## 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 $<0$.

## 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.)

## sheafiness; perversity

$\bullet$ One of the main annoyances in the theory of adic spaces is that, for a given Huber pair $(A,A^+)$, the structure presheaf on $\mathrm{Spa}(A,A^{+})$ is not always a sheaf. One usually remedies this by restricting attention to various classes of Huber rings, e.g. strongly Noetherian Tate rings, perfectoid Tate rings, Noetherian adic rings, etc. However, the following class of rings doesn’t seem to be addressed in the literature:

Definition. Let $A$ be an adic ring with finitely generated ideal of definition $I \subset A$. We say $A$ is strongly Noetherian outside $I$ if, for all $n \geq 0$, the scheme $\mathrm{Spec}\,A\left\langle X_1,\dots,X_n \right\rangle \smallsetminus V(IA\left\langle X_1,\dots,X_n \right\rangle )$ is Noetherian.

Here $A\left\langle X_1,\dots,X_n \right\rangle$ denotes the usual ring of restricted power series. Note that if $A$ is a Tate ring and $(A_0,(\varpi))$ is any couple of definition, then $A$ is strongly Noetherian if and only if $A_0$ is strongly Noetherian outside $(\varpi)$. I should also point out that the condition of being strongly Noetherian outside $I$ is already considered in a very interesting paper of Fujiwara-Gabber-Kato; they use the terminology “topologically universally rigid-Noetherian”, but I prefer my terminology on account of the previous sentence. Anyway, the following conjecture seems reasonable:

Conjecture. If $A$ is strongly Noetherian outside $I$, the structure presheaf on $\mathrm{Spa}(A,A)$ is a sheaf.

This implies that any strongly Noetherian Tate ring is sheafy (which of course is already known), but it also implies e.g. that if $A$ is topologically finitely presented over $\mathcal{O}_K$ for some nonarchimedean field $K$, then $A$ is sheafy. Sheafiness in the latter situation is known when $K$ is discretely valued, but to the best of my knowledge it’s open for general $K$.

I’m sure this conjecture is within reach, and maybe it’s an easy exercise for experts on sheafiness (*cough* Kiran *cough*). Note that FGK already proved some interesting consequences of this definition, which are probably relevant to proving this conjecture. Precisely, they show that if $A$ is strongly Noetherian outside $I$, then:
1. The $I$-power-torsion submodule of any finitely generated $A$-module is killed by a power of $I$.
2. If $N \subset M$ is any inclusion of $A$-modules, with $M$ finitely generated, then the subspace topology on $N$ induced by the $I$-adic topology on $M$ coincides with the $I$-adic topology on $N$.

$\bullet$ Let $j:U \to X$ be some immersion of varieties over a separably closed field. Everyone knows that the intermediate extension functor $j_{!\ast}$ on perverse sheaves (say with coefficients in $\Lambda = \mathbf{Q}_{\ell}$) is pretty great: it’s totally canonical, it commutes with Verdier duality, it preserves irreducibility, it preserves monic and epic maps of perverse sheaves, etc.

Recently I noticed that if $f: Z \to X$ is any map of varieties, with $X$ smooth and $Z$ irreducible, there’s still a natural functor $f^{\ast !}:\mathrm{Perv}(X) \to \mathrm{Perv}(Z)$ which commutes with Verdier duality. To define this functor, note that for any $f$ and any $\mathcal{F} \in D^b_c(X)$, there is a natural map $f^\ast \mathcal{F} \otimes Rf^! \Lambda \to Rf^! \mathcal{F}$, obtained by adjunction from the composite map $Rf_!( f^\ast \mathcal{F} \otimes Rf^! \Lambda) \cong \mathcal{F} \otimes Rf_! Rf^! \Lambda \to \mathcal{F}$ (the first isomorphism here is the projection formula). Since $X$ is smooth, the dualizing complex $\omega_X$ is just $\Lambda[2 \dim X]$, so then $Rf^! \Lambda = Rf^!\omega_X[-2 \dim X] = \omega_Z[-2 \dim X]$. Thus we get a natural map $\alpha: f^\ast \mathcal{F} \otimes \omega_Z[-2 \dim X] \to Rf^! \mathcal{F}$.

Next, note that the complex $\omega_Z$ is concentrated in degrees $[-2 \dim Z,0]$, and in the lowest of these degrees it’s just the constant sheaf, i.e. $\tau^{ \leq -2 \dim Z} \omega_Z \cong \Lambda[2 \dim Z]$. In particular, there is a canonical map $\Lambda[2 \dim Z] \to \omega_Z$. Shifting by $-2 \dim X$ and tensoring with $f^\ast \mathcal{F}$ gives a map $\beta: f^\ast \mathcal{F}[2 \dim Z - 2 \dim X] \to f^\ast \mathcal{F} \otimes \omega_Z[-2 \dim X]$. Putting things together, we get a natural map $\alpha \circ \beta: f^\ast \mathcal{F}[2 \dim Z - 2 \dim X] \to Rf^! \mathcal{F}$. Set $c= \dim X - \dim Z,$ so after shifting this becomes a natural map

$\gamma=\gamma_{\mathcal{F}}: f^{\ast}\mathcal{F}[-c] \to Rf^!\mathcal{F}[c]$.

This shifting has the advantage that Verdier duality exchanges the functors $f^{\ast}[-c]$ and $Rf^![c]$ on $D^b_c$, and one can check that the Verdier dual of $\gamma_{\mathcal{F}}$ identifies with $\gamma_{\mathbf{D}\mathcal{F}}$.

Definition. The functor $f^{\ast !}: \mathrm{Perv}(X) \to \mathrm{Perv}(Z)$ sends any $\mathcal{F}$ to the image of the map $^p\mathcal{H}^0(\gamma):$ $^p\mathcal{H}^0(f^{\ast} \mathcal{F}[-c]) \to$ $^p\mathcal{H}^0(Rf^!\mathcal{F}[c])$.

Here of course $^p\mathcal{H}^0(-)$ denotes the zeroth perverse cohomology sheaf.

Exercise. Show that $f^{\ast !}(\Lambda[\dim X]) \cong \mathcal{IC}_Z$.

It might be interesting to compute this functor in some other examples. Note that it can be quite stupid: if $f$ is a closed immersion (with $c > 0$) and $\mathcal{F} \in \mathrm{Perv}(X)$ is already supported on $Z$, then $f^{ \ast !} \mathcal{F} = 0$. On the other hand, if $f$ is smooth and surjective, then $f^{\ast !} \cong f^{\ast}[-c] \cong Rf^![c]$ is faithful.

## 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 $D^{b}_{c}(Y,\mathbf{F}_{\ell})$.