## 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.)

## 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!

## A counterexample

Let $C/\mathbf{Q}_p$ be a complete algebraically closed nonarchimedean field extension, and let $X$ be any proper rigid space over $C$. Let $\mathbf{L}$ be any $\mathbf{Z}_p$-local system on $X_{\mathrm{proet}}$. By the main results in Scholze’s p-adic Hodge theory paper, the pro-etale cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{L})$ are always finitely generated $\mathbf{Z}_p$-modules. It also seems likely that Poincare duality holds in this setting (and maybe someone has proved this?).

Suppose instead that we’re given a $\mathbf{Q}_p$-local system $\mathbf{V}$. By analogy, one might guess that the cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{V})$ are always finitely generated $\mathbf{Q}_p$-vector spaces. Indeed, this (and more) was claimed as a theorem by Kedlaya-Liu in a 2016 preprint. However, it is false. The goal of this post is to work out an explicit counterexample.

So, consider $X=\mathbf{P}^1$ as a rigid space over $C$. This is the target of the Gross-Hopkins period map $\pi_{\mathrm{GM}}: \mathcal{M} \to X$, where $\mathcal{M}$ is (the rigid generic fiber of the base change to $\mathcal{O}_C$ of) the Lubin-Tate deformation space of some fixed connected p-divisible group $G_0/\overline{\mathbf{F}_p}$ of dimension 1 and height 2. The rational p-adic Tate module of the universal p-divisible group $G/\mathcal{M}$ descends along $\pi_{\mathrm{GM}}$ to a rank two $\mathbf{Q}_p$ local system $\mathbf{V}_{LT}$ on $X$.

Theorem. Maintain the above setup. Then
i. For any $i \neq 1,2$ the group $H^i_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is zero.

ii. The group $H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is a Banach-Colmez space over $C$ of Dimension $(1,-2)$.
iii. The group $H^2_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ is a Banach-Colmez space over $C$ of Dimension $(1,2)$.

Recall that a Banach-Colmez space is a special kind of topological $\mathbf{Q}_p$-vector space (really it’s a functor valued in such things, but I’ll be a little sloppy about this point). Morally, it’s something like a finite-dimensional $C$-vector space defined up to a finite-dimensional $\mathbf{Q}_p$-vector space. In particular, any such space has a well-defined Dimension, which is a pair in $\mathbf{Z}_{\geq 0} \times \mathbf{Z}$ whose entries record the $C$-dimension and the $\mathbf{Q}_p$-dimension of the space, respectively. So for example the space $C^2$ has Dimension $(2,0)$, and the space $C/\mathbf{Q}_p$ has Dimension $(1,-1)$. Unsurprisingly, any Banach-Colmez space whose $C$-dimension is positive will be disgustingly infinitely generated as a $\mathbf{Q}_p$-vector space, so the Theorem really does give us an example of the desired type. Note also that Poincare duality fails in this example.

Proof. Let $\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}$ be the evident sheaf on $X_{\mathrm{proet}}$, where e.g. $\mathbf{B}_{\mathrm{crys}}^{+}$ is the crystalline period sheaf defined in Tan-Tong’s paper on crystalline comparison. The key observation is that there is a short exact sequence

$(\ast)\;\;\; 0 \to \mathbf{V}_{LT} \to \mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p} \to \mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X \to 0$

of sheaves on $X_{\mathrm{proet}}$. This is a sheaf-theoretic version of the sequence (1.0.1) in Scholze-Weinstein, and it can be constructed using the methods in their paper. One can also construct it directly using the modern fancy-pants interpretation of $X$ in this setup as the period domain parametrizing admissible length one modifications of the bundle $\mathcal{O}(1/2)$ on the Fargues-Fontaine curve. (Nb. The mysterious middle term in the sequence is really the sheaf of $\varphi$-equivariant maps from $H^1_{\mathrm{crys}}(G_0/W(\overline{\mathbf{F}_p}))[\tfrac{1}{p}]$ to $\mathbf{B}_{\mathrm{crys}}^{+}$.)

Anyway, this reduces us to computing the groups $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p})$ and $H^i_{\mathrm{proet}}(X,\mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X)$. This might look a bit terrifying, but it’s not so bad. In fact, the first of these can be handled by a general lemma.

Lemma. Let $\mathbf{M}$ be any Banach-Colmez space over $C$. For any proper rigid space $X/C$, we may regard $\mathbf{M}$ as a (pre)sheaf on $X_{\mathrm{proet}}$, so in particular we can talk about the pro-etale cohomology groups $H^i_{\mathrm{proet}}(X,\mathbf{M})$. In this notation, the natural map $H^i_{\mathrm{proet}}(X,\mathbf{Q}_p) \otimes_{\mathbf{Q}_p} \mathbf{M}(C) \to H^i_{\mathrm{proet}}(X,\mathbf{M})$ is an isomorphism.

(Proof sketch: Use the 5 lemma to reduce to the case of effective Banach-Colmez spaces, and then to the cases of the space $\mathbf{Q}_p$, where it’s a tautology, and the space Colmez notates $\mathbb{V}^1$, where it follows from the primitive comparison theorem, cf. Theorem 3.17 in Scholze’s CDM survey.)

Applied to our problem, this immediately gives that $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}) \cong H^i_{\mathrm{proet}}(X,\mathbf{Q}_p) \otimes_{\mathbf{Q}_p} B_{\mathrm{crys}}^{+,\varphi^2 = p}$. By the standard easy computation of $H^i_{\mathrm{proet}}(X,\mathbf{Q}_p)$, we get that $H^i_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p})$ is a copy of $B_{\mathrm{crys}}^{+,\varphi^2=p}$ for either of $i \in \{ 0,2 \}$, and it vanishes otherwise. In particular, in degrees 0 and 2, it has Dimension $(1,2)$ by some of the calculations in Colmez’s original paper.

Next, we need to compute the pro-etale cohomology of $\mathcal{E} = \mathrm{Lie}(G)[\tfrac{1}{p}] \otimes_{\mathcal{O}_X} \widehat{\mathcal{O}}_X$. For this, we use the fact (already in Gross and Hopkins’s original article) that $\mathrm{Lie}(G)[\tfrac{1}{p}] \simeq \mathcal{O}_X(1)$. Let $\lambda : X_{\mathrm{proet}} \to X_{\mathrm{an}}$ be the evident projection of sites. Combining the description of $\mathrm{Lie}(G)[\tfrac{1}{p}]$ with an easy projection formula gives an isomorphism $E\overset{def}{=}R\lambda_{\ast} \mathcal{E} \cong \mathcal{O}_X (1) \otimes_{\mathcal{O}_X} R\lambda_{\ast} \widehat{\mathcal{O}}_X$. Moreover, $R^i \lambda_{\ast} \widehat{\mathcal{O}}_X \simeq \Omega_{X}^i$ identifies with $\mathcal{O}_X$ in degree zero and $\mathcal{O}_X(-2)$ in degree 1, and vanishes otherwise. In particular, the only nonvanishing cohomology sheaves of $E$ are $\mathcal{O}_X(1)$ in degree 0 and $\mathcal{O}_X(-1)$ in degree 1; moreover, the latter has no global cohomology in any degree. Feeding this information into the Leray spectral sequence for $\lambda$, we get that $H^i_{\mathrm{proet}}(X,\mathcal{E}) \simeq H^i(X,E) \simeq H^i(X, \tau^{\leq 0} E) \simeq H^i(X,\mathcal{O}_X(1))$, so this is $C^2$ for $i=0$ and zero otherwise.

Finally, we can put everything together and take the long exact sequence in pro-etale cohomology associated with $(\ast)$. It’s easy to check that $\mathbf{V}_{LT}$ doesn’t have any global sections, and the middle term of $(\ast)$ has no cohomology in degree one, so we get a short exact sequence $0 \to H^0_{\mathrm{proet}}(X,\mathbf{B}_{\mathrm{crys}}^{+,\varphi^2=p}) \to C^2 \to H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT}) \to 0$. We’ve already identified the $H^0$ here as something of Dimension $(1,2)$, so by the additivity of Dimensions in short exact sequences, we deduce that $H^1_{\mathrm{proet}}(X,\mathbf{V}_{LT})$ has Dimension $(1,-2)$, as desired. By a similar argument, we get an isomorphism $H^2_{\mathrm{proet}}(X,\mathbf{V}_{LT}) \simeq B_{\mathrm{crys}}^{+,\varphi^2=p}$, which we already observed has Dimension $(1,2)$. The vanishing of all the other cohomologies of $\mathbf{V}_{LT}$ also follows easily. $\square$

BTW, there is nothing special about height $2$ in this story; I just stuck with it for convenience. For any heght $h \geq 2$, there is an analogous rank $h$ $\mathbf{Q}_p$-local system $V_{LT,h}$ on $\mathbf{P}^{h-1}$, and one can check that e.g. $H^1_{\mathrm{proet}}(\mathbf{P}^{h-1},\mathbf{V}_{LT,h})$ has Dimension $(h-1,-h)$.

Let me briefly explain the genesis of these calculations. Several months ago Shizhang Li pointed out to me that the primitive comparison theorem doesn’t obviously generalize to $\mathbf{Q}_p$-local systems without globally defined lattices, and he also suggested that the cohomologies of $\mathbf{V}_{LT}$ might be strange. I promptly forgot about this conversation until Monday, when Shizhang reported that Ruochuan Liu had told him in Oberwolfach several weeks ago that the Kedlaya-Liu “theorem” was false. It was then natural to double down on $\mathbf{V}_{LT}$ as a possible counterexample. Immediately after I gave a lecture on the Tan-Tong paper the next day, the sequence $(\ast)$ entered my head randomly. Then everything just came out. Also, luckily, we were at a restaurant with paper tablecloths.

(I’m not sure whether Ruochuan also had this particular counterexample in mind.)

There are lots of interesting questions here, I think. Are there other natural examples of this type? Are the pro-etale cohomology groups of $\mathbf{Q}_p$-local systems on proper rigid spaces always Banach-Colmez spaces?