Diamonds for all!

Regular readers of this blog probably know that I’m obsessed with diamonds.  They can thus imagine my happiness when Peter posted an official foundational reference for diamonds a few weeks ago.

I want to use this occasion to make a remark aimed at graduate students etc. who might be wondering whether they should bother learning this stuff: in my opinion, spending time with difficult* manuscripts like the one above usually pays off in the long run. Of course, this only works if you invest a reasonable amount of time, and there’s some initial period where you’re completely befuddled, but after some months the befuddlement metamorphoses into understanding, and then you have a new set of tools in your toolkit! This shouldn’t be so surprising, though; after all, papers like this are difficult precisely because they are so rich in new ideas and tools.

Really, I’ve had this experience many times now – with the paper linked above and its precedent, with the Kedlaya-Liu “Relative p-adic Hodge theory” series, with Kato’s paper on p-adic Hodge theory and zeta functions of modular forms, etc. – and it was the same every time: for some period of months (or years) I would just read the thing for its own sake, but then at some point something in it would congeal with the rest of the swirling fragments in my head and stimulate me to an idea which never would’ve occurred to me otherwise. It’s the most fun thing in the world. Try it yourself.


*Here by “difficult” I don’t mean anything negative, but rather some combination of dense/forbidding/technical – something with a learning curve.  Of course, there are plenty of papers which are difficult for bad reasons, e.g. because they’re poorly written.  Don’t read them.

Report from Tucson

Just back from the 2017 Arizona Winter School on perfectoid spaces.  First of all, I should say that everything was impressively well-organized, and that the lecturers did a fantastic job, especially considering the technical weight of this material. (Watch the videos if you don’t believe me.)  Jared Weinstein, in particular, has an almost supernatural ability to make a lecture on some technical thing feel comforting.

Now to the jokes.

  • In his opening lecture, Scholze called perfectoid spaces a “failed theory”, on account of his inability to completely settle weight-monodromy. “You see, I’m Prussian, and when a Prussian says he wants to do something, he really feels responsible for doing it.”
  • Audience member: “Why are they called diamonds?”
    Scholze: “[oral explanation of the picture on p. 63 of the Berkeley notes]”
    Weinstein: “Also, diamonds are hard.”
  • Anon.: “When you’re organizing a conference, the important thing is not to give in and be the first one who actually does stuff.  Because then you’ll end up doing everything!  Don’t do that!  Don’t be the dumb one!”
    Me: “Didn’t you organize [redacted] a couple of years ago?”
    Anon.: “Yeah… It turned out that Guido Kings was the dumb one.”
  • Mazur: “It just feels like the foundations of this area aren’t yet… hmm…”
    Me: “Definitive?”
    Mazur: “Yes, exactly.  I mean, if Grothendieck were here, he would be screaming.”
  • “Do you ever need more than two legs?”
  • During the hike, someone sat on a cactus.
  • Finally, here is a late night cartoon of what a universal cohomology theory over \mathbb{Z} might look like (no prizes for guessing who drew this):

Elliptic curves over Q(i) are potentially automorphic

This spectacular theorem was announced by Richard Taylor on Thursday, in a lecture at the joint meetings.  Taylor credited this result and others to Allen-Calegari-Caraiani-Gee-Helm-Le Hung-Newton-Scholze-Taylor-Thorne (!), as an outcome of the (not so) secret mini-conference which took place at the IAS this fall.  The key new input here is work in progress of Caraiani-Scholze on the cohomology of non-compact unitary Shimura varieties, which can be leveraged to check (at least in some cases) the most difficult hypothesis in the Calegari-Geraghty method: local-global compatibility at l=p for torsion classes.

The slides from my talk can be found here. Naturally I managed to say “diamond” a bunch of times.


w-local spaces are amazing

Let X be a spectral space.  Following Bhatt-Scholze, say X is w-local if the subset X^c of closed points is closed and if every connected component of X has a unique closed point.  This implies that the natural composite map X^c \to X \to \pi_0(X) is a homeomorphism (cf. Lemma 2.1.4 of BS).

For the purposes of this post, a w-local adic space is a qcqs analytic adic space whose underlying spectral topological space is w-local.  These are very clean sorts of spaces: in particular, each connected component of such a space is of the form \mathrm{Spa}(K,K^+), where K is a nonarchimedean field and K^+ is an open and bounded valuation subring of K, and therefore has a unique closed point and a unique generic point.

I’ve been slowly internalizing the philosophy that w-local affinoid perfectoid spaces have a lot of amazing properties.  Here I want to record an example of this sort of thing.

Given a perfectoid space \mathcal{X} together with a subset T \subseteq |\mathcal{X}|, let’s say T is perfectoid (resp. affinoid perfectoid) if there is a pair (\mathcal{T},f) where \mathcal{T} is a perfectoid space (resp. affinoid perfectoid space) and f: \mathcal{T} \to \mathcal{X} is a map of adic spaces identifying |\mathcal{T}| homeomorphically with T and which is universal for maps of perfectoid spaces \mathcal{Y} \to \mathcal{X} which factor through T on topological spaces. Note that if the pair (\mathcal{T},f) exists, it’s unique up to unique isomorphism.

Theorem. Let \mathcal{X} be a w-local affinoid perfectoid space. Then any subset T of X = |\mathcal{X}| which is closed and generalizing, or which is quasicompact open, is affinoid perfectoid.

Proof when T is closed and generalizing. The key point here is that the map \gamma: X \to \pi_0(X) defines a bijection between closed generalizing subsets of X and closed subsets of the (profinite) space pi_0(X), by taking preimages of the latter or images of the former. To check that this is true, note that if T is closed and generalizing, then its intersection with a connected component X' of X being nonempty implies (since T is generalizing) that T \cap X' contains the unique rank one point of X'. But then T \cap X' contains all specializations of that point (since T \cap X' is closed in X'), so T \cap X' = X', so any given connected component of X is either disjoint from T or contained entirely in T.  This implies that T can be read off from which closed points of X it contains.  Finally, one easily checks that \gamma(T) is closed in \pi_0(X), since \pi_0(X) is profinite and \gamma(T) is quasicompact.  Therefore T = \gamma^{-1}(\gamma(T)).

Returning to the matter at hand, write \gamma(T) as a cofiltered intersection of qc opens V_i \subset \pi_0(X), i \in I. But qc opens in \pi_0(X) are the same as open-closed subsets, so each V_i pulls back to an open-closed subset U_i \subset X, and its easy to check that any such U_i comes from a unique rational subset \mathcal{U}_i \subset \mathcal{X}.  Then \mathcal{T} := \lim_{\leftarrow i \in I} \mathcal{U}_i is the perfectoid space we seek.

Proof when T is quasicompact open. 

First we prove the result when X is connected, i.e. when \mathcal{X} = \mathrm{Spa}(K,K^+) as above.  We claim that in fact T is a rational subset of X. When T is empty, this is true in many stupid ways, so we can assume T is nonempty. Since T is a qc open, we can find finitely many nonempty rational subsets \mathcal{W}_i = \mathrm{Spa}(K,K^{+}_{(i)}) \subset \mathcal{X} such that T = \cup_i |\mathcal{W}_i|.  But the \mathcal{W}_i‘s are totally ordered, since any finite set of open bounded valuation subrings of K is totally ordered by inclusion (in the opposite direction), so T = |\mathcal{W}| where \mathcal{W} is the largest \mathcal{W}_i.

Now we turn to the general case. For each point x \in \pi_0(X), we’ve proved that T \cap \gamma^{-1}(x) is a rational subset (possibly empty) of the fiber \gamma^{-1}(x).  Since \gamma^{-1}(x) = \lim_{\substack{\leftarrow}{V_x \subset \pi_0(X) \mathrm{qc\,open}, x\in V_x}} \gamma^{-1}(V_x) and each \gamma^{-1}(V_x) is the topological space of a rational subset \mathcal{U}_x of \mathcal{X}, it’s now easy to check* that for every x and for some small V_x as above, there is a rational subset \mathcal{T}_x \subset \mathcal{U}_x such that |\mathcal{T}_x| = U_x \cap T. Choose such a V_x for each point in \pi_0(X).  Since \pi_0(X) = \cup_x V_x, we can choose finitely many x‘s \{x_i\}_{i\in I} such that the V_{x_i}‘s give a covering of \pi_0(X).  Since each of these subsets is open-closed in \pi_0(X), we can refine this covering to a covering of \pi_0(X) by finitely many pairwise-disjoint open-closed subsets W_j, j \in J where W_j \subseteq V_{x_{i(j)}} for all j and for some (choice of) i(j) \in I. Then \gamma^{-1}(W_j) again comes from a rational subset \mathcal{S}_j of \mathcal{X}, so the intersection |\mathcal{T}_{x_{i(j)}}| \cap \gamma^{-1}(W_j) comes from the rational subset \mathcal{T}_j := \mathcal{T}_{x_{i(j)}} \times_{\mathcal{U}_{x_{i(j)}}} \mathcal{S}_j of X, and since |\mathcal{T}_j| = T \cap \gamma^{-1}(W_j) by design, we (finally) have that \mathcal{T} = \coprod_{j} \mathcal{T}_j \subset \coprod_{j} S_j = \mathcal{X} is affinoid perfectoid. Whew! \square

*Here we’re using the “standard” facts that if X_i is a cofiltered inverse system of affinoid perfectoid spaces with limit X, then |X| = \lim_{\leftarrow i} |X_i|, and any rational subset W \subset X is the preimage of some rational subset W_i \subset X_i, and moreover if we have two such pairs (i,W_i) and (j,W_j) with the W_{\bullet}‘s both pulling back to W then they pull back to the same rational subset of X_k for some large k \geq i,j.

Let T be a subset of a spectral space X; according to the incredible Lemma recorded in Tag 0A31 in the Stacks Project, the following are equivalent:

  • T is generalizing and pro-constructible;
  • T is generalizing and quasicompact;
  • T is an intersection of quasicompact open subsets of X.

Moreover, if T has one of these equivalent properties, T is spectral. (Johan tells me this lemma is “basically due to Gabber”.) Combining this result with the Theorem above, and using the fact that the category of affinoid perfectoid spaces has all small limits, we get the following disgustingly general statement.

Theorem. Let \mathcal{X} be a w-local affinoid perfectoid space. Then any generalizing quasicompact subset T \subset |\mathcal{X}| is affinoid perfectoid.

By an easy gluing argument, this implies even more generally (!) that if T \subset |\mathcal{X}| is a subset such that every point t\in T has a qc open neighborhood U_t in |\mathcal{X}| such that T \cap U_t is quasicompact and generalizing, then T is perfectoid (not necessarily affinoid perfectoid).  This condition* holds, for example, if T is locally closed and generalizing; in that situation, I’d managed to prove that T is perfectoid back in May (by a somewhat clumsy argument, cf. Section 2.7 of this thing if you’re curious) after Peter told me it was so.  But the argument here gives a lot more.

*Johan’s opinion of this condition: “I have no words for how nasty this is.”