(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?
Best Answer
You seem to mean two slightly different things by the $p=\ell$ case. Your first theorem is about a field of characteristic $p$, and the second is about $\mathbb C_p$, which is a field of characteristic zero. Let's talk first about the significance of this.
For a field of actual characteristic $p$, the proof of "Artin vanishing" for mod $p$ coefficients is even easier than you gave. One doesn't need any perfectoid spaces, as you can just use the Artin-Schreier sequence $0 \to \mathbb F_p \to \mathcal O_X \to \mathcal O_X \to 0$ to handle the case of affine schemes.
By a Lefschetz principle argument, your proof over $\mathbb C_p$ can be adapted to work over an arbitrary field of characteristic zero. Of course, by the same argument, and the comparison theorem between etale and analytic cohomology, the Artin affine theorem can also be proved by analytic arguments over fields of characteristic zero (say, using a Morse function).
Characteristic zero really is pretty much always easier than characteristic $p$.
In actual characteristic $p$, the $\ell=p$ case is not necessarily always easier. I think a good description is that it is more degenerate. The cohomology groups are usually much smaller, in fewer degrees. It's not surprising that vanishing results are easier to prove in this setting! But other results we want, like Poincare duality or Euler characteristic formulas, are simply not true for mod $p$ coefficients.
There is certainly something unsatisfying about these somewhat ad hoc arguments, but they're not THAT bad. I think getting over some of your discomfort with them might be easier than finding a better argument.
It's perhaps worthwhile to point out that Grothendieck famously wanted to solve problems by setting up theories that make the problems easy rather than the concrete calculations favored by mathematicians like Serre and Deligne, which were later necessary to solve problems like the Weil conjectures. If you find the arguments in the part of etale cohomology worked out by Grothendieck to have too many concrete calculations and not enough big theories then I am tempted to say you have gone too far, but I perhaps shouldn't because many fields of mathematics have trended in that direction since Grothendieck's day, and that perspective does work for at least some mathematicians.
Anyways, the use of $\mathbb P^1$ in this argument is certainly not a pure application of general theory, but it is also not ad hoc. It is part of a systematic strategy in the foundations of etale cohomology, which is to first prove the desired statement for curves using the powerful theorems available for curves (precise descriptions of $H^1$ and $H^2$, together with dimension estimates), and then find an inductive approach to the general case based on varieties mapping to a curve with fibers mapping to a curve, etc., suitable to the particular problem.
For different problems, it makes sense to try slightly different curves, and put them together in slightly different ways.
For the affine vanishing theorem, we of course want to look at affine curves, so we might look first at the simplest such, $\mathbb A^1$, and then observe we can prove the theorem for arbitrary products of $\mathbb A^1$, or $\mathbb A^n$. Can we reduce from the general case to $\mathbb A^n$? Yes, by Noether normalization.
For the comparison of etale and analytic cohomology, it's convenient to work with affine curves, and fibrations of affine curves by affine curves, so that all our cohomology will match the cohomology of the fundamental group, as then we can take advantage of the comparison of the etale and analytic fundamental groups. In this case, it suffices to check that smooth varieties are covered by open subsets isomorphic to an iterated fibration of curves, and this we can do.
For the proper base change theorem, we of course want to start with proper curves, and so we try the simplest such, $\mathbb P^1$. There's no clear reason to take nontrivial fibrations of $\mathbb P^1$ over $\mathbb P^1$, so we just look at powers, $(\mathbb P^1)^n$. Can we reduce from a general projective variety to $(\mathbb P^1)^n$? Not directly, as we only know that projective varieties map to $\mathbb P^n$, but using the map $(\mathbb P^1)^n \to \mathbb P^n$ we can close the gap.
So in some sense it's all the same idea, just adapated in different ways to different contexts.
The Weil conjectures are one place this strategy failed, as Grothendieck tried to prove them by showing every smooth projective variety is covered by a product of curves, but this is false. (Deligne was later, in Weil II, able to solve the problem by an inductive argument, using the theory of weights of constructible sheaves.)