- After two years and many excellent talks, we’ve decided to officially end the RAMpAGe seminar. Bhargav Bhatt gave the very first talk, way back in June 2020 – now Bhargav has generously agreed to also give the final talk, on August 10 at 12 noon EST. See you there!
- In December I’ll move to a tenured position at the National University of Singapore. To my colleagues in Asia – I’m really looking forward to travelling more easily around the region, beginning new collaborations, and engaging with the arithmetic geometry community!
In this post I want to talk about my recent paper with Bhargav Bhatt, which you can find here. This paper was a lot of fun to write, and I hope the toolkit we built will be useful for other researchers in this area. In this post I want to make some random remarks on this paper, which probably won’t mean anything if you don’t go read the real introduction to the paper first.
One funny point is that the proof of Theorem 1.6 leans on Theorem 1.7 fairly heavily, but in fact you can prove Theorem 1.6 without appealing to Theorem 1.7, at the price of much more intricate arguments. This was actually the state of the manuscript until mid-December, when we finally figured out how to prove Theorem 1.7.
Another funny point is that the discussion of the “standard” / “constructible” t-structure on turned out to be surprisingly subtle, cf. Theorem 3.39. Note that is by definition a full subcategory of , and the latter carries an obvious t-structure. Nevertheless, we weren’t able to settle the question of whether these t-structures are compatible:
Question. Do the cohomological functors on produced by Theorem 3.39 agree with the usual cohomology sheaves on ?
I would be extremely interested to know the answer to this.
One thing missing from the paper is any discussion of ULA sheaves. (See Fargues-Scholze for the foundations of ULA sheaves in p-adic geometry. In what follows I take , but the case should actually also be OK.) The first basic point to make is that for any rigid space , any object is ULA for the structure map . Sketch: The claim is local on , so we can assume is quasicompact. By Proposition 3.6 and stability of ULA sheaves under proper pushforward, we reduce to the special case where is constant. By an argument with resolution of singularities, we now reduce further to the case where is constant and is smooth, which is handled in Fargues-Scholze. Identical remarks apply with -coefficients, or with general coefficients (but then only for objects of “finite tor-dimension”).
This is already enough to show that the Lefschetz trace formula works as expected for proper rigid spaces in characteristic zero. More precisely, suppose given a correspondence of proper rigid spaces over an algebraically closed field, and a cohomological correspondence on some . Then the usual recipe to define local terms applies, and the expected equality holds true. (Note that is a perfect -complex by Theorem 3.35.(3).) This can be proved by imitating the unpleasant arguments with giant diagrams in SGA5, or by the amazing categorical magic of Lu-Zheng. Of course, the local terms are just as mysterious as in the case of schemes.
It’s also natural to guess that an analogue of Deligne’s generic ULA theorem holds in this setting. Let me formally state this as a conjecture.
Conjecture. Let be a proper map of characteristic zero rigid spaces, and let be any given object. Then there is a dense Zariski-open subset of over which is -ULA.
This should be within reach, but I didn’t think about it very much.
Finally, I want to highlight the open problems mentioned in Remark 4.10, Remark 4.14 and Section 4.5. Conjecture 4.16 is probably (for whatever reason) my favorite open problem right now. Actually, I don’t even know how to prove that is Hodge-Tate, or the even weaker statement that it has integral Hodge-Tate-Sen weights.
MH once pointed out the “linguistic trap” Grothendieck created when he defined the notion of an excellent ring: “Suppose somebody finds an even better class of rings? Then what?”
It turns out there IS an even better class of rings/schemes, which occurs naturally in some contexts.
Definition. A scheme is marvelous if it is Noetherian and excellent, and if for every irreducible component and every closed point . A ring is marvelous if is marvelous.
You can easily check that any marvelous scheme is finite-dimensional. Moreover, it turns out that a Noetherian quasi-excellent scheme is marvelous if and only if the function is a true dimension function for (in a certain technical sense). This function is of course the most naive and clean possibility for a dimension function on any given scheme, but it doesn’t always have the right properties.
Unfortunately, marvelous schemes are so marvelous that, unlike excellent schemes, they aren’t stable under many natural operations, not even under passing to an open subscheme! In fact, is marvelous if it is covered by marvelous open affines, but the converse fails. You can check that a scheme as simple as isn’t marvelous, even though is marvelous. So regular excellent schemes aren’t always marvelous, and adjoining a polynomial variable can kill marvelousity. I briefly entertained the hope that any Jacobson excellent scheme is marvelous, but this fails too (the scheme considered in EGAIV3 (10.7.3) is a counterexample).
It’s not all bad news, though:
- anything of finite type over or a field is marvelous,
- any excellent local ring is marvelous,
- any ring of finite type over an affinoid -algebra in the sense of rigid geometry is marvelous,
- any scheme proper over a marvelous scheme is marvelous; more generally, if is marvelous and is a finite type morphism which sends closed points to closed points, then is marvelous,
- if is a marvelous domain, then the dimension formula holds: for all prime ideals . (Recall that the dimension formula can fail, even for excellent regular domains.)
You might be wondering why I would care about such a stupid and delicate property. The reason is the following. Fix any marvelous scheme and any invertible on . Then there is a canonical potential dualizing complex (in the sense of Gabber) which restricts to on the regular locus of . Here is the (locally constant) dimension of the regular locus, so this numerology is the same as in the case of varieties. Moreover, for any prime invertible on , there is a good theory of -adic perverse sheaves on with the same numerology as in the case of varieties; in particular, the IC complex restricts to on the regular locus. (See sections 2.2 and 2.4 of Morel’s paper for more. Note in particular the hypothesis on in the first sentence of section 2.2: it is exactly the condition that is marvelous.) This discussion all applies, in particular, when for any -affinoid ring . This turns out to be an important ingredient in my forthcoming paper with Bhargav…
(One more comment: Most real-life examples of marvelous schemes, e.g. examples 1. and 3. above, are also Jacobson. It might be more reasonable to consider the class of marvelous Jacobson schemes, because these are permanent under finite type maps. But on the other hand we lose excellent local rings when we do this.)
In this post I’ll talk about a really fun trick Bhargav explained to me yesterday.
Let be a field with separable closure . Algebraic variety over means separated -scheme of finite type. Let be a prime invertible in . Perverse sheaf means perverse -sheaf.
If is a proper map of algebraic varieties over , the decomposition theorem tells you that after base extension to there is a direct sum decomposition
in . Here is some finite set of closed subvarieties, and is a lisse -sheaf on the smooth locus of . (My convention is that where is the natural map, so . Some people have different conventions for shifts here.)
The decomposition is non-canonical. In particular, it is not -equivariant, so it has no reason to descend to an analogous direct sum decomposition of in . Indeed, typically there is no such decomposition! However, as Bhargav explained to me, one can still descend a certain piece of to when is projective. This turns out to be good enough for some interesting applications.
To present Bhargav’s trick, let me make a definition. (What follows is a slight reinterpretation of what Bhargav told me, all mistakes are entirely due to me.)
Definition. Let be a perverse sheaf on a variety . Let be the inclusion of the maximal dense open subvariety such that is a (shifted) lisse sheaf. Then we define the generic part of as the perverse sheaf .
Note that is zero if and only if is supported on a nowhere-dense closed subvariety. Also, in general there is no map between and . However, in some cases is a direct summand of :
Proposition. Let be a perverse sheaf on a -variety , and suppose that the pullback of to is a direct sum of IC sheaves. Then is a direct summand of .
Proof. Let be as in the definition of the generic part of , with closed complement . Our assumptions together with the definition of the generic part guarantee that for some closed subvarieties contained in .
Now look at the natural maps . Set and . Since does not admit any nonzero quotient supported on , the composite map is zero. Thus factors over an inclusion . Moreover, has the same generic part as . This is enough to imply that , so we have a natural inclusion . A dual argument shows that factors over a surjection . It is now easy to see that the composite map is an isomorphism, so is a direct summand of .
Corollary 0. Let be a projective map of -varieties. Then is a direct summand of .
Proof. The decomposition theorem and the relative hard Lefschetz theorem give a decomposition in . Then is a direct sum of IC sheaves after pullback to , so we can apply the previous proposition.
Corollary 1. Let be a projective alteration of -varieties with smooth. Then is a direct summand of .
Proof. Check that is a direct summand of by playing with trace maps. Now apply the previous corollary.
Corollary 2. Let be a finite extension. Then for any -variety , the -adic intersection cohomology is a de Rham -representation.
Proof. Let be a resolution of singularities. The previous corollary shows that is a direct summand of compatibly with the -actions. Since is de Rham and the de Rham condition is stable under passing to summands, we get the result.
Note that we can’t prove this corollary by applying the decomposition theorem directly out of the box: the decomposition theorem does immediately give you a split injection , but this map is not guaranteed a priori to be -equivariant!
Corollary 3. Let be a finite extension of or . If satisfies the weight-monodromy conjecture for all smooth projective -varieties , then satisfies the weight-monodromy conjecture for all proper -varieties . In particular, the weight-monodromy conjecture holds for the -adic intersection cohomology of all proper -varieties for finite.
Proof. Entirely analogous to the previous proof.
It would be interesting to know whether Corollary 1 has a “motivic” incarnation. Here I will pretend to understand motives for a minute. Suppose we have an assignment on quasi-projective -varieties, where is a suitable triangulated category of constructible motivic sheaves on with -coefficients. This should come with the formalism of (at least) the four operations , and with faithful exact -adic realization functors compatible with the four operations. I think this has all been constructed by Ayoub, maybe with some tiny additional hypothesis on ? Let denote the symmetric monoidal unit. It then makes sense to ask:
Question. In the setting of Corollary 1, is there an idempotent such that for all ?
This would imply that the split injections provided by Corollary 1 can be chosen “independently of ”, i.e. that they are the -adic realizations of some split injection in .
I’m excited to announce a new weekly online-only research seminar on p-adic geometry and related topics, organized by Arthur-César Le Bras, Jared Weinstein, and myself. We will “meet” on Zoom on Thursdays at 16:00 UTC (that’s 9 am in California, noon in Boston, 5 pm in London, 6 pm in Bonn…). Bhargav Bhatt will give the first talk on June 18.
All credit to Jared for the name!
- Try and fail to buy toilet paper at Edeka.
- Make ketchup from scratch.
- Watch 30 episodes of Iron Chef.
- Annoy your wife by following her around the apartment, or by listening to old blues songs too loudly.
- Annoy VP* with basic questions about the p-adic Langlands program.
- Try and fail to buy toilet paper at Aldi’s and at Lidl.
- Meet with your masters and PhD students on Zoom.
- Buy a car.
- Gain a proper appreciation for Auslander-Gorenstein rings.
- Buy 72 rolls of Polish toilet paper on Amazon and pay extra for it to be absurdly shipped from Britain, because that’s the only option for some reason.
- Start a cool new joint project with BB and spend way too much time thinking about it (more on this soon!).
- Promise CJ you’ll finish writing the proof of a certain result in a certain nearly final version of a certain paper and then somehow don’t finish doing it (yet). See previous item.
- Move one kilometer to a much cheaper and much nicer apartment.
- Write silly blog posts that (hopefully) no one will read.
*The Lithuanian VP, not the French VP.
In p-adic geometry, what should it mean for a morphism to be a Zariski-closed immersion? For locally Noetherian adic spaces, the usual notion of a closed immersion of locally ringed spaces works just fine. For general analytic adic spaces, though, one quickly runs into annoying foundational issues. The issue is roughly as follows. Let be an (analytic) affinoid adic space. We can certainly define a reasonable notion of Zariski-closed subset, just by following our nose: a subset should be Zariski-closed if there is an ideal such that iff . These are exactly the subsets obtained by pulling back closed subsets of along the natural map . The problem, however, is that such a need not come from an actual closed immersion of an affinoid adic space into , because the quotient could just be some junky non-sheafy ring, and maybe there’s no canonical tweak (like replacing by its closure, or replacing by its uniform completion, or…) which will make it sheafy. And even if we can tweak to make it sheafy, how do we know that is still surjective after going to some rational subset ? You get the picture.
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!
A few years ago I started thinking about whether there was a natural rigid analytic version of the Artin-Grothendieck vanishing theorem. Last summer this grew into an obsession, and I managed to prove some general results. In particular, I showed that if is an affinoid rigid space over a complete algebraically closed field, AND comes via base change from an affinoid defined over a discretely valued subfield, then for all and all prime to the residue characteristic. I also proved a similar result with a non-constant coefficient sheaf, assuming moreover that the base field is of characteristic zero. This all got written up here.
Now, the hypothesis of definability over a discretely valued field is clearly stupid and shouldn’t be there, but I wasn’t able to remove it. So I was extremely happy this morning when Akhil Mathew and Bhargav Bhatt sent me an expanded version of their paper on arc-descent, in which they give a beautiful proof of rigid analytic Artin-Grothendieck vanishing without any superfluous assumptions. Their arguments are phrased in terms of algebraic geometry, rather than rigid analysis; in this post I want to recast (mostly for my own benefit I guess) the essential point of their argument in rigid analytic language.
The key is to prove the following.
Theorem (Bhatt-Mathew). Let be an affinoid rigid space over a complete algebraically closed nonarchimedean field . Set where is any integer prime to the residue characteristic. Then .
This implies the characteristic zero case of Conjecture 1.2 in my paper.
The proof proceeds in three steps.
Step One: Treat the case where is smooth. This was already done by Berkovich in the 90’s and I’ll take it for granted, although BM give their own nice argument for it. (Both arguments eventually appeal to the classical Artin-Grothendieck vanishing theorem.)
Step Two: Prove the weaker statement that in general.
For this we use induction on . I’ll assume for simplicity that has characteristic zero. Without loss of generality we can assume that is reduced. Then by excellence of affinoid algebras, we can pick some non-zero-divisor such that is regular. Fix a nonzero nonunit , and for any consider the rational subsets and inside . Set , so we get a Mayer-Vietoras distinguished triangle
for any . Note that and are smooth affinoids, so their etale cohomology is concentrated in degrees by Step One. Therefore, truncating the above Mayer-Vietoras sequence we get a quasi-isomorphism
for any . Moreover, in the sense of adic spaces, which implies that the etale cohomology of the left-hand side is the colimit of the etale cohomologies of the right-hand sides. Therefore, passing to the colimit over , the previous quasi-isomorphism gives a quasi-isomorphism
But now we win, because is an affinoid of dimension , so by the induction hypothesis its etale cohomology is concentrated in degrees .
Step Three. Bootstrap from the result of Step Two by a trick. More precisely, let and be as in the statement of the main theorem. By Step Two, we just have to show that . By another application of Step Two, the complex has cohomology in degree given by , and its enough to show that the latter module is zero. But
by the Kunneth formula*, and is a -dimensional affinoid, so its cohomology is concentrated in degrees by yet another application of Step Two. This gives the result.
*The necessary result is that if and are -affinoid spaces, then . I’m not sure if this is in the literature; Bhargav and Akhil prove (an algebraic form of) it in their paper. However, it is easy to deduce this from the results in Huber’s book. The point is that have canonical adic compactifications , and etale cohomology (with constant coefficients) doesn’t change if you replace or by its compactification. But then and are proper over (in the sense of Huber’s book), so it’s easy to show that
by the usual combination of proper base change and the projection formula.
Let be a spectral space. Following Bhatt-Scholze, say is w-local if the subset of closed points is closed and if every connected component of has a unique closed point. This implies that the natural composite map 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 , where is a nonarchimedean field and is an open and bounded valuation subring of , 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 together with a subset , let’s say is perfectoid (resp. affinoid perfectoid) if there is a pair where is a perfectoid space (resp. affinoid perfectoid space) and is a map of adic spaces identifying homeomorphically with and which is universal for maps of perfectoid spaces which factor through on topological spaces. Note that if the pair exists, it’s unique up to unique isomorphism.
Theorem. Let be a w-local affinoid perfectoid space. Then any subset of which is closed and generalizing, or which is quasicompact open, is affinoid perfectoid.
Proof when is closed and generalizing. The key point here is that the map defines a bijection between closed generalizing subsets of and closed subsets of the (profinite) space , by taking preimages of the latter or images of the former. To check that this is true, note that if is closed and generalizing, then its intersection with a connected component of being nonempty implies (since is generalizing) that contains the unique rank one point of . But then contains all specializations of that point (since is closed in ), so , so any given connected component of is either disjoint from or contained entirely in . This implies that can be read off from which closed points of it contains. Finally, one easily checks that is closed in , since is profinite and is quasicompact. Therefore .
Returning to the matter at hand, write as a cofiltered intersection of qc opens , . But qc opens in are the same as open-closed subsets, so each pulls back to an open-closed subset , and its easy to check that any such comes from a unique rational subset . Then is the perfectoid space we seek.
Proof when is quasicompact open.
First we prove the result when is connected, i.e. when as above. We claim that in fact is a rational subset of . When is empty, this is true in many stupid ways, so we can assume is nonempty. Since is a qc open, we can find finitely many nonempty rational subsets such that . But the ‘s are totally ordered, since any finite set of open bounded valuation subrings of is totally ordered by inclusion (in the opposite direction), so where is the largest .
Now we turn to the general case. For each point , we’ve proved that is a rational subset (possibly empty) of the fiber . Since and each is the topological space of a rational subset of , it’s now easy to check* that for every and for some small as above, there is a rational subset such that . Choose such a for each point in . Since , we can choose finitely many ‘s such that the ‘s give a covering of . Since each of these subsets is open-closed in , we can refine this covering to a covering of by finitely many pairwise-disjoint open-closed subsets where for all and for some (choice of) . Then again comes from a rational subset of , so the intersection comes from the rational subset of , and since by design, we (finally) have that is affinoid perfectoid. Whew!
*Here we’re using the “standard” facts that if is a cofiltered inverse system of affinoid perfectoid spaces with limit , then , and any rational subset is the preimage of some rational subset , and moreover if we have two such pairs and with the ‘s both pulling back to then they pull back to the same rational subset of for some large .
Let be a subset of a spectral space ; according to the incredible Lemma recorded in Tag 0A31 in the Stacks Project, the following are equivalent:
- is generalizing and pro-constructible;
- is generalizing and quasicompact;
- is an intersection of quasicompact open subsets of .
Moreover, if has one of these equivalent properties, 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 be a w-local affinoid perfectoid space. Then any generalizing quasicompact subset is affinoid perfectoid.
By an easy gluing argument, this implies even more generally (!) that if is a subset such that every point has a qc open neighborhood in such that is quasicompact and generalizing, then is perfectoid (not necessarily affinoid perfectoid). This condition* holds, for example, if is locally closed and generalizing; in that situation, I’d managed to prove that 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.”