## The six functors for Zariski-constructible sheaves in rigid geometry

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 $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ turned out to be surprisingly subtle, cf. Theorem 3.39. Note that $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ is by definition a full subcategory of $D(X_v,\mathbf{Z}_{\ell})$, 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 $^c \mathcal{H}^n(-)$ on $D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})$ produced by Theorem 3.39 agree with the usual cohomology sheaves on $D(X_v,\mathbf{Z}_{\ell})$?

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 $\ell \neq p$, but the case $\ell = p$ should actually also be OK.) The first basic point to make is that for any rigid space $X/K$, any object $A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell})$ is ULA for the structure map $X \to \mathrm{Spa}K$. Sketch: The claim is local on $X$, so we can assume $X$ is quasicompact. By Proposition 3.6 and stability of ULA sheaves under proper pushforward, we reduce to the special case where $A = \mathbf{F}_{\ell}$ is constant. By an argument with resolution of singularities, we now reduce further to the case where $A$ is constant and $X$ is smooth, which is handled in Fargues-Scholze. Identical remarks apply with $\mathbf{Z}_{\ell}$-coefficients, or with general $\mathbf{Z}/n$ 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 $c=(c_1,c_2): C \to X \times X$ of proper rigid spaces over an algebraically closed field, and a cohomological correspondence $u: c_1^{\ast}A \to Rc_2^{!}A$ on some $A \in D^b_{zc}(X,\mathbf{Z}_{\ell})$. Then the usual recipe to define local terms applies, and the expected equality $\mathrm{tr}(u|R\Gamma(X,A)) = \sum_{\beta \in \pi_0 \mathrm{Fix}(c)} \mathrm{loc}_{\beta}(u,A)$ holds true. (Note that $R\Gamma(X,A)$ is a perfect $\mathbf{Z}_{\ell}$-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 $\mathrm{loc}_{\beta}(u,A)$ 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 $f:X \to Y$ be a proper map of characteristic zero rigid spaces, and let $A \in D^{(b)}_{zc}(X,\mathbf{F}_{\ell})$ be any given object. Then there is a dense Zariski-open subset of $Y$ over which $A$ is $f$-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 $IH^{\ast}(X_C,\mathbf{Q}_p)$ is Hodge-Tate, or the even weaker statement that it has integral Hodge-Tate-Sen weights.

## Better than excellent

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 $X$ is marvelous if it is Noetherian and excellent, and if $\dim \mathcal{O}_{Y,y} = \dim Y$ for every irreducible component $Y \subset X$ and every closed point $y \in Y$. A ring $A$ is marvelous if $\mathrm{Spec}(A)$ 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 $x \in |X| \mapsto \dim \overline{ \{ x \} }$ is a true dimension function for $X$ (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, $X$ is marvelous if it is covered by marvelous open affines, but the converse fails. You can check that a scheme as simple as $\mathrm{Spec}\mathbf{Z}_p[x]$ isn’t marvelous, even though $\mathrm{Spec}\mathbf{Z}_p$ 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 $S$ considered in EGAIV3 (10.7.3) is a counterexample).

It’s not all bad news, though:

1. anything of finite type over $\mathbf{Z}$ or a field is marvelous,
2. any excellent local ring is marvelous,
3. any ring of finite type over an affinoid $K$-algebra in the sense of rigid geometry is marvelous,
4. any scheme proper over a marvelous scheme is marvelous; more generally, if $X$ is marvelous and $f: Y \to X$ is a finite type morphism which sends closed points to closed points, then $Y$ is marvelous,
5. if $A$ is a marvelous domain, then the dimension formula holds: $\dim (A/\mathfrak{p}) + \mathrm{ht}\,\mathfrak{p} = \dim A$ for all prime ideals $\mathfrak{p} \subset A$. (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 $X$ and any $n$ invertible on $X$. Then there is a canonical potential dualizing complex $\omega_{X} \in D^{b}_{c}(X,\mathbf{Z}/n)$ (in the sense of Gabber) which restricts to $\mathbf{Z}/n[2\dim ](\dim)$ on the regular locus of $X$. Here $\dim$ 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 $\ell$ invertible on $X$, there is a good theory of $\ell$-adic perverse sheaves on $X$ with the same numerology as in the case of varieties; in particular, the IC complex restricts to $\mathbf{Q}_{\ell}[\dim]$ on the regular locus. (See sections 2.2 and 2.4 of Morel’s paper for more. Note in particular the hypothesis on $X$ in the first sentence of section 2.2: it is exactly the condition that $X$ is marvelous.) This discussion all applies, in particular, when $X=\mathrm{Spec}(A)$ for any $K$-affinoid ring $A$. 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.)

## A trick and the decomposition theorem

In this post I’ll talk about a really fun trick Bhargav explained to me yesterday.

Let $K$ be a field with separable closure $C$. Algebraic variety over $K$ means separated $K$-scheme of finite type. Let $\ell$ be a prime invertible in $K$. Perverse sheaf means perverse $\mathbf{Q}_\ell$-sheaf.

If $f:X \to Y$ is a proper map of algebraic varieties over $K$, the decomposition theorem tells you that after base extension to $C$ there is a direct sum decomposition

$Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus_i IC_{Z_i}(\mathcal{L}_i)[n_i]\,\,\,\,\,\,\,\,(\dagger)$

in $D^b_c(Y_C,\mathbf{Q}_\ell)$. Here $Z_i \subset Y_{C}$ is some finite set of closed subvarieties, and $\mathcal{L}_i$ is a lisse $\mathbf{Q}_\ell$-sheaf on the smooth locus of $Z_i$. (My convention is that $IC_{Z}(\mathcal{L}) = j_{!\ast} (\mathcal{L}[\dim Z])$ where $j:Z^{sm} \to X$ is the natural map, so $IC_{X,\mathbf{Q}_\ell} = IC_{X}(\mathbf{Q}_\ell)$. Some people have different conventions for shifts here.)

The decomposition $(\dagger)$ is non-canonical. In particular, it is not $\mathrm{Aut}(C/K)$-equivariant, so it has no reason to descend to an analogous direct sum decomposition of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Indeed, typically there is no such decomposition! However, as Bhargav explained to me, one can still descend a certain piece of $(\dagger)$ to $D^b_c(Y,\mathbf{Q}_\ell)$ when $f$ 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 $\mathcal{F}$ be a perverse sheaf on a variety $X$. Let $j:U \to X$ be the inclusion of the maximal dense open subvariety such that $j^\ast \mathcal{F}$ is a (shifted) lisse sheaf. Then we define the generic part of $\mathcal{F}$ as the perverse sheaf $\mathcal{F}^{gen} = j_{!\ast} j^{\ast} \mathcal{F}$.

Note that $\mathcal{F}^{gen}$ is zero if and only if $\mathcal{F}$ is supported on a nowhere-dense closed subvariety. Also, in general there is no map between $\mathcal{F}^{gen}$ and $\mathcal{F}$. However, in some cases $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$:

Proposition. Let $\mathcal{F}$ be a perverse sheaf on a $K$-variety $X$, and suppose that the pullback of $\mathcal{F}$ to $X_{C}$ is a direct sum of IC sheaves. Then $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$.

Proof. Let $j:U \to X$ be as in the definition of the generic part of $\mathcal{F}$, with closed complement $Z \subset X$. Our assumptions together with the definition of the generic part guarantee that $\mathcal{F}|X_C \simeq \mathcal{F}^{gen}|X_C \bigoplus \oplus_i IC_{Z_i}(\mathcal{L}_i)$ for some closed subvarieties $Z_i \subset X_C$ contained in $Z_C$.

Now look at the natural maps $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F} \overset{\alpha}{\to} \mathcal{F} \overset{\beta}{\to} \phantom{}^{\mathfrak{p}}j_{\ast} j^{\ast} \mathcal{F}$. Set $\mathcal{G} = \mathrm{im}\,\alpha$ and $\mathcal{H} = \mathrm{im}\,\beta$. Since $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F}$ does not admit any nonzero quotient supported on $Z$, the composite map $\mathcal{G}|X_C \hookrightarrow \mathcal{F}|X_C \to \oplus_i IC_{Z_i}(\mathcal{L}_i)$ is zero.  Thus $\alpha$ factors over an inclusion $\mathcal{G}|X_C \subset \mathcal{F}^{gen}|X_C$. Moreover, $\mathcal{G}$ has the same generic part as $\mathcal{F}$. This is enough to imply that $\mathcal{G} = \mathcal{F}^{gen}$, so we have a natural inclusion $\mathcal{F}^{gen} \simeq \mathcal{G} \subset \mathcal{F}$. A dual argument shows that $\beta$ factors over a surjection $\mathcal{F} \twoheadrightarrow \mathcal{H} \simeq \mathcal{F}^{gen}$. It is now easy to see that the composite map $\mathcal{F}^{gen} \hookrightarrow \mathcal{F} \twoheadrightarrow \mathcal{F}^{gen}$ is an isomorphism, so $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$. $\square$

Corollary 0. Let $f:X \to Y$ be a projective map of $K$-varieties. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})^{gen}$ is a direct summand of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$.

Proof. The decomposition theorem and the relative hard Lefschetz theorem give a decomposition $Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus \phantom{}^{\mathfrak{p}}\mathcal{H}^i(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})[-i]$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})$ is a direct sum of IC sheaves after pullback to $Y_C$, so we can apply the previous proposition. $\square$

Corollary 1. Let $f:X \to Y$ be a projective alteration of $K$-varieties with $X$ smooth. Then $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $Rf_{\ast}\mathbf{Q}_{\ell}[\dim X]$.

Proof. Check that $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}\mathbf{Q}_{\ell}[\dim X])^{gen}$ by playing with trace maps. Now apply the previous corollary. $\square$

Corollary 2. Let $K/\mathbf{Q}_p$ be a finite extension. Then for any $K$-variety $X$, the $p$-adic intersection cohomology $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a de Rham $G_K$-representation.

Proof. Let $X' \to X$ be a resolution of singularities. The previous corollary shows that $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a direct summand of $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ compatibly with the $G_K$-actions. Since $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ is de Rham and the de Rham condition is stable under passing to summands, we get the result. $\square$

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 $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p) \to H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$, but this map is not guaranteed a priori to be $G_K$-equivariant!

Corollary 3. Let $K$ be a finite extension of $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$. If $H^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all smooth projective $K$-varieties $X$, then $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all proper $K$-varieties $X$. In particular, the weight-monodromy conjecture holds for the $\ell$-adic intersection cohomology of all proper $K$-varieties for $K/\mathbf{F}_p((t))$ finite.

Proof. Entirely analogous to the previous proof. $\square$

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 $X \mapsto D_{mot}(X)$ on quasi-projective $K$-varieties, where $D_{mot}(X)$ is a suitable triangulated category of constructible motivic sheaves on $X$ with $\mathbf{Q}$-coefficients. This should come with the formalism of (at least) the four operations $f^{\ast}_{mot}, Rf_{mot\ast}, \otimes, R\mathcal{H}\mathrm{om}$, and with faithful exact $\ell$-adic realization functors $\mathcal{R}_{\ell}: D_{mot}(X) \to D^b_c(X,\mathbf{Q}_{\ell})$ compatible with the four operations. I think this has all been constructed by Ayoub, maybe with some tiny additional hypothesis on $K$? Let $\mathbf{Q}_{X} \in D_{mot}(X)$ denote the symmetric monoidal unit. It then makes sense to ask:

Question. In the setting of Corollary 1, is there an idempotent $e \in \mathrm{End}_{D_{mot}(Y)}(Rf_{mot \ast} \mathbf{Q}_{X}[\dim X])$ such that $\mathcal{R}_{\ell}( e Rf_{mot \ast} \mathbf{Q}_{X}[\dim X]) \simeq IC_{Y,\mathbf{Q}_{\ell}}$ for all $\ell$?

This would imply that the split injections $IH^{\ast}(Y_{\overline{K}},\mathbf{Q}_\ell) \to H^{\ast}(X_{\overline{K}},\mathbf{Q}_\ell)$ provided by Corollary 1 can be chosen “independently of $\ell$”, i.e. that they are the $\ell$-adic realizations of some split injection in $D_{mot}(\mathrm{Spec}\,K)$.

## Rampage!

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.

Please follow the instructions at the seminar website here to get the Zoom link. We will also keep an up-to-date schedule on researchseminars.org here.

All credit to Jared for the name!

## Things to do during a pandemic

• 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.

## 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!

## Artin-Grothendieck vanishing, again

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 $X$ is an affinoid rigid space over a complete algebraically closed field, AND $X$ comes via base change from an affinoid defined over a discretely valued subfield, then $H^i(X,\mathbf{Z}/n)=0$ for all $i > \mathrm{dim}(X)$ and all $n$ 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 $\mathrm{Spa}A$ be an affinoid rigid space over a complete algebraically closed nonarchimedean field $K$. Set $\Lambda = \mathbf{Z}/n$ where $n$ is any integer prime to the residue characteristic. Then $R\Gamma(\mathrm{Spa}A,\Lambda) \in D^{\leq \mathrm{dim}A}(\Lambda)$.

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 $\mathrm{Spa}A$ 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 $R\Gamma(\mathrm{Spa}A,\Lambda) \in D^{\leq 1+\mathrm{dim}A}(\Lambda)$ in general.

For this we use induction on $\mathrm{dim}A$. I’ll assume for simplicity that $K$ has characteristic zero. Without loss of generality we can assume that $A$ is reduced. Then by excellence of affinoid algebras, we can pick some non-zero-divisor $f \in A$ such that $A[1/f]$ is regular. Fix a nonzero nonunit $\pi \in \mathcal{O}_K$, and for any $n \geq 1$ consider the rational subsets $U_n = \{ x\,with\,|f(x)| \geq |\pi|^n \}$ and $V_n = \{x\,with\,|f(x)| \leq |\pi|^n \}$ inside $\mathrm{Spa}A$. Set $W_n = U_n \cap V_n$, so we get a Mayer-Vietoras distinguished triangle

$R\Gamma(\mathrm{Spa}A,\Lambda) \to R\Gamma(U_n,\Lambda)\oplus R\Gamma(V_n,\Lambda) \to R\Gamma(W_n,\Lambda)\to$

for any $n$. Note that $U_n$ and $W_n$ are smooth affinoids, so their etale cohomology is concentrated in degrees $\leq \mathrm{dim}A$ by Step One. Therefore, truncating the above Mayer-Vietoras sequence we get a quasi-isomorphism

$\tau^{\geq \mathrm{dim}A+2}R\Gamma(\mathrm{Spa}A,\Lambda) \simeq \tau^{\geq \mathrm{dim}A+2}R\Gamma(V_n,\Lambda)$

for any $n$. Moreover, $\mathrm{Spa}(A/f) \sim \lim_{n} V_n$ 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 $n$, the previous quasi-isomorphism gives a quasi-isomorphism

$\tau^{\geq \mathrm{dim}A+2} R\Gamma(\mathrm{Spa}A,\Lambda) \simeq \tau^{\geq \mathrm{dim}A+2}R\Gamma(\mathrm{Spa}(A/f),\Lambda)$.

But now we win, because $A/f$ is an affinoid of dimension $\dim(A)-1$, so by the induction hypothesis its etale cohomology is concentrated in degrees $\leq \mathrm{dim}A$.

Step Three. Bootstrap from the result of Step Two by a trick. More precisely, let $X=\mathrm{Spa}A$ and $\Lambda=\mathbf{Z}/n$ be as in the statement of the main theorem. By Step Two, we just have to show that $H^{\mathrm{dim}+1}(X,\Lambda)=0$. By another application of Step Two, the complex $R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(X,\Lambda)$ has cohomology in degree $2\mathrm{dim}A+2$ given by $H^{\mathrm{dim}A+1}(X,\Lambda)^{\otimes 2}$, and its enough to show that the latter module is zero. But

$R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(X,\Lambda) \simeq R\Gamma(X \times X,\Lambda)$

by the Kunneth formula*, and $X \times X$ is a $2\mathrm{dim}A$-dimensional affinoid, so its cohomology is concentrated in degrees $\leq 2\mathrm{dim}A+1$ by yet another application of Step Two. This gives the result.

*The necessary result is that if $X$ and $Y$ are $K$-affinoid spaces, then $R\Gamma(X \times Y, \Lambda) \simeq R\Gamma(X,\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(Y,\Lambda)$. 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 $X, Y$ have canonical adic compactifications $\overline{X},\overline{Y}$, and etale cohomology (with constant coefficients) doesn’t change if you replace $X$ or $Y$ by its compactification. But then $\overline{X}$ and $\overline{Y}$ are proper over $\mathrm{Spa}K$ (in the sense of Huber’s book), so it’s easy to show that

$R\Gamma(\overline{X} \times \overline{Y},\Lambda) \simeq R\Gamma(\overline{X},\Lambda) \otimes_{\Lambda}^{\mathbf{L}} R\Gamma(\overline{Y},\Lambda)$

by the usual combination of proper base change and the projection formula.

## w-local spaces are amazing

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

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

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

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

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

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

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

Proof when $T$ is quasicompact open.

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

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

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

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

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

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

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

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

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