Zariski closed immersions

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 X=\mathrm{Spa}(A,A^+) be an (analytic) affinoid adic space. We can certainly define a reasonable notion of Zariski-closed subset, just by following our nose: a subset Z \subset |X| should be Zariski-closed if there is an ideal I \subset A such that x \in |Z| iff |f|_x = 0\,\forall f \in I. These are exactly the subsets obtained by pulling back closed subsets of \mathrm{Spec}(A) along the natural map |\mathrm{Spa}(A,A^+)| \to |\mathrm{Spec}(A)|. The problem, however, is that such a Z need not come from an actual closed immersion of an affinoid adic space into X, because the quotient A/I could just be some junky non-sheafy ring, and maybe there’s no canonical tweak (like replacing I by its closure, or replacing A/I by its uniform completion, or…) which will make it sheafy. And even if we can tweak A/I to make it sheafy, how do we know that A \to A/I is still surjective after going to some rational subset U \subset X? You get the picture.

Perhaps surprisingly, the situation for affinoid perfectoid spaces is a lot better. In particular, if (A,A^+) is a perfectoid Tate-Huber pair, there are canonical bijections (satisfying some obvious compatibilities) between
1) closed subsets of \mathrm{Spec}(A),
2) Zariski-closed subsets of X=\mathrm{Spa}(A,A^+),
3) (isomorphism classes of) maps of Tate-Huber pairs (A,A^+) \to (B,B^+) where B is a perfectoid Tate ring, A \to B is surjective, and B^+ is the integral closure of the image of A^+ in B.

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

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 X \to Y is a Zariski-closed immersion if for any affinoid perfectoid space W with a map W \to Y, the base change X \times_{Y} W \to W 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 X \to Y is a Zariski-closed immersion and Y is (a small v-sheaf, a diamond, a locally spatial diamond, qc or qs or separated or partially proper over a base S), then so is X.

Example 1. Let f: X \to Y be a closed immersion of locally Noetherian adic spaces. If Y is affinoid (so X is too), then the map of diamonds f^{\lozenge} : X^{\lozenge} \to Y^{\lozenge} is a Zariski-closed immersion. This is easy.

Example 2. Let f: X \to Y be a closed immersion of locally Noetherian adic spaces again, but now assume that f is the analytification of a closed immersion of quasiprojective varieties.  Then f^{\lozenge}: X^{\lozenge} \to Y^{\lozenge} is a Zariski-closed immersion.  For this, we can use the assumption on f to choose a vector bundle \mathcal{E} on Y together with a surjection \mathcal{E} \twoheadrightarrow \mathcal{I}_{X} \subset \mathcal{O}_Y. Then for any map g: W \to Y from an affinoid perfectoid, the pullback g^{\ast}\mathcal{E} (in the usual sense of ringed spaces) is a vector bundle on W, hence generated by finitely many global sections e_1,\dots,e_n by Kedlaya-Liu. The images of e_1,\dots,e_n along the natural map (g^{\ast}\mathcal{E})(W) \to \mathcal{O}_{W}(W) generate an ideal, and the associated closed immersion of affinoid perfectoids V \to W represents the fiber product X^{\lozenge} \times_{Y^{\lozenge}} W. (Hat tip to PS for suggesting this vector bundle trick.)

Example 3. Let X^{\ast} be a minimally compactified Hodge-type Shimura variety with infinite level at p. Then the boundary Z \to X^{\ast} is a Zariski-closed immersion, and so is the diagonal X^{\ast} \to X^{\ast} \times X^{\ast}. (These both reduce to the previous example, using a small limit argument in the second case.) In particular, if U,V \subset X^{\ast} are any open affinoid perfectoid subsets, then U \cap V 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 p is perfectoid.

Example 4. Fix a perfectoid base field K of characteristic zero. Then the inclusions \mathrm{Fil}^n \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}} \subset \mathrm{B}_{\mathrm{dR}} are Zariski-closed immersions of (ind-)diamonds over \mathrm{Spd} K. This can be proved by induction on n, and the base case reduces to the fact that the inclusion \mathrm{Fil}^1 \mathrm{B}_{\mathrm{dR}}^{+} \subset \mathrm{B}_{\mathrm{dR}}^{+} is the pullback of \{ 0 \} \to \mathbf{A}^{1}_{K} along \theta. (To make the induction work, you need to pick an element \xi \in \mathrm{B_{dR}}^+(K) generating \ker \theta.)

Example 5. Fix a complete algebraically closed extension C / \mathbf{Q}_p. Fix a reductive group G / \mathbf{Q}_p and a geometric conjugacy class of G-valued cocharacters \mu. Then \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C} is a Zariski-closed immersion. Also, if \nu \leq \mu, then \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \nu, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, \leq \mu, C} is a Zariski-closed immersion. These claims can be reduced to the case G = \mathrm{GL}_n, which in turn reduces to Example 4 by some trickery.

Example 6. Fix a complete algebraically closed nonarchimedean field C of residue characteristic p, and let \mathcal{E} \to \mathcal{F} be any injective map of coherent sheaves on the Fargues-Fontaine curve X_C. Then the associated map of Banach-Colmez spaces \mathbb{V}(\mathcal{E}) \to \mathbb{V}(\mathcal{F}) 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 G \to H is a closed immersion of reductive groups, the induced map \mathrm{Gr}^{\mathrm{B_{dR}}}_{G, C} \to \mathrm{Gr}^{\mathrm{B_{dR}}}_{H, C} is a Zariski-closed immersion, although it is surely true. The problem here is (roughly) that an H-torsor over some affinoid perfectoid X can only be reduced to a G-torsor locally in the analytic topology on X, 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 X is affinoid perfectoid with a covering by rational subsets U_i, and Z is a closed subset such that Z \cap U_i is Zariski-closed in U_i for all i, is Z 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 0 \to H^1(\mathcal{O}(-1)) = \mathbf{A}^{1,\lozenge}_{C} / \underline{\mathbf{Q}_p} is a Zariski-closed immersion, because then pulling back would imply that \underline{\mathbf{Q}_p} \to \mathbf{A}^{1,\lozenge}_{C} 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 X/C and a closed subset S \subset X such that C maps isomorphically to the completed residue field at every point in S and such that S 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 f: X \to Y be a monomorphism of locally Noetherian adic spaces. If f^{\lozenge} is a Zariski-closed immersion, is f actually a closed immersion?

Happy new year!




spectral spaces; snark

(Update 12/24: Unfortunately the argument below doesn’t work as written. I overlooked the following detail in my “proof” that X^{wl} \to X is open: if S \to S' is a surjective continuous map of finite T_0 spaces, it’s not necessarily true that S^Z \to S'^Z is surjective. For instance, if S'= \{ \eta \rightsquigarrow s \} is the spectrum of a DVR and S=\{ \eta, s \} is S' equipped with the discrete topology, then S'^Z has three points but S=S^Z has two.

For the argument below to work, it would be enough to know that for any open subset V \subset X^{wl}, its image in X 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 X be a spectral space, and let Z be a closed generalizing nowhere-dense subset of X. Then Z is nowhere-dense for the constructible topology on X, i.e. Z doesn’t contain any nonempty constructible subset of X.

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

Corollary. Let (A,A^+) be some Tate-Huber pair with adic spectrum X=\mathrm{Spa}(A,A^+), and let Z \subset X be a Zariski-closed nowhere-dense subset. Suppose U_1 and U_2 are quasicompact open subsets of X such that U_1 \cap (X - Z) = U_2 \cap (X - Z). Then U_1 = U_2.

Proof of Corollary. We need to check that V= (U_1 \cup U_2 ) - (U_1 \cap U_2) is empty. But V is a constructible subset of X contained in Z, 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 X^{wl} of X. Is there a more direct approach? Am I missing something obvious?

(Sketch of actual argument: the profinite set of closed points X^{wl}_{c} maps homeomorphically onto X equipped with the constructible topology, so if V \subset X is constructible it is clopen when viewed as a subset of X^{wl}_{c}. The key point is then to check that Z is nowhere-dense when viewed as a subset of X^{wl}_c. This can be done, using that the natural surjection t: X^{wl} \to X is open and that t^{-1}(Z) (which is then closed, generalizing and nowhere-dense in X^{wl}, the last point by openness of t) is the preimage of its image in \pi_0(X^{wl}) \cong X^{wl}_c.

The openness of t 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 T_0 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.

Hard to believe

In this brief post, I want to draw attention to an amazing theorem which deserves to be well-known.  Probably many readers are familiar with Nagata’s compactification theorem: if S is any qcqs scheme and f: X \to S is a separated morphism of finite type, then f can be factored as an open embedding followed by a proper morphism. This is a by-now-classical result, and has many applications.

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

Temkin’s Factorization Theorem. Let f:X \to Y be any separated morphism of qcqs schemes. Then f 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 i: \mathbf{A}^2 - \{ 0,0 \} \to \mathbf{A}^2 should give a counterexample. But then they realize (or I point out) that i can be factored as p \circ j, where p: X \to \mathbf{A}^2 is the blowup of \mathbf{A}^2 at the origin and j is the natural (affine!) open immersion of \mathbf{A}^2 - \{ 0,0 \} into X. Then they are convinced.

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

Some historical snippets

Probably everyone is familiar with the MacTutor History of Mathematics website. While browsing through their additional material, I came across some wonderful things:

  • 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. …”

p-adic Kahler manifolds

In complex geometry, the most interesting class of complex manifolds is probably the Kahler class. In the non-archimedean world, say over a fixed p-adic base field K, the analogue of a compact complex manifold is a smooth proper rigid analytic space. In some ways, these are already surprisingly “close” to being Kahler – in particular, the Hodge-de Rham spectral sequence of such a space always degenerates at E_1. However, Hodge symmetry can definitely fail. A standard example is the non-archimedean Hopf surface X = \mathbf{A}^2_{K} \smallsetminus \{ (0,0) \} / p^{\mathbf{Z}} where p^n acts through diagonal multiplication. By a fun direct calculation, one checks that H^0(X,\Omega^1_X)=0 and H^1(X,\mathcal{O}_X) = K, so Hodge symmetry fails in degree one.

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 X satisfies (*) if it admits a formal model \mathfrak{X} over \mathcal{O}_K 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 X be a smooth proper rigid space satisfying (*). Then h^{1,0}(X) = h^{0,1}(X).

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 D be the complex disk, with D^\times =D \smallsetminus \{0 \} the punctured disk. Let f:Y \to D be a proper map of complex analytic spaces. Suppose that f^{-1}(D^\times) \to D^\times is a submersion, and that the central fiber Y_0=f^{-1}(0) is the analytification of a projective (and not necessarily smooth) algebraic variety. Then for all t \in D^\times with |t| \ll 1, the fiber Y_t satisfies Hodge symmetry and Hodge-de Rham degeneration.

Of course, the analogy is that \mathfrak{X} \to \mathrm{Spf} \mathcal{O}_K is analogous to Y \to D, and X is analogous to the “nearby” fibers Y_t with 0<|t| \ll 1.

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.