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 $<0$ .

2 thoughts on “Families of perverse sheaves”

mayorliatmath says:

March 6, 2021 at 11:37 pm

I guess this question you must have thought about: is there an analogue of decomposition theorem in this relative setting?

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1. arithmetica says:
  
  March 7, 2021 at 1:21 pm
  
  Not in a naive way. One problem is that $\mathrm{Perv}(X/S)$ is not Artinian in general, and its simple objects are “silly”, unlike in the setting of usual perverse sheaves. On a closely related note, there’s no good relative analogue of IC sheaves.
  
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2 thoughts on “Families of perverse sheaves”

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