/t^n$.

In algebraic topology, the first fact follows immediately from (the proof of) the second, using the CW complex decomposition of $\mathbf{CP}^\infty$.

Is there a sense in which $B\mathbf{G}_m=\lim\mathbf{P}^n$ as stacks, and if so does this allow us to compute its cohomology?

Answer 1

This is a classic computation that one can do, for any $\mathrm{GL}_n$, essentially using the natural map $\mathrm{Grass}_n(h)\to B\mathrm{GL}_n$ defined by the universal quotient. It's shown with spectacular clarity in Lemma 2.3.1 of Kai Behrend's article The Lefschetz trace formula for algebraic stacks. Specifically, you are interested in the subsequent corollary when $n=1$.

For $\mathbb{G}_m$ what this essentially means that is that for all $i\geqslant 0$ you have a natural map $\mathbb{P}^n\to B\mathbb{G}_m$ defined by the universal quotient which induces a map

$$H^i(B\mathbb{G}_m,\mathbb{Q}_\ell)\to H^i(\mathbb{P}^n_k,\mathbb{Q}_\ell)$$

which is an isomorphism for $i\leqslant 2(n-1)$. One then gets the desired result by passing to the limit in $i$.