# Perfectoid universal covers

It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this is hinted at (although never said directly) in section 5.7 of Jared Weinstein's Reciprocity laws and Galois representations: recent breakthroughs (pdf).

I was wondering if someone could answer the following questions which, while perhaps naive, are of great interest to me.

If this is too many questions I can break it up into multiple posts.

1) Let $X$ be the `perfectoid closed unit disk' given by $\text{Spa}(\mathbb{C}_p\langle T^{\frac{1}{p^\infty}}\rangle,\mathcal{O}_{\mathbb{C}_p}\langle T^{\frac{1}{p^\infty}}\rangle)$. Is it true that $X-\{0\}$ is 'simply connected' in the sense that $\pi_1^{\acute{e}\text{t}}(X-\{0\})=0$? One might imagine that this is something like a pro-etale universal cover of the usual punctured closed unit disk over $\mathbb{C}_p$ since it's at least $\sim$ (in the language of Scholze) to an inverse limit of the finite etale covers $x\mapsto x^{p^n}$ of the punctured closed disk over $\mathbb{C}_p$. I suspect not since covers of the disk (like Artin-Schreier covers) exist. So, is there a 'universal cover' in this case?

EDIT: As mentioned below, I was being hasty and should have said something like 'the maximal $p$-quotient of $\pi_1^{\acute{e}\text{t}}$ is zero.

2) If $X$ is any rigid space over $K$ (a $p$-adic field, or perhaps $\widehat{\overline{K}}$ for a $p$-adic field) then is there a 'universal cover' of $X$. The precise definition of this is open for me--is there a particularly good notion, and if so, when does it exist. For example, if there is a pro-etale cover $\{U_i\}$ of $X$ and an adic space $\widetilde{X}$ such that $\widetilde{X}\sim \varprojlim U_i$ and $\widetilde{X}$ is 'simply connected' (i.e. that $\pi_1^{\acute{e}{t}}(\widetilde{X})=0$ or, perhaps even better, $\pi_1^{\text{pro}\acute{e}\text{t}}(\widetilde{X})=0$).

EDIT: As mentioned below, I was being hasty and should have said something like 'the maximal $p$-quotient of $\pi_1^{\acute{e}\text{t}}$ is zero and similarly for the proetale fundamental group.

3) I have the sense that perfectoid spaces have 'simpler geometry' (for example, I think this can be seen by the almost zeroness of their cohomology with values in $\mathcal{O}_X^+$ and how this relates to their cohomology with values in $\mathbb{F}_p$ by the AS sequence), but I don't know a precise statement of this. Namely, how is the etale topology of a perfectoid space simpler than, say, a general rigid variety?

4) It is sometimes said that one can use perfectoid geometry to try and compute things like the etale cohomology of some rigid variety by computing the Cech cohomology of some pro-etale perfectoid cover. What is the precise statement of this? I think that one can compute the cohomology of $\mathcal{O}_X^+$ almost (in the technical sense) from a pro-etale perfectoid cover. Does one then try to compute cohomology with coefficients in $\mathbb{F}_p$ almost-ly by using the AS sequence again?

Any answers to any of these questions would be greatly appreciate--as well as any other insight someone might want to add.

Thanks!

EDIT: I guess I should add that one probably doesn't expect universal covers in terms of 'pure topological trivialness' (or, rather, pure topological trivialness in-so far as $\pi_1$ or other '1-dimensional topology' is concerned) but rather in the sort of topological trivialness (again in degree $1$) concerned with $p$-torsion or pro-$p$-torsion coefficients.

So, as Will Sawin mentions below, you can still likely make covers with prime-to-$p$ degree of the 'universal cover'.

In summary, perhaps '$p$-universal cover' is better...

• 1. I suspect not - isn't there a nontrivial covering space that comes from adjoining a $k$th root of $T$ for $k$ prime to $p$? 2. Presumably we could take $U_i$ to be the inverse system of all connected etale covering spaces of $X$. Sep 13 '16 at 21:04
• @WillSawin Yeah, or that. Also, for your second comment, I don't see why that really works. For example, what adic space (let alone perfectoid) would you have that is $\sim$ to that? Second, unless I'm being silly, it's not obvious to me that this is a pro-etale cover. What precisely is the system you're taking? I also edited my question as (is clear from your first, helpful comment) it's non-obvious that I meant 'universal with respect to $p$' triviality. Sep 14 '16 at 3:25
• 2. Most likely you are aware of this, but check the proof Thm 4.9. in math.uni-bonn.de/people/scholze/pAdicHodgeTheory.pdf. There Scholze defines the universal cover of an affinoid $X$ over $\mathbb{Q}_p$ to be a perfectoid space over $\mathbb{C}_p$ for which every finite etale cover admits a section. This should be equivalent to the triviality of $\pi_1^{\text{proet}}$ for such a perfectoid space. Dec 22 '16 at 0:56

Lol @"varying degrees of enthusiasm" ;-). And sorry for the late answer...

Let me try to answer your questions. First, for any connected analytic adic space $$X$$, say, with a geometric point $$\overline{x}\to X$$, one can define $$\pi_1^{\mathrm{et}}(X,\overline{x})$$ just like in SGA1 for schemes, by looking at the Galois category of finite etale covers of $$X$$. In particular, passing to an inverse limit of all such finite etale covers equipped with a lift of $$\overline{x}$$, one can define a (profinite) "universal cover" $$\tilde{X}\to X$$. If $$X$$ lives over $$\mathbb Z_p$$ and is affinoid (probably Stein is enough) then $$\tilde{X}$$ is perfectoid; see for example Lemma 15.3 here (the funny phrasing there is solely due to the desire to also handle the case that $$X$$ is not connected).

This largely answers question 2). Unfortunately, I don't know how to define a pro-etale fundamental group in the spirit of my paper with Bhatt. There we handle the case of schemes that locally have only a finite number of irreducible components. This is a very mild condition for schemes, but for analytic adic spaces, the condition is much too strong, see Example 7.3.12 of our paper. That example shows that the formalism actually does not work in the same way for analytic adic spaces, and I don't know how to correct it. So I will only use the usual $$\pi_1^{\mathrm{et}}$$.

For question 1), the answer is actually No. Using Artin-Schreier covers, there are lots and lots of finite etale covers beyond the ones one might think about, so in particular the perfectoid closed unit disc has very large $$\pi_1^{\mathrm{et}}$$ (even (or especially) pro-$$p$$). What one might reasonably hope is that any finite etale cover of degree $$p$$ of the punctured perfectoid closed unit disc extends to a finite etale cover of the perfectoid closed unit disc. For this precise question, I'm actually confused: If the finite etale covers comes from some finite stage, it follows from some classical results in rigid geometry that it extends to a finite, possibly ramified cover, over the puncture, and then by Abhyankar's lemma this becomes trivial after passing to the perfectoid cover. However, I believe that at infinite level, one will get new, more nasty covers, that do not come from finite level.

About question 3): One key fact is that affinoid perfectoid spaces have etale $$p$$-cohomological dimension $$\leq 1$$, i.e. for etale $$p$$-torsion sheaves, etale cohomology sits in degrees $$\leq 1$$. This in fact reduces by tilting to the case of characteristic $$p$$, where it follows from Artin-Schreier theory. Combining this with some interesting examples of perfectoid towers, one can get interesting vanishing results. In fact, these can usually be slightly improved upon by using $$\mathcal O_X^+$$-cohomology, the primitive comparison theorem, and the (almost) vanishing of $$\mathcal O_X^+$$-cohomology on affinoid perfectoids. This has been applied for example to Shimura varieties, abelian varieties [Well, the written version of that paper actually doesn't use this method, but our original approach did use it, see the discussion on page 1], and moduli spaces of curves.

This ties in with question 4). What one usually does is the following. Say $$\ldots\to X_2\to X_1\to X_0$$ is some tower of proper rigid-analytic varieties over $$\mathbb C_p$$ with perfectoid limit $$X_\infty$$. For each $$X_n$$, the primitive comparison theorem says that $$H^i(X_n,\mathbb F_p)\otimes \mathcal O_{\mathbb C_p}/p\to H^i(X_n,\mathcal O_{X_n}^+/p)$$ is an almost isomorphism, where both sides are etale cohomology. (The proof of this uses some Artin-Schreier theory, and one could also formulate an Artin-Schreier sequence, but this tends to give weaker results.) Passing to the colimit over $$n$$ (so the limit on spaces), one sees that also $$H^i(X_\infty,\mathbb F_p)\otimes \mathcal O_{\mathbb C_p}/p\to H^i(X_\infty,\mathcal O_{X_\infty}^+/p)$$ is an isomorphism. Now on perfectoid $$X_\infty$$, the group on the right behaves like coherent cohomology, in particular it can (almost) be computed on the analytic side, and in fact by a Cech complex. This shows in particular that it (almost) vanishes in degrees larger than $$\dim X_\infty$$. In particular, $$H^i(X_\infty,\mathbb F_p)$$ vanishes in degrees larger than $$\dim X_\infty$$, which gives the vanishing theorems I mentioned.

This is far too long for a comment, and it won't fully answer your questions. I invite anyone who knows more about perfectoid spaces than me to edit this to clarify or correct anything. I am not in any sense an expert on perfectoid spaces.

We first have to fix what we mean by $\pi_1$. In general if I have some notion of geometry and good morphisms(for example, étale, étalé, smooth, pro-étale, fppf...) I can do two things.

1. I can take the category of good covers, pick a point and take the fiber functor to sets and do the Grothendieck dance to define a pro-group.

2. I can take the Grothendieck topology generated by these, and consider the category of locally constant sheaves of groups for the topology, take a fiber functor(a point of the topos), and do the Grothendieck dance to define a pro-group.

One way to interpret representability results for locally constant sheaves / sheaf-theoretic principal homogeneous spaces is that they compare the two strategies.

As examples,

1. The fundamental group of SGA1 uses strategy (1) for schemes and étale covers.
2. The fundamental group of SGA3 uses strategy (2) for schemes and étale principal homogeneous spaces.

There is a general representability result for principal $G$-spaces when $G$ is a finite group, which tells us that the two agree after profinite completion(equivalently, they have equivalent categories of finite quotients). A stronger result compares the two for Noetherian and geometrically unibranch schemes.

1. The pro-étale fundamental group (in the Bhatt-Scholze paper) is a souped up version of (2). Their category is schemes and their topology is the pro-étale topology, but they are not only consider l.c. sheaves of groups, but l.c. sheaves of topological groups, they've also modified the "Grothendieck dance" with a couple more shimmies, twists, and jazz hands.

None of the above examples directly applies to perfectoid spaces, and I am far from an expert on them. However I think it should give you a clearer picture of exactly how to phrase your question.

Is it true that $X\setminus\{0\}$ is 'simply connected' in the sense that $\pi_1^{\acute{e}t}(X\setminus\{0\})=0$ [after taking maximal pro-$p$ quotients]?

It cannot be pro-$p$-simply connected because of the Artin-Schreier coverings. I believe that a finite étale cover of $\operatorname{spec} A$ analytifies to a finite étale cover of $\operatorname{sp} A$, as the almost finite étale condition of Definition 7.2 seems to be a corollary of Theorem 1.10(both in Scholze's Perfectoid Spaces paper). This means that the maximal pro-$p$ quotient will still see the Artin-Schreier coverings.

If $X$ is any rigid space over $K$ (a $p$-adic field, or perhaps $\hat{\overline{K}}$ for a p-adic field) then is there a 'universal cover' of $X$.

Let's just restrict to the affinoid case for simplicity. It would seem there is some pro-étale covering space, but it's not clear what category the thing lives in. Rigid spaces have some amount of finite-type conditions built into them and so taking a limit over finite étale maps might not be a rigid space. The nice thing about Huber's formalism of adic spaces is that it is much more flexible, so you may be able to just take the limit of adic spaces and/or the colimit of the underlying $f$-adic rings. But the colimit of $f$-adic rings needn't be $f$-adic so you'll have to do some work to figure out if the structure presheaves are still sheaves. For all I know it might be a direct corollary of Buzzard-Verberkmoes paper.

how is the etale topology of a perfectoid space simpler than, say, a general rigid variety?

I don't know that it's simpler, but the major thing is that you've killed off Frobenius in a suitable sense, meaning that you can easily pass between characteristic $0$ and characteristic $p$. Taking the laziest of glances at Scholze's proof (which I do not claim to understand!), it seems like his proof of the acyclicity of $\mathscr{O}_X^+$ on affinoids goes by first reducing to the case of pure characteristic $p$, then recognizing the sheaf $\mathscr{O}_X^+$ as the completion of a filtered colimit of things that are of finite type in a suitable sense. The case for things of finite type is a 'classical' statement of rigid geometry. Cohomology commutes with filtered colimits, yadda yadda yadda, and you get the result.

I don't have an answer for (4), but I am also interested in reading one!