Let $k$ be an algebraically closed field, $X$ an affine connected smooth $k$ curve, and $\mathscr{F}$ a locally constant sheaf on $X_{et}$. Then $H^0_c (X, \mathscr{F}) = 0$? (This is 2.10.(a) of Deligne's Weil I)
To show this, I showed:
Let $X$ be an irreducible scheme, $j : U \to X$ an open immersion such that $U \neq X$, and $\mathscr{F}$ be a locally constant sheaf on $U$. Then $\Gamma(X, j_! \mathscr{F}) = 0$.
I think that what I want to show originally (i.e., 2.10.(a) of Deligne) follows form this, since for a compactification $j : X \to \overline{X}$, we have $H^0_c(X, \mathscr{F}) = \Gamma(\overline{X}, j_!\mathscr{F})$. Is my opinion true? I'm not confident about it because in this "proof" I used only $X \neq \overline{X}$, and did not use the affineness and smoothness of $X$.
Thank you very much!