Geometric realization of $B{\mathbb G}_{\mathfrak m}({\mathbb C})$ is ${\mathbb C}{\mathbb P}^\infty=\varinjlim_n~ {\mathbb C}{\mathbb P}^n_k$; what if one considers a separable field $k\neq {\overline k}$? Sheaf-theoretically, $B{\mathbb G}_{\mathfrak m}$ represents the simplicial sheaf $B{\mbox{hom}}_{k}(-,{\mathbb G}_{\mathfrak m})$ on the big etale site $Sch_k$; should I think of the analogue for ${\mathbb C}{\mathbb P}^\infty$ as $\varinjlim_n~ {\mbox{hom}}_{k}(-,{\mathbb P}^n_k)$?

Context: calculating etale cohomology of $B{\mathbb G}_{\mathfrak m}(k)$.

Note: For separably closed field $k$, theorem of Friedlander-Parshall says that $H^\ast_{et}(BG(k),{\mathbb Z}/m)\cong H^\ast_{et}(BG({\mathbb C}),{\mathbb Z}/m)$ for complex reductive group scheme $G({\mathbb C})$.