*The money will be spent on higher categories for the needy.

]]>Perhaps surprisingly, the situation for affinoid perfectoid spaces is a lot better. In particular, if is a perfectoid Tate-Huber pair, there are canonical bijections (satisfying some obvious compatibilities) between

1) closed subsets of ,

2) Zariski-closed subsets of ,

3) (isomorphism classes of) maps of Tate-Huber pairs where is a perfectoid Tate ring, is surjective, and is the integral closure of the image of in .

We’ve already discussed the bijection 1) <–> 2). For 3) –> 1) or 2), just send to the closed subset cut out by the ideal . The miracle is the association 2) –> 3), which holds by an amazing theorem of Bhatt: if is a closed ideal in a perfectoid Tate ring , then the uniform completion of is perfectoid and the natural map is surjective, cf. Theorem 2.9.12 in Kedlaya’s notes here. Moreover, the map remains surjective after rational localization on . In particular, if is a Zariski-closed subset, then 2) –> 3) gives an honest closed immersion of locally ringed spaces, and maps homeomorphically onto .

The point of all this is that Zariski-closed immersions of affinoid perfectoid spaces behave as well as one could ever dream (with one caveat, which I’ll get to later). The following definition then suggests itself.

**Definition. **A map of small v-stacks is a Zariski-closed immersion if for any affinoid perfectoid space with a map , the base change is a Zariski-closed immersion of affinoid perfectoid spaces.

Now of course we’re free to make any definition we want in mathematics, but if it doesn’t capture some essential idea or experimentally observed phenomenon, then who cares? Let me now give some evidence that this definition passes this test.

**Example 0.** The property of being a Zariski-closed immersion is preserved under composition and base change. If is a Zariski-closed immersion and is (a small v-sheaf, a diamond, a locally spatial diamond, qc or qs or separated or partially proper over a base ), then so is .

**Example 1. **Let be a closed immersion of locally Noetherian adic spaces. If is affinoid (so is too), then the map of diamonds is a Zariski-closed immersion. This is easy.

**Example 2. **Let be a closed immersion of locally Noetherian adic spaces again, but now assume that is the analytification of a closed immersion of quasiprojective varieties. Then is a Zariski-closed immersion. For this, we can use the assumption on to choose a vector bundle on together with a surjection . Then for any map from an affinoid perfectoid, the pullback (in the usual sense of ringed spaces) is a vector bundle on , hence generated by finitely many global sections by Kedlaya-Liu. The images of along the natural map generate an ideal, and the associated closed immersion of affinoid perfectoids represents the fiber product . (Hat tip to PS for suggesting this vector bundle trick.)

**Example 3. **Let be a minimally compactified Hodge-type Shimura variety with infinite level at . Then the boundary is a Zariski-closed immersion, and so is the diagonal . (These both reduce to the previous example, using a small limit argument in the second case.) In particular, if are any open affinoid perfectoid subsets, then is also affinoid perfectoid. This small observation plays a non-negligible role in my forthcoming paper with Christian Johansson, where (among other things) we prove that any minimally compactified Shimura variety of pre-abelian type with infinite level at is perfectoid.

**Example 4. **Fix a perfectoid base field of characteristic zero. Then the inclusions are Zariski-closed immersions of (ind-)diamonds over . This can be proved by induction on , and the base case reduces to the fact that the inclusion is the pullback of along . (To make the induction work, you need to pick an element generating .)

**Example 5. **Fix a complete algebraically closed extension . Fix a reductive group and a geometric conjugacy class of -valued cocharacters . Then is a Zariski-closed immersion. Also, if , then is a Zariski-closed immersion. These claims can be reduced to the case , which in turn reduces to Example 4 by some trickery.

**Example 6. **Fix a complete algebraically closed nonarchimedean field of residue characteristic , and let be any injective map of coherent sheaves on the Fargues-Fontaine curve . Then the associated map of Banach-Colmez spaces is a Zariski-closed immersion. This can also be reduced to Example 4.

Let me end with some caveats. First of all, I wasn’t able to prove that if is a closed immersion of reductive groups, the induced map is a Zariski-closed immersion, although it is surely true. The problem here is (roughly) that an -torsor over some affinoid perfectoid can only be reduced to a -torsor locally in the analytic topology on , and we then run into the following open question:

**Question. **Is the property of being Zariski-closed local for the analytic topology? More precisely, if is affinoid perfectoid with a covering by rational subsets , and is a closed subset such that is Zariski-closed in for all , is Zariski-closed?

There are also naturally occurring closed things which probably aren’t Zariski-closed immersions. For instance, I don’t think the map of Banach-Colmez spaces is a Zariski-closed immersion, because then pulling back would imply that is a Zariski-closed immersion, which seems extremely unlikely to me. (But I didn’t manage to disprove it! Actually, can one give an explicit example of an affinoid perfectoid and a closed subset such that maps isomorphically to the completed residue field at every point in and such that is NOT Zariski-closed? Surely such examples exist.) I also don’t think (closures of) Newton strata in flag varieties are Zariski-closed immersions – they are just too weird.

I also wasn’t able to settle the following compatibility (but admittedly I didn’t try very hard).

**Question. **Let be a monomorphism of locally Noetherian adic spaces. If is a Zariski-closed immersion, is actually a closed immersion?

Happy new year!

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For the argument below to work, it would be enough to know that for any open subset , its image in contains an open subset. Is this true?

In any case, the Corollary is still true, although by a totally different argument.)

*****

I spent about six hours yesterday and today proving the following thing.

**Lemma. **Let be a spectral space, and let be a closed generalizing nowhere-dense subset of . Then is nowhere-dense for the constructible topology on , i.e. doesn’t contain any nonempty constructible subset of .

This has the following concrete consequence, which is what I really needed.

**Corollary**. Let be some Tate-Huber pair with adic spectrum , and let be a Zariski-closed nowhere-dense subset. Suppose and are quasicompact open subsets of such that . Then .

*Proof of Corollary.* We need to check that is empty. But is a constructible subset of contained in , so this is immediate from the lemma.

Amusingly, even though this corollary is pretty down-to-earth, I only managed to prove it by proving the lemma, and I only managed to prove the lemma by exploiting the structure of the w-localization of . Is there a more direct approach? Am I missing something obvious?

(Sketch of actual argument: the profinite set of closed points maps homeomorphically onto equipped with the constructible topology, so if is constructible it is clopen when viewed as a subset of . The key point is then to check that is nowhere-dense when viewed as a subset of . This can be done, using that the natural surjection is open and that (which is then closed, generalizing and nowhere-dense in , the last point by openness of ) is the preimage of its image in .

The openness of doesn’t seem to be stated in the literature, but it can be deduced from the proof of Lemma 2.1.10 in Bhatt-Scholze, using the fact that it’s obviously true for finite spaces.)

You may have noticed that RIMS is hosting a series of four workshops next year under the umbrella of a “RIMS Research Project” entitled Expanding Horizons of Inter-universal Teichmuller Theory. The first of the workshops looks pretty reasonable, the other three not so much. In case you’re wondering (as I did) how much money RIMS is ponying up for this, it seems to be capped at 5 million yen, or about $41k (according to e.g. this document). This doesn’t seem like very much money to support four workshops; I guess some funding is also coming from that infamous EPSRC grant.

Anyway, when you’re inside a black hole, your horizons might seem quite expansive indeed, but I doubt you’ll have much luck convincing others to join you.

]]>Less well-known, however, is the following result of Temkin (cf. Theorem 1.1.3 here):

**Temkin’s Factorization Theorem.** *Let be any separated morphism of qcqs schemes. Then can be factored as an affine morphism followed by a proper morphism.*

Telling other people about this theorem is an amusing experience. Invariably, their first reaction is that it simply cannot be true, and that the inclusion map should give a counterexample. But then they realize (or I point out) that can be factored as , where is the blowup of at the origin and is the natural (affine!) open immersion of into . Then they are convinced.

Unrelated: JW pointed out to me that I am now a professional writer of appendices. Maybe this should worry me?

]]>- Hardy writing to Veblen from Princeton, ca. 1928:

“…However, I suppose my present passion for the soda fountain will abate by degrees.” … “I do find myself regretting that Wiener’s not here: but no doubt if he were I should very quickly revise my opinion.” - Pedoe on Hodge: “One fine morning Hodge and I were inside the grounds of Pembroke College when we met J A Todd, an excellent geometer, the author of a fine textbook on projective geometry, a University lecturer – and a pipe smoker who spent more time striking matches than actually smoking. As we stood talking, Todd struck match after match and dropped them on the ground at Hodge’s feet. Hodge, as the Acting Bursar, was responsible for the proper maintenance of the grounds of Pembroke, so as Todd dropped each match, Hodge bent down to pick it up. Todd, who wore eyeglasses with strong lenses, was completely unaware of what was going on. The spectacle of the very thin Todd unconsciously dropping matches, and the rotund Hodge bending down every few seconds – while becoming more and more exasperated – is one I shall never forget.”

…

“Hodge became Master of Pembroke and President of the Royal Society. He was very shrewd and usually tactful, but had definite ideas on certain matters. He thought, for example, that a mathematical paper should be just that, with no embellishment. When Patrick Du Val, a contemporary of Donald Coxeter, a good geometer, and a lover of the arts, submitted a paper to the Cambridge Philosophical Society for publication, with a quotation from Dante following its title, Hodge suggested that this was not “appropriate.” He was badly flustered when a furious Du Val withdrew the paper.” - Schwartz on Maurice Audin’s thesis
- Hermann Weyl’s speech at Emmy Noether’s funeral
- Hardy again, this time on Waring’s problem
- Dedekind attending a lecture course by Gauss in the winter of 1850: “…The lecture room, separated from Gauss’ office by an anteroom, was quite small. We sat at a table which had room for three people comfortably at each side, but not for four. Gauss sat opposite the door at the top end, at a reasonable distance from the table, and when we were all present, the two who came in last had to sit quite close to him with their notebooks on their laps. Gauss wore a lightweight black cap, a rather long brown coat and grey trousers. He usually sat in a comfortable attitude, looking down, slightly stooped, with his hands folded above his lap. He spoke quite freely, very clearly, simply and plainly; but when he wanted to emphasise a new point of view, for which he used a particularly characteristic word, then he would raise his head, turn to one of those sitting beside him, and gazed at the student with his beautiful, penetrating blue eyes during his emphatic speech. That was unforgettable. …”
- Thue on mathematics in Berlin in 1891-92: “Fuchs, whom I heard lecture on analytical mechanics, did not at first make much of an impression on me. The concepts he employed were, as far as I could see, surrounded by a mist of vagueness. When I heard him in a seminar, however, I got a strong presentiment that he can excel when he wants to do so. He lectures with his eyes shut and looks thoroughly tired and peevish. He can also be rather absent-minded. I remember how he was once talking about differentials, and quite unconsciously he picked up a handful of bits of chalk which he waved in illustration before our wondering eyes. Afterwards he carefully laid his differentials down again on his desk, with his eyes still closed. Professor Fuchs, like Kronecker, is a very prepossessing man, but not overly talkative. I was at a ball at his home this winter. It was a delightful affair. We danced so energetically that the floor cracked in a couple of places.”

…

“The mathematical seminar down here functions in much the same way as yours does in Oslo. It is an established university institution. Fuchs and Kronecker preside in turn. Meetings are held between 5 and 7. No report is circulated. I have requested Kronecker to permit my highly attractive voice to be heard at the aforementioned place, but so far he hasn’t paid any attention. …”

We now see a natural question: is there is some non-archimedean analogue of the Kahler condition which restores Hodge symmetry? Two years ago, Shizhang Li hit upon the following candiate condition:

A smooth proper rigid space satisfies (*) if it admits a formal model over whose special fiber is *projective* (as opposed to merely proper).

Using fantastic ideas due to Shizhang, we managed to prove the following suggestive result.

**Theorem. ***Let be a smooth proper rigid space satisfying (*). Then .*

Of course, one can then guess that (*) implies Hodge symmetry in all degrees. This speculation seems to have caught the imagination of others in the field, but until recently I personally regarded it as not much more than wishful thinking. However, my perspective completely changed a month ago, when I learned from Shizhang that, according to Robert Friedman, the *archimedean* analogue of “(*) implies Hodge symmetry” is a *theorem! *More precisely, we have the following result:

**Theorem.** *Let be the complex disk, with the punctured disk. Let be a proper map of complex analytic spaces. Suppose that is a submersion, and that the central fiber is the analytification of a projective (and not necessarily smooth) algebraic variety. Then for all with , the fiber satisfies Hodge symmetry and Hodge-de Rham degeneration.*

Of course, the analogy is that is analogous to , and is analogous to the “nearby” fibers with .

The proof of this theorem uses the full power of mixed Hodge theory. In fact the claim about Hodge-de Rham degeneration is exactly Corollary 11.24 in the book of Peters-Steenbrink. Hodge symmetry is even more subtle, and the argument for this doesn’t seem to be written down anywhere; Friedman explained it to Shizhang, who explained it to me, but the details entailed such a horrible explosion of gradings, filtrations, and multi-indices that I can’t hope to reproduce it here.

Anyway, I’m now completely convinced that Shizhang’s condition (*) implies Hodge symmetry in all degrees, and that this is truly the “right” p-adic analogue of the Kahler condition.

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**Definition. **Let be an adic ring with finitely generated ideal of definition . We say is *strongly Noetherian outside * if, for all , the scheme is Noetherian.

Here denotes the usual ring of restricted power series. Note that if is a Tate ring and is any couple of definition, then is strongly Noetherian if and only if is strongly Noetherian outside . I should also point out that the condition of being strongly Noetherian outside is already considered in a very interesting paper of Fujiwara-Gabber-Kato; they use the terminology “topologically universally rigid-Noetherian”, but I prefer my terminology on account of the previous sentence. Anyway, the following conjecture seems reasonable:

**Conjecture. **If is strongly Noetherian outside , the structure presheaf on is a sheaf.

This implies that any strongly Noetherian Tate ring is sheafy (which of course is already known), but it also implies e.g. that if is topologically finitely presented over for some nonarchimedean field , then is sheafy. Sheafiness in the latter situation is known when is discretely valued, but to the best of my knowledge it’s open for general .

I’m sure this conjecture is within reach, and maybe it’s an easy exercise for experts on sheafiness (*cough* Kiran *cough*). Note that FGK already proved some interesting consequences of this definition, which are probably relevant to proving this conjecture. Precisely, they show that if is strongly Noetherian outside , then:

1. The -power-torsion submodule of any finitely generated -module is killed by a power of .

2. If is any inclusion of -modules, with finitely generated, then the subspace topology on induced by the -adic topology on coincides with the -adic topology on .

Let be some immersion of varieties over a separably closed field. Everyone knows that the intermediate extension functor on perverse sheaves (say with coefficients in ) is pretty great: it’s totally canonical, it commutes with Verdier duality, it preserves irreducibility, it preserves monic and epic maps of perverse sheaves, etc.

Recently I noticed that if is any map of varieties, with smooth and irreducible, there’s still a natural functor which commutes with Verdier duality. To define this functor, note that for any and any , there is a natural map , obtained by adjunction from the composite map (the first isomorphism here is the projection formula). Since is smooth, the dualizing complex is just , so then . Thus we get a natural map .

Next, note that the complex is concentrated in degrees , and in the lowest of these degrees it’s just the constant sheaf, i.e. . In particular, there is a canonical map . Shifting by and tensoring with gives a map . Putting things together, we get a natural map . Set so after shifting this becomes a natural map

.

This shifting has the advantage that Verdier duality exchanges the functors and on , and one can check that the Verdier dual of identifies with .

**Definition. **The functor sends any to the image of the map .

Here of course denotes the zeroth perverse cohomology sheaf.

**Exercise. **Show that .

It might be interesting to compute this functor in some other examples. Note that it can be quite stupid: if is a closed immersion (with ) and is already supported on , then . On the other hand, if is smooth and surjective, then is faithful.

]]>- Gabber wasn’t there, but there were some Gabberesque moments anyway. In particular, during Xuhua He’s talk, Goertz observed that a point is an example of a Deligne-Lusztig variety, so any variety is a union of Deligne-Lusztig varieties. Gotta be careful…
- The food was about the same as usual. Worst Prize was tied between two dishes: a depressing vegetable soup which somehow managed to be flavorless and bitter simultaneously, and a dessert which looked like a lovely innocent custard but tasted like balsamic vinegar. The best dishes were all traditional German fare.
- Best Talks (in no particular order): Jean-Stefan Koskivirta, Miaofen Chen, Ben Howard, Timo Richarz.
- Apparently this paper can be boiled down to a page or two.
- There was (not surprisingly) some late-night discussion of the Stanford Mystery. [Redacted] proposed a theory so mind-bogglingly outrageous that it certainly won’t fit in this margin.
- “Fun was never really my goal.” – A representative UChicago alum.
- On Thursday it snowed, and a snowball fight broke out after dinner. This was a lot of fun, but I’m still glad we didn’t follow Pilloni’s suggestion of a match between Team Europe (Pilloni, Stroh, Morel, Anschutz, Richarz, Mihatsch, etc.) and Team USA (me).
- Here’s an innocent problem which turns out to be pretty tricky. Let be a (separated, smooth) rigid analytic space over , and let be a map from a perfectoid space which is a -torsor for some profinite group . In shorthand, you should think that with G acting freely (this is all literally true in the category of diamonds). It’s easy to cook up examples of this scenario: for instance, you can take and , so then is a torsor for the group . However, there are also much more complicated examples which arise in nature. In particular, if is a Rapoport-Zink space or abelian-type Shimura variety at some finite level, and is the associated infinite level perfectoid guy over it, then we’re in the situation above, with open in the -points of some auxiliary reductive group.
Anyway, supposing we’re in the situation above, we can ask the following complementary questions:

**Q1**. Suppose that is affinoid perfectoid. Does this imply that is an affinoid rigid space?

**Q2**. Suppose that is an affinoid rigid space. Does this imply that is affinoid perfectoid?It seems like both of these questions are actually really hard! For Q1, we can (by assumption) write for some perfectoid Tate-Huber pair , and then one might guess that coincides with . There is certainly a map , but now one is faced with the problem of showing that is “big enough” for this map to be an isomorphism. This can be reduced to any one of a handful of auxiliary problems, but they all seem hard (at least to me). For instance, as a warmup one could try to prove either of the following implications:

**W1**. Under the hypothesis of Q1, vanishes.

**W2**. Under the hypothesis of Q1, is killed by a fixed power ofBoth of these conclusions would certainly hold if we already knew that was affinoid: the first is just (a consequence of) Tate acyclicity, while the fact that is killed by some power of for smooth affinoids is a non-trivial theorem of Bartenwerfer. But I have totally failed to prove either W1 or W2.

In any case, the essential point with Q1 seems to be the following. If is some open subgroup, then will always have plenty of elements, and indeed taking the direct limit as shrinks recovers . But the obstruction to lifting an element of to an element of is a torsion class in , and the latter group seems hard to control.

For Q2, there is maybe a slightly clearer path through the forest: it would follow from the following conjecture, which I explained during my talk in the workshop.

To set things up, let be any uniform Tate-Huber pair over , and let be the associated pre-adic space. Let denote the site given by perfectoid spaces over with covers given by v-covers, and let and be the obvious structure sheaves on . Set and , so the association is an endofunctor on the category of uniform Tate-Huber pairs over . One can check that breve’ing twice is the same as breve’ing once, and that the natural map induces an isomorphism of diamonds. If is a smooth (or just seminormal) affinoid -algebra for some , or if is perfectoid, then breve’ing doesn’t change .

**Conjecture.**Let be a uniform Tate-Huber pair over such that every completed residue field of is a perfectoid field. Then is a perfectoid Tate ring.Aside from disposing of Q2, this conjecture would settle another notorious problem: it would imply that if is a uniform sheafy Huber ring and is a perfectoid space, then is actually perfectoid.

It may be instructive to see an example of a non-perfectoid (uniform) Tate ring which satisfies the hypothesis of this conjecture. To make an example (with ), set , and let with the obvious topology. Set , so there are natural maps . Then and are perfectoid, but isn’t: the requisite -power roots of mod don’t exist. Nevertheless, every completed residue field of is perfectoid (exercise!), and the map induces an isomorphism .

OK, this bullet point turned out pretty long, but these things have been in my head for the last couple months and it feels good to let them out. Besides, Yoichi Mieda asked me about Q1 during the workshop, so despite the technical nature of these questions, I might not be the only one who cares.

- Oberwolfach continues to be one of the best places in the world to do mathematics.

Thanks to the organizers for putting together such an excellent week!

]]>Let be an irreducible geometric Galois representation. Suppose that is a direct sum of characters. Is induced from a character?

Emerton found a nice argument which proves Greenberg’s conjecture conditionally on some kind of -adic variational Hodge conjecture. Is there any similar evidence for this question in some higher-dimensional cases?

Do separated etale maps of schemes satisfy effective descent with respect to fpqc covers? This is known if one restricts to quasi-compact separated etale maps. An analogous result is true for perfectoid spaces.

Here’s a funny story I heard from Glenn Stevens a while back:

At some point in the early ’90s, before he announced his proof of Fermat, Wiles came to Boston and gave a seminar talk at BU. He spoke about what is now known as the Greenberg-Wiles duality formula. However, he didn’t mention his main motivations for this formula. The upshot is that Stevens came away from the talk with the sad feeling that Wiles had lost his touch.

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