I was reading Milne's proof of Weak Lefschetz theorem from his book on Étale cohomology [VI.7] and certain parts of theorem 7.9 did not make sense to me. I shall give some definitions first.
For any finite type morphism $\pi:Y\rightarrow X$ and any $y\in Y$ with image $x\in X$ we define $\delta(y) = d(x) + \operatorname{trdeg}_{k(x)} k(y)$, where $d(x)$ is the dimension of the closure of the point $x$. And for any étale sheaf $F$ on $Y$, he defines $\delta(F)=\operatorname{sup}\{\delta(y)|F_{\overline{y}}\ne 0\}$
Now, in the proof of [7.9] he takes $X$ to be the spectrum on a strictly local ring and $F$ a torsion sheaf on $\mathbb{A}^1_X$ such that $\delta(F)\leq d$ (assume $d\geq 2$). Let $j:\mathbb{A}^1_X\rightarrow \mathbb{P}^1_X$ be an embedding. Then Milne claims that $H^i(\mathbb{P}^1_X, j_*F)=0$ for all $i> 2$ as a consequence of proper base change theorem. I believe the proper map being considered here is $\mathbb{P}^1_X\rightarrow X$, but I don't see how this can be used to prove that the cohomology is zero! Could someone please help me with that?
A second question I had in the same proof was towards the end. Let $g:\mathbb{A}^{n}_X\rightarrow \mathbb{A}^{n-1}_X$ be the morphism that identifies $\mathbb{A}^{n}_X$ with the affine line over $\mathbb{A}^{n-1}_X$ and let $F$ be a torsion sheaf on $\mathbb{A}^{n}_X$ with $\delta(F)\leq d$. Then using the case $n=1$ shown in the first half of the proof at the level of stalks helps solve the general case. I agree with this as long as we can show that $\delta(R^jg_*F)\leq d-j$. But why should $\delta(R^jg_*F)\leq d-j$?
Thanks in advance for any help!