A stupid remark on cohomological dimensions

Let Y be a finite-dimensional Noetherian scheme, and let \ell be a prime invertible on Y. Gabber proved that if f:X \to Y is any finite type morphism, then there is some integer N such that R^n f_{\ast} F vanishes for all \ell-torsion etale sheaves F and all n > N, cf. Corollary XVIII.1.4 in the Travaux de Gabber volume. In particular, if the \ell-cohomological dimension \mathrm{cd}_{\ell}(Y) is finite, then so is \mathrm{cd}_{\ell}(X). It’s natural to ask for a quantitative form of this implication.

Claim. One has the bound \mathrm{cd}_{\ell}(X) \leq \mathrm{cd}_{\ell}(Y)+2\mathrm{dim}f+2\mathrm{dim}(X)+s(X).

Here \mathrm{dim}f is the supremum of the fibral dimensions of f, and s(X) \in \mathbf{Z}_{\geq 0} is defined to be one less than the minimal number of separated open subschemes required to cover X. In particular, s(X)=0 iff X is separated.

Can this be improved?

Anyway, here’s a cute consequence:

Corollary. Let Y be a finite-dimensional Notherian scheme, and let \ell be a prime invertible on Y such that \mathrm{cd}_{\ell}(Y) < \infty. Then all affine schemes X \in Y_{\mathrm{et}} have uniformly bounded \ell-cohomological dimension. In particular, the hypotheses of section 6.4 of Bhatt-Scholze hold, so the arguments therein show that D(Y,\mathbf{F}_{\ell}) is compactly generated and that its compact objects are exactly the objects of D^{b}_{c}(Y,\mathbf{F}_{\ell}).

The one-point compactification of a scheme, part 1

In this series of posts I want to talk about some joint work in progress with Johan de Jong. The goal here is to give a new and totally canonical definition of the functor Rf_! in the etale cohomology of schemes, and to do some really fun geometry along the way. In fact we go beyond the usual situation where f is assumed to be compactifiable, and give a canonical definition of Rf_! for f any locally finite type morphism of locally Noetherian schemes. Such a definition has also been proposed by Liu-Zheng, by a method which requires a very heavy use of \infty-categories. Our approach does not involve \infty-categories at all.

For motivation, let X be a locally compact Hausdorff space. The one-point compactification of X is obtained by suitably topologizing the set \overline{X}=X \cup \{\infty\}: precisely, one takes the open subsets to be all the open subsets of X together with all subsets of the form V \cup \{\infty \} where V \subset X is such that X \smallsetminus V is a closed compact subset of X. Then \overline{X} is a compact Hausdorff space, and X is a dense open subset of \overline{X} if X is non-compact. Quite generally, one can check that the inclusion X \to \overline{X} is final among all open embeddings of locally compact Hausdorff spaces X \to Y.

Our first goal is to imitate this compactification in the setting of schemes. The correct definition turns out to be as follows.

Definition 1.1. Fix a base scheme S, and let f: X \to S be a morphism of schemes which is separated and of finite type (for brevity, we say X is a good S-scheme if its structure map is separated and of finite type). The one-point compactification of X over S, denoted \overline{X}^S, is the contravariant functor from S-schemes to sets sending an S-scheme T\to S to the set of closed subschemes Z \subset X_T = X\times_S T such that the composite map Z \to X_T \to T is an open immersion. Equivalently, \overline{X}^S(T) is the set of pairs (Z,\varphi) where Z \subset T is an open subscheme and \varphi : Z \to X is an S-scheme map whose graph \Gamma_{\varphi}: Z \to X\times_S T is a closed immersion.

Usually S will be clear from context, and we’ll abbreviate \overline{X}^S to \overline{X}. Let \overline{f} : \overline{X} \to S denote the “structure map”.

(Here and in what follows, we write \mathrm{Sch}/S for the category of S-schemes, and we freely “do geometry” in the category of presheaves of sets on \mathrm{Sch}/S in the modern style, since S-schemes embed fully faithfully into this category in the usual way. There are some small issues here with “big categories” which I will totally ignore; suffice it to say that we “fix a choice of a category \mathrm{Sch}/S” in the sense of the Stacks Project.)

Anyway, here are some immediate observations on this thing. First of all, there is a canonical map j^X : X \to \overline{X} sending a T-point \varphi: T \to X to the pair (T,\varphi); indeed, the separateness of f guarantees that \Gamma_{\varphi} : T \to X_T is a closed immersion. Moreover, the structure map \overline{X} \to S has a canonical “section at infinity” \infty: S \to \overline{X} sending any S-scheme T to the closed subscheme Z= \emptyset \subset X_T, and j^X and \infty are “disjoint” in the evident sense.

Example 1.2. If S is arbitrary and X \to S is proper, then \overline{X} = X \coprod S. (Hint: For any T-point of \overline{X}, the map Z \to T is a proper open immersion.)

Example 1.3. If S is arbitrary and X= \mathbf{A}^1_S, then \overline{X} is the ind-scheme obtained as an “infinite pinching” of \mathbf{P}^1_S along the section at infinity. More precisely, we can assume (e.g. by Theorem 1.5.i below) that S=\mathrm{Spec}A is affine. Let B_n \subset A[t^-1] be the ring of polynomials \sum a_i t^{-i} such that a_i=0 for all 0 < i < n. Set  U=\mathrm{Spec}A[t] and V_n = \mathrm{Spec}B_n; gluing these along their common open \mathbf{G}_{m,S} in the obvious way, we get an inductive system of schemes X_1=\mathbf{P}^1_S \to X_2 \to X_3 \to \cdots, with compatible maps X_i \to \overline{X}. In the colimit this gives a map \mathrm{colim} X_n \to \overline{X}. This map turns out to be an isomorphism, but this is not so obvious.

Example 1.4. If S is arbitrary and X = \mathbf{A}^2_S, then \overline{X} is NOT an ind-scheme or ind-algebraic space.

This last example is typical: for almost all X \to S, the functor \overline{X} will not be an object of classical algebraic geometry. Nevertheless, it turns out to have very reasonable properties, which we collect in the following theorem:

Theorem 1.5. Fix a base scheme S, and let X \to S and \overline{X}=\overline{X}^S \to S be as above. Then:
i. If S' \to S is any scheme map, there is a canonical isomorphism \overline{X}^S \times_S S' \cong \overline{X \times_S S'}^{S'}.
ii. The map j^X : X \to \overline{X} is (representable in schemes and) an open immersion.
iii. The functor \overline{X} is a sheaf for the fpqc topology.
iv. The structure map \overline{X} \to S satisfies the valuative criterion of properness.
v. If X \to S is of finite presentation, then \overline{X} \to S is limit-preserving.
vi. The diagonal \Delta: \overline{X} \to \overline{X} \times_S \overline{X} is representable in formal algebraic spaces: for any scheme with a map T \to \overline{X} \times_S \overline{X}, the fiber product W= \overline{X} \times_{\Delta, \overline{X} \times_S \overline{X}} T is representable by a formal algebraic space whose underlying reduced subspace is a closed subscheme of T. If S is locally Noetherian then W is a countably indexed directed colimit of closed subschemes of T along thickenings.
vii. If g: X \to Y is any proper map of good S-schemes, there is a canonical map \overline{g}: \overline{X} \to \overline{Y} such that j^Y \circ g = \overline{g} \circ j^X.
viii. If h: U \to V is any open immersion of good S-schemes, there is a canonical map \tilde{h}: \overline{V} \to \overline{U} such that j^U = \tilde{h} \circ j^V \circ h.

Let me say a few words about the proofs. Part i. is trivial from the definitions. For ii., one checks directly that for a given T-point T \to \overline{X} with associated pair (Z,\varphi), the pullback X \times_{\overline{X}} T is just the open subscheme Z \subset T.

For iii., let T' \to T be an fpqc cover, and suppose given (Z',\varphi') \in \overline{X}(T') lying in the equalizer of \overline{X}(T') \rightrightarrows \overline{X}(T' \times_T T'). One first descends the open subscheme Z' \subset T' to an open subscheme Z \subset T using the fact that |T'| \to |T| is a quotient map, and then one descends the morphism \varphi' to a map Z \to X. To see that (Z,\varphi) has the right properties, note that the graph \Gamma_{\varphi} pulls back to the closed immersion \Gamma_{\varphi'} along the fpqc cover X_{T'} \to X_{T}, so \Gamma_{\varphi} is necessarily a closed immersion.

For iv., one reduces by i. to checking that if S=\mathrm{Spec}A is the spectrum of an arbitrary valuation ring with generic point \eta \in S and X \to S is any good S-scheme, then the evident “restriction” map r: \overline{X}^S(S) \to \overline{X_{\eta}}^{\eta}(\eta) is a bijection. After showing that the points at infinity match up, this reduces to showing that any section s: \eta \to X_{\eta} spreads out to a unique point (Z,\varphi) \in \overline{X}^S(S). For this, let Z \subset X be the scheme-theoretic image of s in X. By the Lemma in my previous post, the composite map Z \to S is an open immersion, and we’re done.

For v., one takes an arbitrary T-point of \overline{X}, where T = \lim T_i is a limit of affine S-schemes, and then uses “standard limit nonsense” to show that everything comes by pullback from a T_i-point for some i – I leave it as an exercise for the reader to fill in the details, making use of Proposition 8.6.3 and Corollaire 8.6.4 in EGA IV_3.

Part vi. is probably the hardest. Let T \to \overline{X} \times_{S} \overline{X} be as in the statement. This corresponds to a pair of T-points of \overline{X}, i.e. a pair of closed subschemes Z_i \subset X_T for i=1,2 such that the induced maps Z_i \to T are open immersions. Let U=Z_1 \cup_T Z_2, so this is an open subscheme of T. Let Z=Z_1 \times_{X_T} Z_2 be the intersection of the Z_i‘s inside X_T, so we get natural closed immersions Z \to Z_i, and composing either of them with the inclusion Z_i \to U realizes Z as a closed subscheme of the open subscheme U \subset T. At this point we make the

Definition. Let T be a scheme, and suppose given an open subscheme U \subset T together with a closed subscheme Z \subset U. Let T_{Z \to U} be the subfunctor of T whose V-points are given by scheme maps f: V \to T such that f^{-1}(U) \to U factors over the closed immersion Z \to U.

Now, returning to the situation at hand, you can check directly from the definitions (if you dare) that the fiber product W in vi. is given by the functor T_{Z \to U}, for the specific T,Z,U above. This reduces us to a general result:

Lemma. Notation as in the previous definition, the functor T_{Z\to U} is a formal algebraic space whose underlying reduced subspace is a closed subscheme of T, namely the reduced closed subscheme corresponding to the closed subset |Z| \cup (|T| \smallsetminus |U|) \subset |T|. If T is Noetherian, \mathcal{I} \subset \mathcal{O}_T is the coherent ideal sheaf corresponding to the scheme-theoretic closure \overline{Z} \subset T of Z, and \mathcal{J} is any coherent ideal sheaf whose vanishing locus cuts out the closed subspace |T| \smallsetminus |U|, then T_{Z \to U} \cong \mathrm{colim}\, \underline{\mathrm{Spec}}\mathcal{O_X}/(\mathcal{I}\cdot \mathcal{J}^n).

Intuitively, T_{Z \to U} is the “union” inside T of the locally closed subscheme Z and the formal completion of T along the complement of U.

For vii., one takes the scheme-theoretic image of Z \subset X_T along the map Z \to X_T \to Y_T and then checks that the resulting closed subscheme Z' \subset Y_T has the right properties; in fact Z' \simeq Z.

For viii., one takes the pullback of Z \subset V_T along the open immersion U_T \to V_T. This clearly has the right properties.

Note that the contravariant functoriality in part viii. might seem strange at first, since it has no analogue for schematic compactifications.  However, this is totally analogous with the situation for one-point compactifications of topological spaces: if U \to V is an open embedding of locally compact Hausdorff spaces, then \overline{U} is obtained from \overline{V} by contracting \overline{V} \smallsetminus U down to the point at infinity, giving a canonical map \overline{V} \to \overline{U}.

In part 2, we’ll discuss the applications to etale cohomology.




Some disconnected thoughts

Sorry for the silence here, the last six months have been pretty busy. Anyway, here are some small and miscellaneous thoughts.

\bullet Dan Abramovich is a really excellent and entertaining writer.  I especially recommend this BAMS review of books by Cutkosky and Kollar, and this survey article for the 2018 ICM; after reading these, you’ll feel like you really understand what “resolution of singularities” looks like as a working field.  His mathscinet review of Fesenko’s IUTT survey is also deeply funny – let us hope that the phrase “resounding partial success” enters common usage.

\bullet Speaking of the 2018 ICM, Scholze’s article is also now available here. I was amused to see a reference to “prismatic cohomology” in the last paragraph – when I visited Bhargav in October, this object had a truly horrible and useless provisional name, so it’s great to see that name replaced by such a perfect and suggestive one.  (This is a new cohomology theory which is closely related to crystalline cohomology, and is also morally related to diamonds – cf. these notes for the actual definition.)

\bullet Recently I had need of the following result:
Lemma. Let S=\mathrm{Spec}\,A be the spectrum of a valuation ring, with generic point \eta \in S. Let X \to S be a separated and finite type map of schemes, and let x: \eta \to X_{\eta} be a section over the generic point of S, with scheme-theoretic image Z \subset X. Then the induced map Z \to S is an open immersion.

Note that if X \to S is proper, then Z \to S is an isomorphism. Anyway, I ended up working out two proofs: a short argument relying on Nagata compactification, and a longer argument which avoided this result but used some other pretty tricky stuff, including Zariski’s main theorem and Raynaud-Gruson’s magic theorem that if R is a domain, then any flat finite type R-algebra is finitely presented. If someone knows a truly elementary proof of this lemma, I’d be interested to see it.

Next time I’ll say something about what I needed this for…

Extending finite etale coverings

Fix a nonarchimedean field K of residue characteristic p, and let X be a normal rigid analytic space over K.  Suppose we’re given a closed nowhere-dense analytic subspace Z \subset X and a finite etale cover Y of X \smallsetminus Z.  It’s natural to ask if Y can be extended to a finite cover Y' \to X, and whether some further conditions on Y' pin such an extension down uniquely.  Although the analogous problem for complex analytic spaces was solved by Stein and Grauert-Remmert in the 50s (cf. Grauert and Remmert’s article here), there isn’t very much literature on this problem in the rigid analytic context, with the notable exception of Lutkebohmert’s paper, about which more in a minute.  Anyway, it turns out that at least for a base field K of characteristic zero, this problem has a very satisfying answer, and the proof is a fun exercise in swinging lots of big hammers.

First, here’s the precise definition of “cover” which we’ll use.

Definition. Let X be a normal rigid analytic space.  A cover of X is a finite surjective map \pi: Y \to X from a normal rigid space Y, such that one of the following two equivalent conditions holds:
1. There exists a closed nowhere-dense analytic subset Z \subset X such that \pi^{-1}(Z) is nowhere-dense and Y \smallsetminus \pi^{-1}(Z) \to X \smallsetminus Z is finite etale.
2.  Each irreducible component Y_i of Y maps surjectively onto an irreducible component X_i of X, and contains a point y_i such that \mathcal{O}_{X,\pi(y_i)} \to \mathcal{O}_{Y_i,y_i} is etale.

Equivalence of these conditions is a fun exercise left to the reader; note that the second requirement in 2. is automatic when K has characteristic 0.

Theorem. Let X be a normal rigid analytic space over a characteristic zero nonarchimedean field K, and let Z \subset X be any closed nowhere-dense analytic subset.  Then any finite etale cover of X \smallsetminus Z extends uniquely to a cover of X.

In other words, the restriction functor from {covers of X etale over X \smallsetminus Z} to {finite etale covers of X \smallsetminus Z} is an equivalence of categories.

The uniqueness holds without any condition on K, and is an easy consequence of a powerful theorem due to Bartenwerfer.  To explain this result, let X be a normal rigid space and let Z \subset X be any closed nowhere-dense analytic subset. Then Barternwerfer proved that any bounded function on X \smallsetminus Z extends (uniquely) to a function on X. In particular, if Y \to X is a cover and U \subset X is any open affinoid subset, then \mathcal{O}_Y(\pi^{-1}(U)) \cong \mathcal{O}_{Y}^{+}(\pi^{-1}(U \smallsetminus U \cap Z))[1/ \varpi] depends only on the restriction of Y to X \smallsetminus Z. Since the affinoids \pi^{-1}(U) cover Y, this gives the desired uniqueness.  More generally, this argument shows that for any closed nowhere-dense analytic subset Z \subset X, the restriction functor from covers of X to covers of X \smallsetminus Z is fully faithful.

The existence of an extension is harder, of course.  Until further notice, assume K has characteristic zero.  Note that by the uniqueness argument, we can always work locally on X when extending a finite etale cover of X \smallsetminus Z.  Now the key input is the following base case, due to Lutkebohmert:

Theorem (Lutkebohmert): If X is a smooth rigid space and Z \subset X is a simple normal crossings divisor, then any finite etale cover of X \smallsetminus Z extends to a cover of X.

This is more or less an immediate consequence of Lemma 3.3 in Lutkebohmert’s paper, although he doesn’t state this result so explicitly (and curiously, he never discusses the uniqueness of extensions).  The main ingredient aside from this Lemma is a result of Kiehl on “tubular neighborhoods”, which says (among other things) that if D \subset X is a snc divisor in a smooth rigid space, then for any point x in D at which r components of D meet, we can find some small affinoid neighborhood U of x in X together with a smooth affinoid S and an isomorphism U \simeq S \times B^r (where B^r = \mathrm{Sp}K \left\langle X_1, \dots, X_r \right\rangle denotes the r-dimensional closed ball) under which the individual components of D meeting x identify with the zero loci of the coordinate functions X_i.

Granted these results, we argue as follows.  Clearly we can assume that X is quasicompact.  We now argue by induction on the maximal number i(D) of irreducible components of D passing through any individual point of X. Let me sketch the induction informally. If i(D)=1, then D is smooth, so Kiehl’s result puts us exactly in the situation covered by the case r=1 of Lemma 3.3. If i(D)=2, then locally on X we can assume that D has two smooth components D_1 and D_2. By the previous case, any finite etale cover Y of X \smallsetminus D extends uniquely to covers Y_i of X \smallsetminus D_i, which then glue to a cover Y_0 of X \smallsetminus D_1 \cap D_2.  But now locally along D_1 \cap D_2, Kiehl’s result puts is in the situation covered by the case r=2 of Lemma 3.3, and then Y_0 extends to a cover of X.  If i(D)=3, then locally on X we can assume that D has three smooth components D_1, D_2, D_3. By the previous case, any finite etale cover of X \smallsetminus D extends to a cover Y_i of X \smallsetminus D_i, for each i \in \{1,2,3\}; here we use the fact that i(D \smallsetminus D_i) \leq 2 for D \smallsetminus D_i viewed as a strict normal crossings divisor in X \smallsetminus D_i.  Again the Y_i‘s glue to a cover Y_0 of X \smallsetminus D_1 \cap D_2 \cap D_3, and again locally along D_1 \cap D_2 \cap D_3 Kiehl’s result puts us in the situation handled by Lemma 3.3, so Y_0 extends to a cover of X.  Etc.

To get existence in the general case, we use some recent results of Temkin on resolution of singularities.  More precisely, suppose X = \mathrm{Sp}(A) is an affinoid rigid space, and Z \subset X is a closed nowhere-dense subset as before; note that Z=\mathrm{Sp}(B) is also affinoid, so we get a corresponding closed immersion of schemes \mathcal{Z} = \mathrm{Spec}(B) \to \mathcal{X} = \mathrm{Spec}(A).  These are quasi-excellent schemes over \mathbf{Q}, so according to Theorem 1.1.11 in Temkin’s paper, we can find a projective birational morphism f: \mathcal{X}' \to \mathcal{X} such that \mathcal{X}' is regular and (\mathcal{X}' \times_{\mathcal{X}} \mathcal{Z})^{\mathrm{red}} is a strict normal crossings divisor, and such that f is an isomorphism away from \mathcal{Z} \cup \mathcal{X}^{\mathrm{sing}}.  Analytifying, we get a proper morphism of rigid spaces g: X' \to X with X' smooth such that g^{-1}(Z)^{\mathrm{red}} is an snc divisor etc.

Suppose now that we’re given a finite etale cover Y of X \smallsetminus Z.   Pulling back along g, we get a finite etale cover of X' \smallsetminus g^{-1}(Z), which then extends to a cover h: Y'\to X' by our previous arguments. Now, since g \circ h is proper, the sheaf (g \circ h)_{\ast} \mathcal{O}_{Y'} defines a sheaf of coherent \mathcal{O}_X-algebras. Taking the normalization of the affinoid space associated with the global sections of this sheaf, we get a normal affinoid Y'' together with a finite map Y'' \to X and a canonical isomorphism Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}} \cong Y|_{(X \smallsetminus Z)^{\mathrm{sm}}}. The cover we seek can then be defined, finally, as the Zariski closure Y''' of Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}} in Y'': this is just a union of irreducible components of Y'', so it’s still normal, and it’s easy to check that Y''' satisfies condition 1. in the definition of a cover. Finally, since Y''' and Y are canonically isomorphic after restriction to (X \smallsetminus Z)^{\mathrm{sm}}, the uniqueness argument shows that this isomorphism extends to an isomorphism Y'''|_{X \smallsetminus Z} \cong Y. This concludes the proof.

Combining this existence theorem with classical Zariski-Nagata purity, one gets a purity theorem for rigid spaces:

Corollary. Let X be a smooth rigid analytic space over a characteristic zero nonarchimedean field, and let Z \subset X be any closed analytic subset which is everywhere of codimension \geq 2.  Then finite etale covers of X are equivalent to finite etale covers of X \smallsetminus Z.

Presumably this result has other fun corollaries.  I’d be happy to know more.

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.