## The dangers of naming things after people

I spent part of last weekend reading Alice Silverberg’s blog, which is simultaneously depressing and hilarious. Everyone should read it, but you probably won’t enjoy reading it unless you enjoy Coen brothers movies or the short stories of Kafka. Anyway, the following thoughts have been in my head for a few months, but I decided to record them here after reading this.

The cohomology of non-basic local Shimura varieties is described by the “Harris-Viehmann conjecture”, which is formally stated as Conjecture 8.4 in Rapoport-Viehmann’s paper. This story started with a daring and beautiful conjecture of Harris (conjecture 5.2 here), whose formulation however turned out to be slightly incorrect in general, cf. Example 8.3 in RV. The conjecture was then modified by Viehmann, and Rapoport named this modified conjecture the Harris-Viehmann conjecture (footnote 5 in RV).

Unfortunately, Conjecture 8.4 in RV is still not correct as stated: the Weil group action on the summands appearing on the right-hand side needs to be modified by certain half-integral Tate twists. As far as I know, Alexander Bertoloni-Meli is the only person who has publicly pointed out the need for this modification, and Conjecture 3.2.1 in his very cool paper is the only correct formulation of the Harris-Viehmann conjecture in print.

Since the need for these Tate twists was overlooked by a lot of very smart people, it only seems fair to me that Alexander should get credit for his contribution here. The obvious way to do this would be to refer to the Harris–Viehmann–Bertoloni-Meli conjecture, or the Bertoloni-Meli–Harris–Viehmann conjecture. You could pick the second option if you’re a stickler for alphabetical name orders in mathematics, or the first option if you feel (as I do) that Harris’s contribution here deserves priority.

But it gets even worse, because Harris also formulated another conjecture along similar lines (conjecture 5.4 in his article linked above), which has gotten somewhat less attention but which is nevertheless extremely interesting.* It turns out that one can formulate a unified conjecture which encompasses both Harris’s conjecture 5.4 and the Harris–Viehmann–Bertoloni-Meli conjecture. What should it be called? The Harris–Viehmann–Bertoloni-Meli—Harris conjecture? I guess not.

*Here’s a comment from MH: “I was (and am) much more attached to this conjecture than to the one that is called the Harris-Viehmann conjecture, because it required some work to find the right formalism (the parabolics that transfer between inner forms), whereas the other conjecture (independently of the incorrect formulation in my paper) was just the obvious extension of Boyer’s result.”

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## Unappreciative

Some things I appreciate but which I used to not appreciate: The Bernstein center, $\infty$-categories, etale cohomology, L-packets, perverse sheaves.

Some things I still don’t appreciate: Monads, endoscopy, rigid cohomology, A-packets, Springer theory.

## Remarks on Fargues-Scholze, part 2

Today I want to talk about section VII.3 in the manuscript. Here they define and study a functor $f_{\natural}$ on solid sheaves which is left adjoint to the usual pullback functor. But why is this called relative homology?

To explain this name, you have to remember that in the usual formalism of etale cohomology, homology is realized as the compactly supported cohomology of the dualizing complex. Symbolically, if $X$ is a variety with structure map $f:X \to \mathrm{Spec} k$, then the homology of $X$ is given by $Rf_! Rf^! \Lambda$. Now, you might ask whether this works in families: if $f:X \to Y$ is some map of varieties, maybe I can find some complex on $Y$ whose stalk at $y$ realizes the homology of $X_y$? If the constant sheaf is $f$-ULA, then formation of $Rf^! \Lambda$ commutes with any base change, and exactly the same formula works, but in general there is no naive sheaf with this property.

The punchline now is that $f_{\natural} \Lambda$ does have this property: when the constant sheaf is $f$-ULA (e.g. if $Y$ is a point) it agrees with $Rf_! Rf^! \Lambda$ by Proposition VII.5.2, and its formation commutes with arbitrary base change, so it really does give a complex on $Y$ whose stalks realize the homology of the fibers of $f$. The only twist is that $f_{\natural}\Lambda$ is a solid sheaf in general, not a classical etale sheaf.

## Remarks on Fargues-Scholze

The Fargues-Scholze geometrization paper is available! In this post, and probably some future posts also, I’ll make some random comments on this paper. These won’t mean anything unless you’ve read (at the very least) the beautifully written introduction to the paper. To be clear, I have nothing of substance to say about the “big picture” – these will be purely technical remarks.

First of all, at the bottom of p. 324, one finds the slightly cryptic claim that although there are no general $i_!$ functors in the $D_{lis}$ setting (for $i$ a locally closed immersion), one can define functors $i^{b}_!$ in the $D_{lis}$ setting, where $i^b: \mathrm{Bun}_{G}^b \to \mathrm{Bun}_G$ is the inclusion of any Harder-Narasimhan stratum into $\mathrm{Bun}_G$. This is stated without proof. However, if you read carefully, you will notice that these lisse $i^{b}_!$ functors are actually used in the paper, in the proof of Proposition VII.7.6. So maybe it’s worth saying something about how to construct them.

To build $i^{b}_!$ in the $D_{lis}$ setting, factor $i^b$ as the composition $\mathrm{Bun}_{G}^b \overset{i}{\to} \mathrm{Bun}_{G}^{ \leq b} \overset{j}{\to} \mathrm{Bun}_G$. Here $\mathrm{Bun}_{G}^{ \leq b}$ is the open substack of bundles which are “more semistable” than $\mathcal{E}_b$. It will also be convenient to write $\mathrm{Bun}_{G}^{ < b} = \mathrm{Bun}_{G}^{ \leq b} - \mathrm{Bun}_{G}^{ b}$. Note that $i$ is a closed immersion, and $j$ is an open immersion, so $j_! = j_{\natural}$ clearly preserves $D_{lis}$. The subtlety is in making sense of $i_!$, since then we can write $i_{!}^{b} = j_! i_!$ as usual.

For $i_!$, we need the local chart $\pi_{b}: \mathcal{M}_b \to \mathrm{Bun}_{G}^{\leq b}$ and its punctured version $\pi_{b}^\circ : \mathcal{M}_{b}^\circ = \mathcal{M}_{b} \times_{\mathrm{Bun}_{G}^{\leq b}} \mathrm{Bun}_{G}^{< b} \to \mathrm{Bun}_{G}^{\leq b}$. Recall that these charts also come with compatible maps $q_b: \mathcal{M}_b \to [\ast / G_b(E)]$ and $q_b^{\circ}: \mathcal{M}_{b}^{\circ} \to [\ast / G_b(E)]$. Then for any $A \in D_{lis}(\mathrm{Bun}_{G}^{b},\Lambda) \cong D_{lis}([\ast / G_b(E)],\Lambda)$, the correct definition turns out to be

$i_! A = \mathrm{Cone}(\pi_{b \natural}^{\circ} q_{b}^{\circ \ast}A \to \pi_{b \natural} q_{b}^{\ast}A)\,\,\,\,(1)$.

The point here is that in the lisse world, the only pushforward functors which come for free are the functors $f_{\natural}$ for cohomologically smooth maps $f$. Since $\pi_{b}$ and $\pi_{b}^{\circ}$ are cohomologically smooth – one of the hardest theorems in the paper! – the above construction preserves $D_{lis}$. Moreover, it’s easy to check that the formula above has the right properties. Indeed, the *-restriction of the RHS of (1) to $\mathrm{Bun}_{G}^{b}$ is just $A$, by Proposition VII.7.2, while its complementary restriction to $\mathrm{Bun}_{G}^{ < b}$ clearly vanishes.

## Families of perverse sheaves

In this post I want to talk about some ongoing joint work with Peter Scholze. Since this came up in Scholze’s geometrization lectures, I thought it would be fun to go into a little more detail here. All inaccuracies below are entirely due to me, and the standard caveats about blog-level rigor apply.

The goal, broadly speaking, is to define a relative notion of perversity in etale cohomology, with respect to any finite type morphism $f:X \to S$ of schemes. In order to not make slightly false statements, I will take my coefficient ring to be $\mathbf{F}_\ell$ for some prime $\ell$ invertible on $S$. Everything below also works with more general torsion coefficients killed by an integer invertible on $S$, but then one has to be mindful of the difference between $D^{b}_{c}$ and $D^{b}_{ctf}$. With mild assumptions on $S$, everything below also works with $\mathbf{Q}_\ell$-coefficients.

When $S=\mathrm{Spec}k$ is a point, $X$ is just a finite type $k$-scheme, and we have the familiar perverse t-structure $( \phantom{}^p D^{\leq 0}(X), \phantom{}^p D^{\geq 0}(X))$ on $D(X)=D(X,\mathbf{F}_\ell)$, with all its wonderful properties as usual. The key new definition is the following.

Definition. Given a finite type map of schemes $f:X \to S$, let $\phantom{}^{p/S}D^{\leq 0}(X) \subset D(X)$ be the full subcategory of objects $A$ such that $A|X_{\overline{s}} \in \phantom{}^p D^{\leq 0}(X_{\overline{s}})$ for all geometric points $\overline{s} \to S$.

It is easy to see that $\phantom{}^{p/S}D^{\leq 0}(X)$ is stable under extensions and (after upgrading to derived $\infty$-categories) under filtered colimits, and is set-theoretically reasonable, so it defines the left half of a t-structure on $D(X)$ by Proposition 1.4.4.11 in Lurie’s Higher Algebra. We denote the right half of this t-structure, unsurprisingly, by $\phantom{}^{p/S}D^{\geq 0}(X)$, and call it the relative perverse t-structure (relative to $X\to S$, of course). We write $\phantom{}^{p/S}\tau^{\leq n}$ and $\phantom{}^{p/S}\tau^{\geq n}$ for the associated truncation functors.

This t-structure satisfies a number of good and fairly obvious formal properties which I won’t get into here (it can be glued from any open-closed decomposition of $X$, various operations are obviously left- or right- t-exact, etc.). Less formally, if $S$ is a finite-dimensional excellent Noetherian scheme, then the relative perverse truncation functors preserve $D^{b}_{c}(X) \subset D(X)$, so we get an induced relative perverse t-structure on $D^{b}_{c}(X)$. This follows from some results of Gabber: roughly, one can check that the relative perverse t-structure is the t-structure associated with the weak perversity function $p(x)=-\mathrm{tr.deg}k(x)/k(f(x))$, and that the conditions in Theorem 8.2 are satisfied for excellent $S$. (Nb. Gabber’s methods also reprove the existence of the relative perverse t-structure for any Noetherian $S$, without appealing to $\infty$-categories.)

However, the right half $\phantom{}^{p/S}D^{\geq 0}(X)$ is defined in a very inexplicit way, and it isn’t clear how to get your hands on this at all. The really shocking theorem, then, is the following result.

Key Theorem. An object $A \in D(X)$ lies in $\phantom{}^{p/S}D^{\geq 0}(X)$ if and only if $A|X_{\overline{s}} \in \phantom{}^p D^{\geq 0}(X_{\overline{s}})$ for all geometric points $\overline{s} \to S$.

Note that I really am taking *-restrictions to geometric fibers here, just as in the definition of $\phantom{}^{p/S}D^{\leq 0}(X)$. One might naively guess that !-restrictions should be appearing instead, but no!

This theorem has a number of corollaries.

Corollary 1. The heart $\mathrm{Perv}(X/S)$ of the relative perverse t-structure consists of objects $A \in D(X)$ which are perverse after restriction to any geometric fiber of $f$. In particular, the objects with this property naturally have the structure of an abelian category.

This fully justifies the choice of name for this t-structure, and shows that the heart of the relative perverse t-structure gives a completely reasonable notion of a “family of perverse sheaves parameterized by $S$”.

Corollary 2. For any map $T\to S$, the pullback functor $D(X) \to D(X_T)$ is t-exact for the relative perverse t-structures (relative to $S$ and $T$, respectively). In particular, relative perverse truncations commute with any base change on $S$, and pullback induces an exact functor $\mathrm{Perv}(X/S) \to \mathrm{Perv}(X_T / T)$.

Corollary 3. If $X\to S$ is any finitely presented morphism of qcqs schemes, then the relative perverse truncation functors on $D(X)$ preserve $D^{b}_{c}(X)$.

Corollaries 1 and 2 are immediate consequences of the Key Theorem. Corollary 3 then follows from the case where $S$ is Noetherian excellent finite-dimensional by Noetherian approximation arguments, using Corollary 2 crucially.

To prove the key theorem, we make some formal reductions to the situation where $S$ is excellent Noetherian finite-dimensional and $A \in D^{b}_{c}(X)$. In this situation, we argue by induction on $\dim S$, with the base case $\dim S=0$ being obvious. In general, this induction is somewhat subtle, and involves playing off the relative perverse t-structure on $D(X)$ against the perverse t-structures on $D(X_{\overline{s}})$ and the (absolute) perverse t-structure on $D(X)$ (which exists once you pick a dimension function on $S$).

However, when $S$ is the spectrum of an excellent DVR, one can give a direct proof of the key theorem, and this is what I want to do in the rest of this post. Let $i: s \to S$ and $j: \eta \to S$ be the inclusions of the special and generic points, with obvious base changes $\tilde{i}:X_s \to X$ and $\tilde{j}: X_\eta \to X$. By definition, $A \in D(X)$ lies in $\phantom{}^{p/S}D^{\leq 0}(X)$ iff $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\leq 0}(X_\eta)$ and $\tilde{i}^{\ast}A \in \phantom{}^{p}D^{\leq 0}(X_s)$. By standard results on gluing t-structures (see chapter 1 in BBDG), this implies that $A$ lies in $\phantom{}^{p/S}D^{\geq 0}(X)$ iff $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta)$ and $R\tilde{i}^{!}A \in \phantom{}^{p}D^{\geq 0}(X_s)$. Thus, to prove the key theorem in this case, we need to show that for any $A \in D(X)$ with $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta)$, the conditions $\tilde{i}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_s)$ and $R\tilde{i}^{!}A \in \phantom{}^{p}D^{\geq 0}(X_s)$ are equivalent.

To show this, consider the triangle $R\tilde{i}^{!}A \to \tilde{i}^{\ast}A \to \tilde{i}^{\ast}R\tilde{j}_{\ast} \tilde{j}^{\ast} A \to$. The crucial observation is that $\tilde{j}^{\ast}A \in \phantom{}^{p}D^{\geq 0}(X_\eta)$ by assumption, and that $\tilde{i}^{\ast}R\tilde{j}_{\ast}$ carries $\phantom{}^{p}D^{\geq 0}(X_\eta)$ into $\phantom{}^{p}D^{\geq 0}(X_s)$. The italicized result follows from some theorems of Gabber generalizing the classical Artin-Grothendieck vanishing theorem for affine varieties, and is closely related to the well-known fact that nearby cycles are perverse t-exact. This immediately gives what we want: we now know that $\tilde{i}^{\ast}R\tilde{j}_{\ast} \tilde{j}^{\ast} A$ only can only have nonzero perverse cohomologies in degrees $\geq 0$, so $R\tilde{i}^{!}A$ and $\tilde{i}^{\ast}A$ have the same perverse cohomologies in degrees $<0$.

## 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.

## What does a general proper rigid space look like?

As the title says. Consider proper rigid spaces $X$ over some nonarchimedean field $K$. The “standard” examples of such things which don’t come from algebraic geometry are i) the Hopf surface $(\mathbf{A}^2 - 0)/p^\mathbf{Z}$, ii) non-algebraizable deformations of K3 surfaces over the residue field of $K$, and iii) generic abeloid varieties (which are analogous to generic compact complex tori).  But there must be gazillions of other examples, right? A “random” proper rigid space is hard to write down, sort of by definition. But there are certainly some natural questions one can ask:

-For every $n \geq 2$, does there exist a proper $n$-dimensional rigid space with no non-constant meromorphic functions, and admitting a formal model whose special fiber has components of general type? Can we find examples of such spaces with arbitrarily large dimension dimension which don’t come from lower-dimensional examples by simple operations (products, quotients by finite groups, etc.)? Same question but with “no non-constant meromorphic functions” replaced by the weaker requirement that $\mathrm{tr.deg}K(X)/K$ is small compared to $\dim X$.

-Do there exist non-algebraizable proper rigid spaces with “arbitrarily bad” singularities?

-Do there exist rigid analytic analogues of Kodaira’s class VII0 surfaces?

## Euler characteristics and perverse sheaves

Let $X$ be a variety over a separably closed field $k$, and let $A$ be some object in $D^b_c(X,\mathbf{Q}_{\ell})$. Laumon proved the beautiful result that the usual and compactly supported Euler characteristics $\chi(X,A)$ and $\chi_c(X,A)$ are always equal. Recently while trying to do something else, I found a quick proof of Laumon’s result, as well as a relative version, and I want to sketch this here.

Pick an open immersion into a compactification $j:X \to X'$; after a blowup, we can assume that $Z=X' - X$ is an effective Cartier divisor. Write $i:Z \to X'$ for the inclusion of the boundary. By the usual triangle $j_!A \to Rj_*A \to i_*i^* Rj_*A \to$, we reduce to showing that $\chi(X',i_*i^* Rj_*A)=0$. Filtering $A$ by its perverse cohomology sheaves, we reduce further to the case where also $A$ is perverse. Cover $X'$ by open affines $X_n'$ such that $Z_n= Z \cap X_n'$ is the divisor of a function $f_n$. By an easy Mayer-Vietoras argument, it’s now enough to show that for every open $U$ contained in some $X_n'$, $\chi(U,(i_* i^{\ast}Rj_{\ast}A)|U) = 0$.

But now we win: for any choice of such $U \subset X_n'$, there is an exact triangle $R\psi_{f_n}(A|U \cap X) \to R\psi_{f_n}(A|U \cap X) \to (i_* i^{\ast}Rj_{\ast}A)|U \to$ in $D^b_c(U,\mathbf{Q}_{\ell})$ where $R\psi_{f_n}:\mathrm{Perv}(U \cap X) \to \mathrm{Perv}( U \cap Z)$ is the unipotent nearby cycles functor associated with $f_n$, and the first arrow is the logarithm of the unipotent part of the monodromy. Since $\chi(U, -)$ is additive in exact triangles and the first two terms agree, we’re done.

A closer reading of this argument shows that you actually get the following stronger statement: for any $A$, the class $[i_*i^* Rj_*A] \in K_0\mathrm{Perv}(X')$ is identically zero. From here it’s easy to get a relative version of Laumon’s result.

Theorem. Let $f:X \to Y$ be any map of $k$-varieties. Then for any $A\in D^b_c(X,\mathbf{Q}_\ell)$, there is an equality $[Rf_! A]=[Rf_\ast A]$ in $K_0\mathrm{Perv}(Y)$.

## Geometry of the B_dR affine Grassmannian

As many readers of this blog already know, one key result in modern p-adic geometry is Scholze’s theorem that the $B_{\mathrm{dR}}$-affine Grassmannian is an ind-spatial diamond. The proof of this given in the Berkeley notes is a bit tricky and technical: it uses covering by infinite-dimensional objects in a crucial way, as well as an abstract Artin-type representability criterion.  So I’m very pleased to report that Bence Hevesi has given a beautiful new proof of this theorem in his Bonn master’s thesis. Bence’s proof avoids representability criteria or coverings by huge objects. Instead, his idea is to reduce to $\mathrm{GL}_n$ and then construct explicit charts for closed Schubert cells, using moduli of local shtukas at infinite level. You can read Bence’s outstanding thesis here.

## 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.)