(In this question, $p$ can be $0$.)

I'm curious if theorems on étale cohomology can be proved by easier way. For example, proper base change theorem. This theorem can be stated as the following way.

Theorem.Let $k$ be a field of charateristic $p$ and $p\ne \ell$. Let's consider that Cartesian diagram $$ \begin{aligned} X'&\overset{g'}{\to} X \\ \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!f'\downarrow& \,\,\,\,\,\,\,\,\downarrow f\\ S'&\overset{g}{\to} S \end{aligned} $$ of $k$-schemes. Then the natural map $$ g^*(R^if_*\mathscr{F})\to R^if'_*((g')^*\mathscr{F})$$ is an isomorphism if $f$ is proper and $\mathscr{F}$ is a $\mathbb{Z}/\ell$-constructible sheaf.

It is an analogue of proper base change theorem for topological spaces and in topological space case, the proof is relatively easy. See https://stacks.math.columbia.edu/tag/09V4. But for étale cohomology, the proof is so hard! The stages in Freitag-Kiehl are the following.

1.Reduce the case of $S=\mathrm{Spec} A$ for a strictly henselian $A$ and $g=s:X=\mathrm{Spec} k\to \mathrm{Spec} A$ for $k=A/\mathfrak{m}$.

2.Prove that if $\dim X_s\le 1$, the natural map $H^i(X,\mathbb{Z}/\ell)\to H^i(X_s,\mathbb{Z}/\ell)$ is a bijection where $i=0$, a surjection where $i\ge 1$ with standard GAGA argument.

3.Prove the above natural maps are actually isomorphism for all $i$ and for all constructible $\mathrm{Z}/\ell$-sheaves using some dimension argument.

4.To prove general case, use $\mathbb{P}^1\times \cdots \times \mathbb{P}^1\to \mathbb{P}^n$ and reduce the general case to the case of $\mathbb{P}^n\to S$.

5.Then we have an inclusion $X\hookrightarrow \mathbb{P}^n_S\to S$ and we win.

This is so complicated! I may understand stage 1,2,3. But I have confused with stage 4,5. To prove the general dimension case, is there the reason for considering $\mathbb{P}^1\times \cdots \times \mathbb{P}^1\to \mathbb{P}^n$? This seems an ad hoc and so different to the proof of proper base change theorem for topological spaces.

In case of $p=\ell$, I have found an elegant proof in https://arxiv.org/pdf/1711.04148.pdf (Corollary 10.6.2) which uses only the fact that higher direct images of well-defined finiteness are also finite and do not use somewhat dimension reduction. so I think $p=\ell$ case proper base change is much easier than $p\ne \ell$ proper base change.

Another example is Artin vanishing theorem.

Theorem.Let $k$ be an algebraically closed field of characteristic $p$ and $X$ be a finite type affine scheme over $k$. Then we have $H^i(X,\mathscr{F})=0$ for all $i>\dim X$ and $\mathbb{Z}/\ell$-constructible sheaf $\mathscr{F}$.

The proof is in https://stacks.math.columbia.edu/tag/0F0P and the stratege of stacks project is to use Noether normalization theorem and Leray spectral sequence to $\mathbb{A}^{i+1}\to \mathbb{A}^i$. Nevertheless that strategy has a clear reason for Artin vanishing, I can't make a conception of this proof and it seems a typical reduction technique. In $p=\ell$ case, Artin vanishing theorem can be stated as

Theorem.For any finite type affine scheme $X$ over $\mathbb{C}_p$, we have $H^i(X,\mathscr{F})=0$ for $i>\dim X$ and $\mathbb{Z}/p$-constructible $\mathscr{F}$.

Proof)(Sketch) Let $X$ be smooth. Then by adjoining $p^n$-th root of unities, we have a perfectoid covering $X_{\infty}\to X$. In $X_{\infty}$, the cohomology dimension of $X_{\infty}$ is $0$ by $$ H^i(X_{\infty},\mathbb{F}_p)\otimes \mathcal{O}^+_{X_{\infty}}/(p)=H^i(X_{\infty},\mathcal{O}^+_{X_{\infty}}/(p))$$ and almost Tate acyclicity on affinoid perfectoid space. Then we can apply Čech-to-derived functor spectral sequence and cohomological dimension property of the group $\mathbb{Z}^{\dim X}_p$ to prove Artin vanishing in case of $X$ smooth and $\mathscr{F}=\mathbb{F}_p$. In general locally constant $\mathscr{F}$, just find a covering that trivializes $\mathscr{F}$ and in general constructible sheaf use dimension induction on $\dim X$ and the definition of constructible sheaf. In general finite type scheme, we can choose a resolution of $X$ and use Leray spectral sequence. The rest is the comparison theorem of étale cohomology with proétale cohomology which is done by Bhatt-Scholze.

It is just a special case of original Artin vanishing. But the proof seems clear than the proof of the original version. So I would prefer that proof to the proof or original version.

As that examples, I would like to think the proof of "Theorem X" in $p\ne \ell$ case is harder than the proof of "Theorem X" in $p=\ell$ case. Why $p\ne \ell$ is much harder than $p=\ell$? I don't know but maybe it seems to be linked to Scholze's global Hodge theory conjecture, that we have somewhat comparison theorem in $p=\ell$ but there is no well-sensed comparison theorem in $p\ne \ell$! But if there is real statement and proof of global Hodge theory conjecture, then I think maybe we can prove "Theorem X" in $p\ne \ell$ case more easily.

The question is

Question.Is $p\ne \ell$ case really hard than $p=\ell$ case?

Of course, sometimes $p=\ell$ case is more hard than $p\ne \ell$. (Think Poincaré duality in case of $p=\ell$ which is not well-defined yet.) But with sufficient background, $p=\ell$ case seems easy than $p\ne \ell$ case. Is this right?

2more comments