Let $V$ be a $k$-vector space and $U$ an open subscheme of $V$.

Let $L$ be an $l$-adic local system on $U$, under which conditions the non-vanishing of the cohomology of $H^{\bullet}(U,L)$ implies the non-vanishing of the cohomology $H^{\bullet}(V, j_{!*}L)$ where $j:U\rightarrow V$ is an open immersion.