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”

    1. 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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