## 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 $<0$.

## 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 $K_0\mathrm{Perv}(Y)$.

## A trick and the decomposition theorem

In this post I’ll talk about a really fun trick Bhargav explained to me yesterday.

Let $K$ be a field with separable closure $C$. Algebraic variety over $K$ means separated $K$-scheme of finite type. Let $\ell$ be a prime invertible in $K$. Perverse sheaf means perverse $\mathbf{Q}_\ell$-sheaf.

If $f:X \to Y$ is a proper map of algebraic varieties over $K$, the decomposition theorem tells you that after base extension to $C$ there is a direct sum decomposition

$Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus_i IC_{Z_i}(\mathcal{L}_i)[n_i]\,\,\,\,\,\,\,\,(\dagger)$

in $D^b_c(Y_C,\mathbf{Q}_\ell)$. Here $Z_i \subset Y_{C}$ is some finite set of closed subvarieties, and $\mathcal{L}_i$ is a lisse $\mathbf{Q}_\ell$-sheaf on the smooth locus of $Z_i$. (My convention is that $IC_{Z}(\mathcal{L}) = j_{!\ast} (\mathcal{L}[\dim Z])$ where $j:Z^{sm} \to X$ is the natural map, so $IC_{X,\mathbf{Q}_\ell} = IC_{X}(\mathbf{Q}_\ell)$. Some people have different conventions for shifts here.)

The decomposition $(\dagger)$ is non-canonical. In particular, it is not $\mathrm{Aut}(C/K)$-equivariant, so it has no reason to descend to an analogous direct sum decomposition of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Indeed, typically there is no such decomposition! However, as Bhargav explained to me, one can still descend a certain piece of $(\dagger)$ to $D^b_c(Y,\mathbf{Q}_\ell)$ when $f$ is projective. This turns out to be good enough for some interesting applications.

To present Bhargav’s trick, let me make a definition. (What follows is a slight reinterpretation of what Bhargav told me, all mistakes are entirely due to me.)

Definition. Let $\mathcal{F}$ be a perverse sheaf on a variety $X$. Let $j:U \to X$ be the inclusion of the maximal dense open subvariety such that $j^\ast \mathcal{F}$ is a (shifted) lisse sheaf. Then we define the generic part of $\mathcal{F}$ as the perverse sheaf $\mathcal{F}^{gen} = j_{!\ast} j^{\ast} \mathcal{F}$.

Note that $\mathcal{F}^{gen}$ is zero if and only if $\mathcal{F}$ is supported on a nowhere-dense closed subvariety. Also, in general there is no map between $\mathcal{F}^{gen}$ and $\mathcal{F}$. However, in some cases $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$:

Proposition. Let $\mathcal{F}$ be a perverse sheaf on a $K$-variety $X$, and suppose that the pullback of $\mathcal{F}$ to $X_{C}$ is a direct sum of IC sheaves. Then $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$.

Proof. Let $j:U \to X$ be as in the definition of the generic part of $\mathcal{F}$, with closed complement $Z \subset X$. Our assumptions together with the definition of the generic part guarantee that $\mathcal{F}|X_C \simeq \mathcal{F}^{gen}|X_C \bigoplus \oplus_i IC_{Z_i}(\mathcal{L}_i)$ for some closed subvarieties $Z_i \subset X_C$ contained in $Z_C$.

Now look at the natural maps $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F} \overset{\alpha}{\to} \mathcal{F} \overset{\beta}{\to} \phantom{}^{\mathfrak{p}}j_{\ast} j^{\ast} \mathcal{F}$. Set $\mathcal{G} = \mathrm{im}\,\alpha$ and $\mathcal{H} = \mathrm{im}\,\beta$. Since $\phantom{}^{\mathfrak{p}}j_! j^{\ast} \mathcal{F}$ does not admit any nonzero quotient supported on $Z$, the composite map $\mathcal{G}|X_C \hookrightarrow \mathcal{F}|X_C \to \oplus_i IC_{Z_i}(\mathcal{L}_i)$ is zero.  Thus $\alpha$ factors over an inclusion $\mathcal{G}|X_C \subset \mathcal{F}^{gen}|X_C$. Moreover, $\mathcal{G}$ has the same generic part as $\mathcal{F}$. This is enough to imply that $\mathcal{G} = \mathcal{F}^{gen}$, so we have a natural inclusion $\mathcal{F}^{gen} \simeq \mathcal{G} \subset \mathcal{F}$. A dual argument shows that $\beta$ factors over a surjection $\mathcal{F} \twoheadrightarrow \mathcal{H} \simeq \mathcal{F}^{gen}$. It is now easy to see that the composite map $\mathcal{F}^{gen} \hookrightarrow \mathcal{F} \twoheadrightarrow \mathcal{F}^{gen}$ is an isomorphism, so $\mathcal{F}^{gen}$ is a direct summand of $\mathcal{F}$. $\square$

Corollary 0. Let $f:X \to Y$ be a projective map of $K$-varieties. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})^{gen}$ is a direct summand of $Rf_{\ast}IC_{X,\mathbf{Q}_\ell}$.

Proof. The decomposition theorem and the relative hard Lefschetz theorem give a decomposition $Rf_{\ast}IC_{X,\mathbf{Q}_\ell} \simeq \oplus \phantom{}^{\mathfrak{p}}\mathcal{H}^i(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})[-i]$ in $D^b_c(Y,\mathbf{Q}_\ell)$. Then $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}IC_{X,\mathbf{Q}_\ell})$ is a direct sum of IC sheaves after pullback to $Y_C$, so we can apply the previous proposition. $\square$

Corollary 1. Let $f:X \to Y$ be a projective alteration of $K$-varieties with $X$ smooth. Then $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $Rf_{\ast}\mathbf{Q}_{\ell}[\dim X]$.

Proof. Check that $IC_{Y,\mathbf{Q}_{\ell}}$ is a direct summand of $\phantom{}^{\mathfrak{p}}\mathcal{H}^0(Rf_{\ast}\mathbf{Q}_{\ell}[\dim X])^{gen}$ by playing with trace maps. Now apply the previous corollary. $\square$

Corollary 2. Let $K/\mathbf{Q}_p$ be a finite extension. Then for any $K$-variety $X$, the $p$-adic intersection cohomology $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a de Rham $G_K$-representation.

Proof. Let $X' \to X$ be a resolution of singularities. The previous corollary shows that $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p)$ is a direct summand of $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ compatibly with the $G_K$-actions. Since $H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$ is de Rham and the de Rham condition is stable under passing to summands, we get the result. $\square$

Note that we can’t prove this corollary by applying the decomposition theorem directly out of the box: the decomposition theorem does immediately give you a split injection $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_p) \to H^{\ast}(X'_{\overline{K}},\mathbf{Q}_p)$, but this map is not guaranteed a priori to be $G_K$-equivariant!

Corollary 3. Let $K$ be a finite extension of $\mathbf{Q}_p$ or $\mathbf{F}_p((t))$. If $H^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all smooth projective $K$-varieties $X$, then $IH^{\ast}(X_{\overline{K}},\mathbf{Q}_{\ell})$ satisfies the weight-monodromy conjecture for all proper $K$-varieties $X$. In particular, the weight-monodromy conjecture holds for the $\ell$-adic intersection cohomology of all proper $K$-varieties for $K/\mathbf{F}_p((t))$ finite.

Proof. Entirely analogous to the previous proof. $\square$

It would be interesting to know whether Corollary 1 has a “motivic” incarnation. Here I will pretend to understand motives for a minute. Suppose we have an assignment $X \mapsto D_{mot}(X)$ on quasi-projective $K$-varieties, where $D_{mot}(X)$ is a suitable triangulated category of constructible motivic sheaves on $X$ with $\mathbf{Q}$-coefficients. This should come with the formalism of (at least) the four operations $f^{\ast}_{mot}, Rf_{mot\ast}, \otimes, R\mathcal{H}\mathrm{om}$, and with faithful exact $\ell$-adic realization functors $\mathcal{R}_{\ell}: D_{mot}(X) \to D^b_c(X,\mathbf{Q}_{\ell})$ compatible with the four operations. I think this has all been constructed by Ayoub, maybe with some tiny additional hypothesis on $K$? Let $\mathbf{Q}_{X} \in D_{mot}(X)$ denote the symmetric monoidal unit. It then makes sense to ask:

Question. In the setting of Corollary 1, is there an idempotent $e \in \mathrm{End}_{D_{mot}(Y)}(Rf_{mot \ast} \mathbf{Q}_{X}[\dim X])$ such that $\mathcal{R}_{\ell}( e Rf_{mot \ast} \mathbf{Q}_{X}[\dim X]) \simeq IC_{Y,\mathbf{Q}_{\ell}}$ for all $\ell$?

This would imply that the split injections $IH^{\ast}(Y_{\overline{K}},\mathbf{Q}_\ell) \to H^{\ast}(X_{\overline{K}},\mathbf{Q}_\ell)$ provided by Corollary 1 can be chosen “independently of $\ell$”, i.e. that they are the $\ell$-adic realizations of some split injection in $D_{mot}(\mathrm{Spec}\,K)$.

## Brain teaser: generic perversity on fibers

Inspired by Shizhang’s Rampage talk last week, here is a brain teaser. Feel free to post your solution in the comments!

Let $f:X \to Y$ be any map of irreducible complex varieties, and let $\mathcal{F}$ be a perverse sheaf on $X$. Prove that there is a dense open subset $U \subset Y$ such that for any closed point $y \in U$, the shifted restriction $(\mathcal{F}|X_y)[-\dim Y]$ is a perverse sheaf on the fiber $X_y$.

## 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.