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 of schemes. In order to not make slightly false statements, I will take my coefficient ring to be for some prime invertible on . Everything below also works with more general torsion coefficients killed by an integer invertible on , but then one has to be mindful of the difference between and . With mild assumptions on , everything below also works with -coefficients.

When is a point, is just a finite type -scheme, and we have the familiar perverse t-structure on , with all its wonderful properties as usual. The key new definition is the following.

**Definition. **Given a finite type map of schemes , let be the full subcategory of objects such that for all geometric points .

It is easy to see that is stable under extensions and (after upgrading to derived -categories) under filtered colimits, and is set-theoretically reasonable, so it defines the left half of a t-structure on by Proposition 1.4.4.11 in Lurie’s *Higher Algebra*. We denote the right half of this t-structure, unsurprisingly, by , and call it the *relative perverse t-structure *(relative to , of course). We write and 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 , various operations are obviously left- or right- t-exact, etc.). Less formally, if is a finite-dimensional excellent Noetherian scheme, then the relative perverse truncation functors preserve , so we get an induced relative perverse t-structure on . 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 , and that the conditions in Theorem 8.2 are satisfied for excellent . (Nb. Gabber’s methods also reprove the existence of the relative perverse t-structure for any Noetherian , without appealing to -categories.)

However, the right half 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 lies in if and only if for all geometric points .

Note that I really am taking *-restrictions to geometric fibers here, just as in the definition of . One might naively guess that !-restrictions should be appearing instead, but no!

This theorem has a number of corollaries.

**Corollary 1. **The heart of the relative perverse t-structure consists of objects which are perverse after restriction to any geometric fiber of . 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 ”.

**Corollary 2. **For any map , the pullback functor is t-exact for the relative perverse t-structures (relative to and , respectively). In particular, relative perverse truncations commute with any base change on , and pullback induces an exact functor .

**Corollary 3.** If is any finitely presented morphism of qcqs schemes, then the relative perverse truncation functors on preserve .

Corollaries 1 and 2 are immediate consequences of the Key Theorem. Corollary 3 then follows from the case where 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 is excellent Noetherian finite-dimensional and . In this situation, we argue by induction on , with the base case being obvious. In general, this induction is somewhat subtle, and involves playing off the relative perverse t-structure on against the perverse t-structures on and the (absolute) perverse t-structure on (which exists once you pick a dimension function on ).

However, when 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 and be the inclusions of the special and generic points, with obvious base changes and . By definition, lies in iff and . By standard results on gluing t-structures (see chapter 1 in BBDG), this implies that lies in iff and . Thus, to prove the key theorem in this case, we need to show that for any with , the conditions and are *equivalent*.

To show this, consider the triangle . The crucial observation is that by assumption, *and that * *carries* *into* . 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 only can only have nonzero perverse cohomologies in degrees , so and have the same perverse cohomologies in degrees .

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

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Not in a naive way. One problem is that 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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