In our text we have several statements of the following sort: for a certain morphism $f:X\to Y$ of varieties over an (algebraically closed) field of characteristic $p$ and some $c>0$ the corresponding etale cohomology homomorpisms $H^i_{et}(f,\mathbb{Z}/n\mathbb{Z})$ are bijective for $i<c$ and $H^c_{et}(f,\mathbb{Z}/n\mathbb{Z})$ is injective; here $n$ is any integer prime to $p$.
Would it be fine to say that $f$ is cohomologically $c$-connected? Or etale $c$-connected? We can probably dualize and pass to etale homology; yet this requires some effort and will not give any new information.
