# Uniform approximation of Betti numbers

Denis-Charles Cisinski  Fakultät für Mathematik, Universität Regensburg, 93040
Regensburg, Germany<EMAIL_ADDRESS>https://cisinski.app.uni-
regensburg.de/

###### Abstract.

We prove that Lefschetz’s principle of approximating the cohomology of a
possibly singular affine scheme of finite type over a field by the cohomology
of a suitable (thickening of a) hyperplane section can be made uniform: in the
affine case, we can choose the hyperplane section independently of the
cohomology. Using Jouanolou’s trick, this gives a new way to bound the Betti
numbers of quasi-projective schemes over a field, independently of the
cohomology. This is achieved through a motivic version of Deligne’s generic
base change formula and an axiomatic presentation of the theory of perverse
sheaves. These methods produce generating families of Voevodsky motivic
sheaves that are realized in perverse sheaves over any base of equal
characteristic.

###### Contents

  1. 1 Motivic shadow of the weak Lefschetz Theorem
  2. 2 Perverse sheaves
  3. 3 Uniformly perverse motivic generators

## Introduction

Since Grothendieck introduced étale cohomology, proving that the Betti numbers
of $\ell$-adic cohomology of algebraic varieties over a separably closed field
are independent of $\ell$ remains a deep open problem in positive
characteristic. The case of smooth and proper algeraic varieties is reached by
weight techniques, using Deligne’s proof of the Weil conjectures, but there is
not any other general result in this direction. The next best thing consists
in providing uniform bounds of Betti numbers. This can be done by dévissage
from the case of smooth and proper algebraic varieties, and then applying
descent techniques together with de Jong’s alteration theorem, as explained by
Katz [Kat01, Theorem 5] who credits the result to de Jong and Berthelot. There
are results for non-constant coefficients as well, working with compatible
families [KL85, Kat01, Org19].

The present notes give new proofs of existence of such uniform bound, using
motivic techniques (in particular, we do not only prove independence of $\ell$
results but rather independence against a large class of Weil cohomologies).
We apply Lefschetz’s principle of approximating a quasi-projective scheme $X$
by a suitable hyperplane section. Of course, this needs adjustments in order
to take into account the singularities of $X$ as well as the fact that we want
the choice of the hyperplane section to be independent of the cohomology we
work with (e.g. $\ell$-adic cohomology of schemes of finite type over a
separably closed field $k$, with $\ell$ distinct from the characteristic of
$k$, or Berthelot’s rigid cohomology if $k$ is perfect of characteristic
$p>0$).

To be more precise, the general strategy goes as follows. In this
introduction, in order to simplify things, we work over an infinite field $k$
of characteristic $p$. Using Jouanolou’s trick [Jou73, Lemme 1.5], since the
formation of Voevodsky motives is $\mathbf{A}^{1}$-homotopy invariant, the
motive of any quasi-projective scheme is isomorphic to the motive of an affine
one. Therefore, we can focus on affine schemes of finite type over $k$. For
such a scheme $X$ of dimension $d$, one defines the notion of Leschetz
subscheme: a closed subscheme $Z$ of $X$ of dimension $d-1$, with affine
complement, that contains all irreducible components of $X$ of dimension $<d$
and the singular locus of $X$, as well as a hyperplane section satisfying a
suitable transversality condition, expressed in the language of the six
operations for motivic sheaves; see Definition 3.2 below. We define the motive
of the pair $(X,Z)$ through the cofiber sequence

$M(Z)\to M(X)\to M(X,Z)$

in Voevodsky’s category $\mathit{DM}_{c}(\mathrm{Spec}(k))$ (of constructible
étale motives). We prove that, for any prime number $\ell\neq p$, the
$\ell$-adic realization of $M(X,Z)$ is concentrated in degree $-d$ and that
its unique non-trivial cohomology group is a free $\mathbf{Z}_{\ell}$-module
of finite type, using the theory of $\ell$-adic perverse sheaves111This is
where the fact that we work with affine schemes matters: the main tool is then
Artin’s vanishing theorem in order to prove Beilinson’s transversality Theorem
2.22 below.; see Lemma 3.4. Since the $\ell$-adic realization of motivic
sheaves is symmetric monoidal, it is compatible with the formation and the
computation of traces of endomorphisms of constructible motives, in particular
of Euler characteristics (these are traces of identities). In this case, this
means that the Euler characteristic of $M(X,Z)[-d]$, seen as a motive,
coincides with the rank $\mathit{rk}(X,Z)$ of the unique non-trivial
cohomology group of its $\ell$-adic realization, proving that the latter is
independent of $\ell$. This yields isomorphisms

$H^{q}_{\mathrm{\acute{e}t}}(\bar{X},\mathbf{Z}_{\ell})\cong
H^{q}_{\mathrm{\acute{e}t}}(\bar{Z},\mathbf{Z}_{\ell})\quad\text{for $q\neq
d,d-1$,}$

together with an exact sequence

$0\to H^{d-1}_{\mathrm{\acute{e}t}}(\bar{X},\mathbf{Z}_{\ell})\to
H^{d-1}_{\mathrm{\acute{e}t}}(\bar{Z},\mathbf{Z}_{\ell})\to\mathbf{Z}_{\ell}^{\mathit{rk}(X,Z)}\to
H^{d}_{\mathrm{\acute{e}t}}(\bar{X},\mathbf{Z}_{\ell})\to 0\,.$

In particular we get a bound of the $d$th Betti number of $X$ by the
motivically defined non-negative integer $\mathit{rk}(X,Z)$, and we observe
that the Betti numbers of $X$ in lower degree are bounded by those of $Z$.
This is meaningful because, following ideas from Beilinson [Bei87], we apply a
motivic variant of Deligne’s generic base change theorem, to deduce a motivic
shadow of the weak Lefschetz theorem (Theorem 1.16 below). This implies that,
for any closed subscheme $Y\subset X$ of dimension $<d$, there is a Lefchetz
subscheme $Z\subset X$ containing $Y$ (this is where we need $k$ to be
infinite: in fact, we prove that the family of eligible hyperplane sections
contains the rational points of an open dense subscheme of the space of all
hyperplane sections); see Prop. 3.3. Therefore, following a longstanding
tradition in Algebraic Topology [Mil64, Tho65], we can produce a Lefschetz
filtration of $X$: a finite sequence of closed subschemes of the form

$\varnothing=X_{-1}\subset X_{0}\subset\ldots\subset X_{d-1}\subset X_{d}=X$

where, for $0\leq i\leq d$, each $X_{i-1}$ is a Lefschetz subscheme of
$X_{i}$. This leads to a filtration of the motive of $X$ whose associated
graded object yields a complex with terms of the form $M(X_{i},X_{i-1})$
computing the Euler characteristic of $X$

$\chi(X)=\sum_{i}(-1)^{i}\mathit{rk}(X_{i},X_{i-1})$

and such that the $i$th $\ell$-adic Betti number of the affine scheme $X$
(both $\mathbf{Q}_{\ell}$\- or with $\mathbf{F}_{\ell}$-coefficients) is
bounded by $\mathit{rk}(X_{i},X_{i-1})$ for all $\ell\neq p$; see Theorem 3.6.

However, our proof does more than providing a bound that is independent of
$\ell$: it provides a uniform bound that is valid in any realization of
Voevodsky’s motivic sheaves that has a theory of perverse sheaves (including
the properties that Verdier duality induces an equivalence on constructible
objects and that Artin’s vanishing theorem holds); see Definition 2.7. By
developing this axiomatic approach, we also try to reach a wide enough level
of generality, geometrically: instead of working over a field, we develop
enough material to prove all the main ingredients over a scheme of finite type
over a field: what is true over an infinite field is in fact true generically,
possibly after pulling back along a flat finite cover of a dense open subset
of any algebraic variety (from this point of view, we can work over any field
$k$). Because what is at stake, after all, is the problem of finding big
enough families of motivic sheaves that are known to be realized into ordinary
sheaves or into perverse sheaves. This is what we do at the end of this
article (Prop. 3.9 and 3.17), thus providing constructions of $t$-structures
on $\mathit{DM}(X,\mathbf{Q})$ that are candidates for the perverse motivic
$t$-structure (3.18). These constructions, by finding nice generators for
categories of motivic sheaves, may be seen as a way to provide uniform bounds
of Betti numbers of all sorts, not only for the cohomology of algebraic
varieties, but for the cohomology of any constructible motivic sheaf (this
includes cohomology with compact support or not, as well as nearby cycles,
thanks to Ayoub’s work [Ayo07]), possibly in relative situations. Finding
explicit generators for such $t$-structures is also part of the standard tools
to develop the theory of Nori motives and to relate them with Voevodsky
motives in characteristic zero. Therefore, the results of the present text
open the way to a uniform theory of Nori motives [HMS17, Ara05, IM19],
possibly over any field: instead of working with a single cohomology theory,
we can now consider the family of all cohomology theories appearing in a six-
functors formalism in which there is a good notion of perverse sheaves. This
is strongly related with Ayoub’s constructions [Ayo23] of new Weil
cohomologies in positive characteristic and their associated Hopf algebroids.

##  1\. Motivic shadow of the weak Lefschetz Theorem

######  1.1.

We consider a $\mathbf{Z}$-linear six-functors formalism $D$ in the sense of
Voevodsky-Ayoub (see [DG22]), taking values in presentable
$\infty$-categories, defined on the category of schemes locally of finite
presentation over a base scheme $S$. We assume furthermore that, for any $X$,
there is a full stable subcategory $D_{c}(X)$ of $D(X)$, whose objects are
called _constructible objects_ with the following properties:

  * (i)

$D_{c}(X)$ contains the $\otimes$-unit $\mathbf{1}_{X}$, is closed under
tensor product, under pullbacks of the form $f^{*}$ for any map $f$, and under
direct image with compact support $f_{!}$ for any separated morphism of finite
presentation $f$;

  * (ii)

for any $X$ and any $F$ in $D_{c}(X)$, there is a dense open subscheme
$U\subset X$ such that $F_{|U}$ is dualizable (in the sense of [Lur17, Section
4.6.1]);

  * (iii)

for any $X$, any dualizable object of $D(X)$ is constructible;

  * (iv)

constructible objects form an hypersheaf of $\infty$-categories for the étale
topology.

Remark that, in the case of schemes with noetherian underlying space, these
properties determine $D_{c}(X)$ uniquely.

######  Example 1.2.

Let $\mathit{DM}(X)=\mathrm{Ind}(\mathit{DM}_{h,lc}(X))$ the $\infty$-category
of ind-objects on the $\infty$-category of locally constructible objects of
the $\infty$-category of $h$-motives in the sense of [CD16, Def. 6.3.1]. Then
$\mathit{DM}$ satisfies all the properties above, at least if we restrict to
noetherian schemes (but this is not even necessary); this follows easily from
see [CD16, Theorem 6.3.26]. For any commutative ring $\Lambda$, there is a
$\Lambda$-linear version $\mathit{DM}(X,\Lambda)$ defined as the
$\Lambda$-modules in $\mathit{DM}(X)$.

If $\Lambda$ is a commutative ring of characteristic $n>0$, there is a
rigidity theorem in the following sense: for $n$ prime to the residue
characteristics of $X$, there is a canonical equivalence between compact
$\Lambda$-modules in $\mathit{DM}(X)$ and ordinary constructible sheaves of
$\Lambda$-modules of finite tor-dimension over the small étale site of $X$:

$\mathit{DM}_{c}(X,\Lambda)\cong D^{b}_{\mathrm{ctf}}(X,\Lambda)$

(see [CD16, Theorem 6.3.11]).

######  Example 1.3.

Let $\ell$ be a prime number and define

$D^{b}_{c}(X,\mathbf{Z}_{\ell})=\varprojlim_{i}\mathit{DM}_{h,lc}(X,\mathbf{Z}/\ell^{i}\mathbf{Z})$

where the limit is taken in the $\infty$-category of (stable)
$\infty$-categories. In the case where $\ell$ is prime to the residue
characteristics of $X$, the rigidity theorem imples that
$D^{b}_{c}(X,\mathbf{Z}_{\ell})$ is the usual derived category of constuctible
$\ell$-adic sheaves on the small étale site of $X$; see [CD16, Prop. 7.2.21].
The assignment

$X\mapsto D(X,\mathbf{Z}_{\ell})=\mathrm{Ind}(D^{b}_{c}(X,\mathbf{Z}_{\ell}))$

is then a six-functors formalism that satisfies all properties listed in
paragraph 1.1. We observe that the dualizable objects are then exactly those
complexes whose cohomology objects are what Deligne calls “faisceaux
localement constants tordus”. Indeed, the functor

$D^{b}_{c}(X,\mathbf{Z}_{\ell})\to D^{b}_{c}(X,\mathbf{F}_{\ell})\ ,\quad
F\mapsto F/\ell$

is conservative (by definition of what “$\ell$-adic” means) and symmetric
monoidal. This implies that it preserves and detects dualizable objects.
Therefore, by virtue of [CD16, Remark 6.3.27], $F$ in
$D^{b}_{c}(X,\mathbf{Z}_{\ell})$ is dualizable if and only if $F/\ell$ is
locally free with perfect values; for a discussion using condensed techniques,
see [BS15, HRS23]. We denote by $D(X,\mathbf{Q}_{\ell})$ the full subcategory
of uniquely divisible objects in $D(X,\mathbf{Z}_{\ell})$. It is compactly
generated with category of compact objects

$D^{b}(X,\mathbf{Q}_{\ell})=D^{b}_{c}(X,\mathbf{Z}_{\ell})\otimes\mathbf{Q}\,.$

######  Example 1.4.

Let $\mathcal{U}$ be an ultrafilter on the set of prime numbers. We define for
any noetherian $X$:

$D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}})^{\prime}=\varinjlim_{A\in\mathcal{U}}\prod_{\ell\notin
A}\mathit{DM}_{h,lc}(X,\mathbf{F}_{\ell})$

(where the colimit and the product are taken in the $\infty$-category of
(stable) $\infty$-categories). We define
$D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}})$ as the full subcategory of
$D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}})^{\prime}$ by noetherian induction: an
object $F$ of $D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}})^{\prime}$ belongs to
$D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}})$ if there is a dense open subscheme $V$
of $U$ such that the restriction $F_{|_{V}}$ is dualizable in
$D^{b}_{c}(V,\mathbf{F}_{\mathcal{U}})^{\prime}$ and $F$ is constructible on a
complement of $V$. We define finally

$D(X,\mathbf{F}_{\mathcal{U}})=\mathrm{Ind}(D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}}))\,.$

If the set of residue characteristics of $X$ is finite,
$D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}})^{\prime}$ coincides with the
$\mathcal{U}$-indexed ultraproduct of the ordinary categories of constructible
sheaves $D^{b}_{c}(X,\mathbf{F}_{\ell})$. If the set of residue
characteristics is finite (e.g. is X is of finite type over a semi-local
scheme), then $D^{b}_{c}(X,\mathbf{F}_{\mathcal{U}})^{\prime}$ coincides with
the ultraproduct of the usual categories of étale sheaves
$D(X,\mathbf{F}_{\ell})$, where $\ell$ runs over all primes distinct from the
residue characteristics of $X$. The corresponding cohomology is then the
ultraproduct in the usual sense of étale cohomology groups:

$\pi_{0}\operatorname{Map}_{D(X,\mathbf{F}_{\mathcal{U}})}(\mathbf{1}_{X},\mathbf{1}_{X}[i])\cong\varinjlim_{A\in\mathcal{U}}\prod_{\ell\notin
A}H^{i}_{\mathrm{\acute{e}t}}(X,\mathbf{F}_{\ell})\,.$

For an arbitrary scheme $X$, by virtue of [CD16, Cor. 4.5.3], we always have

$\mathit{DM}_{h,lc}(X,\mathbf{F}_{\ell})\cong
D^{b}_{c}(X[1/\ell],\mathbf{F}_{\ell})$

where $X[1/\ell]$ is the product of $X$ with the spectrum of
$\mathbf{Z}[\frac{1}{\ell}]$. Therefore, we have

$\pi_{0}\operatorname{Map}_{D(X,\mathbf{F}_{\mathcal{U}})}(\mathbf{1}_{X},\mathbf{1}_{X}[i])\cong\varinjlim_{A\in\mathcal{U}}\prod_{\ell\notin
A}H^{i}_{\mathrm{\acute{e}t}}(X[1/\ell],\mathbf{F}_{\ell})\,.$

The only non-trivial property of 1.1 that is difficult to check is (i). In the
case of quasi-excellent schemes over a semi-local noetherian ring, this
follows right away from Orgogozo’s uniform constructibility theorem [Org19,
Théorème 3.1.1].

######  Example 1.5.

If the base field has a complex embedding $k\hookrightarrow\mathbf{C}$, one
can define the Betti six-functors formalism

$X\mapsto D_{\textit{Betti}}(X)=D(X(\mathbf{C}),\mathbf{Z})$

by taking derived $\infty$-categories of sheaves of abelian groups on the
analytic variety associated to $X$. The associated cohomology is Betti
cohomology.

######  Example 1.6.

If $k$ is of characteristic zero, one can define the de Rham six-functors
formalism over schemes of finite type over $k$ through the theory of algebraic
$D$-modules. The corresponding cohomology is algebraic de Rham cohomology.

######  Example 1.7.

If $k$ is a perfect field of characteristic $p>0$, choosing a complete
discrete valuation ring of characteristic zero with residue field $k$ (such as
the ring of Witt vectors), Berthelot’s theory of arithmetic $D$-modules
defines a six-functors formalism for schemes of finite type over $k$. The
corresponding cohomology is then Berthelot’s rigid cohomology (i.e. it
coincides with Monsky-Washnitzer cohomology for affine smooth $k$-schemes and
with crystalline cohomology for smooth and projective ones).

######  1.8.

If $D$ and $D^{\prime}$ are two six-functors formalisms defined for schemes
over $S$, there is a natural notion of morphism of six-functors formalisms
from $D$ to $D^{\prime}$; see [DG22]. Informally, this consists of colimit-
preserving (hence exact) symmetric monoidal functors

$D(X)\to D^{\prime}(X)$

that are compatible with pullbacks $f^{*}$ for all $f$ as well with direct
image with compact support $f_{!}$ for any $f$ separated and of finite
presentation. In particular, on noetherian schemes, such morphisms must
preserve constructible objets (they preserve dualizable objects, which implies
that they preserve constructible objects by noetherian induction). One can
prove that $\mathit{DM}$ is the initial object among $\mathbf{Z}$-linear six-
functors formalisms satisfying étale descent222One can see that
$\mathit{DM}_{h}$ is initial among $\mathbf{Z}$-linear six-functors formalisms
satisfying étale hyperdescent: since we know what is the initial six-functors
formalism by virtue of Drew and Gallauer’s [DG22, Theorem 7.14], we see easily
that it suffices to prove it for $\mathbf{Q}$-linear coefficients, which
follows right away from [CD19, Theorem 16.1.2 and 16.2.18], and for finite
coefficients, which follows from the combination of rigidity theorems: the one
recalled in Example 1.2 together with those stated in [Ayo14, Bac21]. This
implies that $\mathit{DM}$ is the initial object among $\mathbf{Z}$-linear
six-functors formalisms satisfying the properties of 1.1. For $X$ noetherian
with residue fields that have a uniform bound on their Galois cohomological
dimension, $\mathit{DM}$ and $\mathit{DM}_{h}$ agree anyway.. In particular,
for any other six-functor formalism $D$, there is a unique morphism

$r_{D}\colon\mathit{DM}(-)\to D(-)\,.$

We call an object $F$ of $D(X)$ _strictly of geometric origin_ if there is a
constructible motivic sheaf $M$ in $\mathit{DM}_{c}(X)$ such that $F\cong
r_{D}(M)$ in $D(X)$ (in particular, such an $F$ is constructible). We will say
that $D$ _behaves well_ over $S$ if the functor $r_{D}$ commutes with functors
of the form $f_{*}$ for any morphism of finite presentation $f$. All examples
of six-functors formalisms given in the present notes are known to behave
well: in the case where $D$ is $\mathbf{Q}$-linear and defined on schemes of
finite type over a field, this is subsumed by [CD19, Theorem 4.4.25] – see
Remark 1.13 below to drop the $\mathbf{Q}$-linearity. For $D=D(-,\Lambda)$
with $\Lambda$ of positive characteristic, this is obvious, by rigidity. This
is also true in the case where $D(X)=D(X,\mathbf{Z}_{\ell})$ for any $S$ (see
[CD16, Theorem 7.2.11]), or $D(X)=D(X,\mathbf{F}_{\mathcal{U}})$ (exercise).

The property of being dualizable is at the core of many constructions and
computations. In particular, the fact that any construcible sheaf becomes
dualizable over a dense open subscheme is at the center of everything. For
objects that are strictly of geometric origin, the choice of such a
dualizibility locus can be made motivically, hence, independently of $D$. This
is made precise in the following statement.

######  Proposition 1.9.

Let $D$ be six-functors formalism defined over $S$-schemes. If $F$ is strictly
of geometric origin in $D(X)$, then one can find a dense open subscheme $U$ of
$X$, independently of $D$, such that the restriction $F_{|U}$ is dualizable in
$D(U)$.

###### Proof.

Let $M$ be constructible in $\mathit{DM}(X)$ such that $F\cong r_{D}(M)$. We
choose a dense open immersion $j\colon U\to X$ such that $M_{|U}=j^{*}(M)$ is
dualizable in $\mathit{DM}(X)$. Since the functor

$r_{D}\colon\mathit{DM}(U)\to D(U)$

is symmetric monoidal, it preserves dualizable objects, and therefore
$F_{|U}\cong r_{D}(M_{|U})$ is dualizable. The open subscheme $U$ clearly
depends on $M$ only. ∎

######  1.10.

The following statement, is a generalization of Deligne’s generic base change
theorem for torsion étale sheaves [Del77, Th. Finitude, 1.9]. With this new
degree of generality, it may be seen as an independence of $D$ (in particular,
independence of $\ell$) result as well; see Corollary 1.14.

######  Theorem 1.11 (Deligne’s generic base change formula).

Let $f:X\to Y$ be a morphism between separated schemes of finite type over a
noetherian base scheme $S$. Let $F$ be a constructible object in $D(X)$. Then
there is a dense subscheme $U\subset S$ over which $f_{*}(F)$ becomes
constructible and its formation is compatible with any base-change. More
precisely, for each $w:S^{\prime}\to S$ factoring through $U$ we have an
isomorphism of constructible objects

$v^{*}f_{*}(F)\cong f^{\prime}_{*}u^{*}(F)$

where

${X^{\prime}}$${X}$${Y^{\prime}}$${Y}$${S^{\prime}}$${S}$$\scriptstyle{u}$$\scriptstyle{f^{\prime}}$$\scriptstyle{f}$$\scriptstyle{v}$$\scriptstyle{w}$

is the associated pull-back diagram.

###### Proof.

See [Cis21, Theorem 2.4.2] (the proof is stated for $D(X)=\mathit{DM}(X)$ but
it only uses the properties listed in paragraph 1.1). The statement in _loc.
cit._ does not say that $f_{*}(F)$ becomes constructible over $U$, but this is
what the proof yields anyway. ∎

######  Remark 1.12.

The proof of the generic base change formula really uses étale descent on
constructible sheaves: this implies that, for any quasi-finite morphism $f$.
the functors $f_{!}$ and $f_{*}$ are conservative, which is a crucial
ingredient in Deligne’s proof. It is not known whether the generic base change
formula is true or not for more universal theories of motivic sheaves in which
we only have Nisnevich descent, such as Voevodsky’s and Morel’s $\mathit{SH}$.

######  Remark 1.13.

In fact, a close examination of the proof of the generic base change formula
shows that, for a given constructible $F$ in $D(X)$ and a morphism of finite
presentation $f:X\to Y$ over some base $S$, one can choose the open subset
$U\subset S$ so that the conclusion of the generic base change theorem holds
for $f_{*}(F)$ in $D$ and so that, for any morphism of six-functors formalisms
$\varphi\colon D\to D^{\prime}$ (with $D^{\prime}$ satisfying the assumption
of 1.1 as well), the canonical map

$\varphi(f_{*}(F))\to f_{*}(\varphi(F))$

becomes an isomorphism over $U$ and the conclusion of the generic base change
theorem holds for $\varphi(F)$ with respect to $f$ in $D^{\prime}$ (the idea
is that we can do the proof of the generic base change formula for $F$ in $D$
and for $\varphi(F)$ in $D^{\prime}$ in parallel and observe that $\varphi$ is
compatible with each reduction step, thus giving rise to the same choice of
locus $U$ over which the base change formula holds).

######  Corollary 1.14.

If moreover $F$ is strictly of geometric origin, then the open subscheme
$U\subset S$ in the statement of Theorem 1.11 may be chosen indepently of $D$.

###### Proof.

Let $M$ in $\mathit{DM}_{c}(X)$ such that $F\cong r_{D}(M)$ in $D(X)$. Apply
the generic base change formula to $M$, which gives an open subscheme
$U\subset S$ such that the conclusion of Theorem 1.11 holds in $\mathit{DM}$.
The conclusion of Theorem 1.11 holds for $r_{D}(M)$ in $D$ as well, by the
previous Remark.333For the reader who does not want to verify the claims of
Remark 1.13, Corollary 1.14 can be proved as follows in the case where $D$
behaves well: the conclusion of Theorem 1.11 holds for $F=r_{D}(M)$ in $D$
with respect to the motivically chosen $U\subset S$: since
$r_{D}\colon\mathit{DM}\to D$ commutes with the six operations, all the base
change isomorphisms involving $M$ in $\mathit{DM}$ will yield analogous base
change isomorphisms involving $F$ in $D$. ∎

######  1.15.

We fix a six-functors formalism $D$ defined over $S$-schemes, as in paragraph
1.1. Let $E$ be a vector bundle over $S$ and $\mathbf{P}=\mathbf{P}(E)$ the
associated projective bundle over $S$. We denote by $\mathbf{P}^{\wedge}$ the
projective bundle associated to the dual of $E$, and we assume given a diagram
of $S$-schemes of finite presentation

$X\overset{e}{\to}\bar{X}\overset{\bar{f}}{\to}\mathbf{P}$

(in practice, $e$ will be an open immersion and $\bar{X}$ will be a
compactification of $X$ as an $S$-scheme, but, for the moment, $e$ and
$\bar{f}$ could be any maps). We let

$h_{\mathit{univ}}\colon\mathbf{H}\hookrightarrow\mathbf{P}\times_{S}\mathbf{P}^{\wedge}$

be the universal hyperplane in $\mathbf{P}$, and write

$\mathbf{P}\overset{\pi}{\leftarrow}\mathbf{P}\times_{S}\mathbf{P}^{\wedge}\overset{\pi^{\wedge}}{\to}\mathbf{P}^{\wedge}$

for the canonical projections. We consider the following diagram, in which all
squares are cartesian.

${H_{X}}$${H_{\bar{X}}}$${\mathbf{H}}$${X\times_{S}\mathbf{P}^{\wedge}}$${\bar{X}\times_{S}\mathbf{P}^{\wedge}}$${\mathbf{P}\times_{S}\mathbf{P}^{\wedge}}$${\mathbf{P}^{\wedge}}$${X}$${\bar{X}}$${\mathbf{P}}$${S}$$\scriptstyle{\eta}$$\scriptstyle{h_{X}}$$\scriptstyle{h_{\bar{X}}}$$\scriptstyle{h_{\mathit{univ}}}$$\scriptstyle{\gamma}$$\scriptstyle{p}$$\scriptstyle{\bar{p}}$$\scriptstyle{\pi^{\wedge}}$$\scriptstyle{\pi}$$\scriptstyle{e}$

Observe that both projections $\mathbf{H}\to\mathbf{P}$ and
$\mathbf{H}\to\mathbf{P}^{\wedge}$ are smooth. Given a constructible object
$F$ in $D(X)$, we then have isomorphisms

$\bar{p}^{*}\,e_{*}(F)\cong\gamma_{*}\,p^{*}(F)$

and

$h^{*}_{\bar{X}}\,\gamma_{*}\,p^{*}(F)\cong\eta_{*}\,h^{*}_{X}\,p^{*}(F)\,.$

The first one is an instance of the smooth base change isomorphism, whereas
the second one comes from the following sequence of isomorphisms (using the
smoothness of $\mathbf{H}$ over the projective bundle $\mathbf{P}$):

$\displaystyle h^{*}_{\bar{X}}\,\gamma_{*}\,p^{*}(F)$ $\displaystyle\cong
h^{*}_{\bar{X}}\,\bar{p}^{*}\,e_{*}(F)$
$\displaystyle\cong(\bar{p}h_{\bar{X}})^{*}\,e_{*}(F)$
$\displaystyle\cong\eta_{*}\,(ph_{X})^{*}(F)$
$\displaystyle\cong\eta_{*}\,h^{*}_{X}\,p^{*}(F)\,.$

This leads to the following statement that goes back to Beilinson [Bei87] and
Katz [Kat93].

######  Theorem 1.16 (shadow of the weak Lefschetz theorem).

Under the assumptions above, there exists a dense open subscheme
$U\subset\mathbf{P}^{\wedge}$ with the following property: for any hyperplane
$H_{t}$ in $\mathbf{P}$ defined over an $S$-scheme $T$ corresponding to a map
$t\colon T\to U$ over $S$, if we form the corresponding diagram of cartesian
squares

${X_{T}-H}$${X_{T}}$${H}$${\bar{X}_{T}-\bar{H}}$${\bar{X}_{T}}$${\bar{H}}$$\scriptstyle{j}$$\scriptstyle{e_{T}}$$\scriptstyle{u}$$\scriptstyle{i}$$\scriptstyle{\bar{\jmath}}$$\scriptstyle{\bar{\imath}}$

in which $X_{T}$ and $\bar{X}_{T}$ are the pullbacks of $X$ and $\bar{X}$ over
$T\to S$ respectively, and $\bar{H}$ is the pullback of $H_{t}$ along the
projection $H_{\bar{X}}\to\mathbf{P}^{\wedge}$, we have a canonical
isomorphism

$\bar{\jmath}_{!}\,\bar{\jmath}^{*}\,(e_{T})_{*}(F_{T})\cong(e_{T})_{*}\,j_{!}\,j^{*}(F_{T})$

with $F_{T}$ the pullback of $F$ along the projection $X_{T}\to X$. Moreover,
if ever $F$ is strictly of geometric origin, the subscheme $U$ can be chosen
independently of $D$.

###### Proof.

We apply the generic base change formula to the restriction of $p^{*}(F)$ on
$H_{X}$ and $\eta$, seen over the scheme of hyperplane sections
$\mathbf{P}^{\wedge}$. There is thus a dense open subscheme
$U\subset\mathbf{P}^{\wedge}$ over which $\eta_{*}\,h^{*}_{X}\,p^{*}(F)$ is
constructible and its formation stable under any base change along any map
$T\to U\subset\mathbf{P}^{\wedge}$. Applying the generic base change formula
to $p^{*}(F)$ with respect to the map $\gamma$ over $\mathbf{P}^{\wedge}$, we
may shrink $U$ so that the formation of $\gamma_{*}\,p^{*}(F)$ is stable under
any base change along any map $T\to U\subset\mathbf{P}^{\wedge}$ as well.

Let us consider an $S$-scheme $T$ together with a map $t\colon T\to U$
defining an hyperplane $H_{t}$ in $\mathbf{P}$. We obtain pullback squares of
the form

${H}$${H_{X}}$${X_{T}}$${X\times_{S}\mathbf{P}^{\wedge}}$$\scriptstyle{t_{H}}$$\scriptstyle{i}$$\scriptstyle{h_{X}}$$\scriptstyle{t_{X}}$
and
${\bar{H}}$${H_{\bar{X}}}$${\bar{X}_{T}}$${\bar{X}\times_{S}\mathbf{P}^{\wedge}}$$\scriptstyle{t_{\bar{H}}}$$\scriptstyle{\bar{\imath}}$$\scriptstyle{h_{\bar{X}}}$$\scriptstyle{t_{\bar{X}}}$

inducing base change isomorphisms

$t^{*}_{\bar{X}}\,\gamma_{*}\,p^{*}(F)\cong(e_{T})_{*}\,\bar{\imath}^{*}\,p^{*}(F)=(e_{T})_{*}(F_{T})$

and

$t^{*}_{\bar{H}}\,\eta_{*}\,h^{*}_{X}\,p^{*}(F)\cong
u_{*}\,t^{*}_{H}\,p^{*}(F)=u_{*}\,i^{*}(F_{T})\,.$

We claim that these induce a canonical isomorphism:

$\bar{\imath}_{*}\,\bar{\imath}^{*}\,(e_{T})_{*}(F_{T})\cong(e_{T})_{*}\,i_{*}\,i^{*}(F_{T})\,.$

Indeed, we have:

$\displaystyle\bar{\imath}_{*}\,\bar{\imath}^{*}\,(e_{T})_{*}(F_{T})$
$\displaystyle\cong\bar{\imath}_{*}\,\bar{\imath}^{*}\,t^{*}_{\bar{X}}\,\gamma_{*}\,p^{*}(F)$
$\displaystyle\cong\bar{\imath}_{*}\,t^{*}_{\bar{H}}\,h^{*}_{\bar{X}}\,\gamma_{*}\,p^{*}(F)$
$\displaystyle\cong\bar{\imath}_{*}\,t^{*}_{\bar{H}}\,\eta_{*}\,h^{*}_{X}\,p^{*}(F)$
$\displaystyle\cong\bar{\imath}_{*}\,u_{*}\,i^{*}(F_{T})$
$\displaystyle\cong(e_{T})_{*}\,i_{*}\,i^{*}(F_{T})\,.$

On the other hand, we have a canonical morphism of cofiber sequences:

${\bar{\jmath}_{!}\,\bar{\jmath}^{*}(e_{T})_{*}(F_{T})}$${(e_{T})_{*}(F_{T})}$${\bar{\imath}_{*}\,\bar{\imath}^{*}\,(e_{T})_{*}(F_{T})}$${(e_{T})_{*}\,j_{!}\,j^{*}(F_{T})}$${(e_{T})_{*}(F_{T})}$${(e_{T})_{*}\,i_{*}\,i^{*}(F_{T})}$$\scriptstyle{\cong}$

in which the vertical map on the right hand side is an isomorphism by the
computation we just did. Since the middle vertical map is the identity, this
proves that the vertical map on the left hand side is an isomorphism. The fact
that the open subscheme $U\subset\mathbf{P}^{\wedge}$ may be chosen
independently of $D$ if ever the sheaf $F$ is strictly of geometric origin
follows from Corollary 1.14. ∎

######  Corollary 1.17.

Let $k$ be an infinite field, and $X$ a quasi-projective scheme over $k$. We
choose an open immersion $e:X\to\bar{X}$ into a projective $k$-scheme, and a
constructible motivic sheaf $M$ in $\mathit{DM}(X)$. There exists a dense open
subset $U$ (hence with at least one rational point) of the family of
hyperplane sections of $\bar{X}$, such that, for any $\bar{H}$ in $U$, with
$H=\bar{H}\cap X$ and with corresponding open complements
$\bar{\jmath}\colon\bar{X}-\bar{H}\to\bar{X}$ and $j\colon X-H\to X$, and for
any six functors formalism $D$ as in paragraph 1.1, if we put $F=r_{D}(M)$ for
the realization of $M$ in $D$, then there is a canonical isomorphism

$\bar{\jmath}_{!}\,\bar{\jmath}^{*}\,e_{*}(F)\cong e_{*}\,j_{!}\,j^{*}(F)\,.$

In particular, the cohomology with compact support of $\bar{X}-\bar{H}$ with
coefficients in $e_{*}(F)$ coincides with the cohomology of $X$ with
coefficients in $j_{!}\,j^{*}(F)$.

######  1.18.

A fundamental situation that will come back eventually in these notes is the
following one. Let $F$ be in $D(X)$ for an affine scheme of finite type $X$
over a field $k$. We assume chosen an affine open embedding $e\colon
X\to\bar{X}$ into a projective scheme. A hyperplane section $\bar{H}$ of
$\bar{X}$ with trace $H=X\cap\bar{H}$ is said to be _transverse to $F$_ if by
forming the corresponding pullback squares below

${X-H}$${X}$${H}$${\bar{X}-\bar{H}}$${\bar{X}}$${\bar{H}}$$\scriptstyle{j}$$\scriptstyle{e}$$\scriptstyle{u}$$\scriptstyle{i}$$\scriptstyle{\bar{\jmath}}$$\scriptstyle{\bar{\imath}}$

there is a canonical isomorphism

$\bar{\jmath}_{!}\,\bar{\jmath}^{*}\,e_{*}(F)\cong e_{*}\,j_{!}\,j^{*}(F)\,.$

Theorem 1.16 tells that there is a dense open subspace of such hyperplane
sections and therefore that, possibly after base change along a finite
extension of $k$, one can provide such nice hyperplane sections. Furthemore,
if ever $F$ is strictly of geometric origin, one can choose a dense open
subspace of hyperplane sections in good position with respect to $F$
independently of $D$.

##  2\. Perverse sheaves

######  2.1.

Let $D$ be a six-functors formalism defined on schemes of finite presentation
over a base $S$. A _$t$ -structure_ on $D$ is a family of $t$-structures on
$D(X)$ for each $X$ with the following properties (we adopt cohomological
conventions):

  * (a)

for any scheme $X$ any integer $n\in\mathbf{Z}$, the Tate object
$\mathbf{1}_{X}(n)$ is in the heart of $D(X)$;

  * (b)

for any scheme $X$, the tensor product functor preserves objects concentrated
in degree $\leq 0$ in the sense of the $t$-structure (i.e. tensoring with an
object of the heart is right $t$-exact in $D(X)$);

  * (c)

for any map $f\colon X\to Y$, the functor $f^{*}\colon D(Y)\to D(X)$ is
$t$-exact;

  * (d)

for any separated morphism of finite presentation $f:X\to Y$, the functor
$f_{!}\colon D(X)\to D(Y)$ is right $t$-exact;

  * (e)

the truncation functors preserve dualizable objects and all dualizable objects
of $D(X)$ are bounded for the $t$-structure;

  * (f)

producing the dual of dualizable objects $M\mapsto M^{\wedge}$ is a $t$-exact
functor.

As a consequence, Tate twists $M\mapsto M(n)$ are $t$-exact functors. Any
constructible object $F$ of $D(X)$ is bounded. Indeed, if we define $f$ as the
coproduct of the embeddings of the strata over which $F$ becomes dualizable,
then $f^{*}(F)$ is dualizable in $D(X^{\prime})$, and since $f^{*}$ is
conservative (by localization) and $t$-exact, this proves the claim. Remark
that condition (f) is only reasonnable with $\mathbf{F}$-linear coefficients
for some field $\mathbf{F}$.444Condition (f) can be avoided but we would need
two $t$-structures whose restrictions to dualizable objects would be exchanged
through ordinary duality, and that would only make notations worse. We denote
by $D^{\leq i}_{c}(X)$ (by $D^{\geq i}_{c}(X)$ respectively) the full
subcategory of $D_{c}(X)$ that consists of constructible objects with
cohomology concentrated in degrees $\leq i$ (in degrees $\geq i$,
respectively) with respect to the $t$-structure above. Such a $t$-structure
will be called _ordinary_.

######  Example 2.2.

It is well known that, to say the least, the existence of such a $t$-structure
is very conjectural as far as $\mathit{DM}$ is concerned. However,
$D(X)=D(X,\mathbf{Q}_{\ell})$, $D(X)=D(X,\mathbf{F}_{\ell})$ and
$D(X)=D(X,\mathbf{F}_{\mathcal{U}})$ have canonical $t$-structures coming from
the usual $t$-structures on the derived category of sheaves of
$\Lambda$-modules on the small étale site of $X$, with $\Lambda$ and finite
commutative ring. If moreover $X$ is noetherian, all the properties listed
above will be fulfilled in these cases.

######  2.3.

Let $k$ be a field and $D$ a six-functors formalism as in 1.1 defined on
schemes of finite type over $k$ and equipped with a $t$-structure in the sense
of 2.1. We can then reproduce word for word section 2.2 of [BBDG18]. We will
always consider here the autodual perversity. The perverse $t$-structure on
$D(X)$ is characterized by noetherian induction, using the following facts:

  * (i)

The perverse $t$-structure coincides with the usual one on $D(X)$ for any
$k$-scheme $X$ of dimension $\leq 0$;

  * (ii)

For any closed immersion $i\colon Z\to X$, the functor $i_{!}\colon D(Z)\to
D(X)$ is $t$-exact with respect to the perverse $t$-structure, and for any
open immersion $j\colon U\to X$, the functor $j^{*}\colon D(X)\to D(U)$ is
$t$-exact with respect to the perverse $t$-structure;

  * (iii)

For any regular scheme $X$ of pure dimension $d$, any dualizable object $F$ of
$D(X)$ that is also in the heart of the usual $t$-strucure defines an object
$F[-\dim(X)]$ in the heart of the perverse $t$-structure of $D_{c}(X)$.

More precisely, we define the perverse $t$-structure on $D(X)$ by induction on
the dimension of $X$ (together with properties (i), (ii) and (iii) above). For
$X$ of dimension $\leq 0$, this is the ordinary $t$-structure (the one from
2.1). For $X$ of dimension $d>0$, assuming that the perverse $t$-structure is
defined on $D(Z)$ for any scheme $Z$ of dimension $<d$, we define the perverse
$t$-structure on $D(X)$ as follows. For any nowhere dense closed subscheme
$i\colon Z\hookrightarrow X$ with open complement $j\colon U\hookrightarrow X$
such that $U_{\mathit{red}}$ is regular, we denote by $D_{U}(X)$ the full
subcategory of $D(X)$ spanned by constructible objects $F$ with the property
that $j^{*}(F)$ is dualizable. If $D^{\wedge}(U)$ denotes the full subcategory
of dualizable objects of $D(U)$, the functors $i_{*}$ and $j^{*}$ expressing
$D(X)$ as a recollement of $D(U)$ and $D(Z)$ in the sense of [BBDG18, 1.4.3]
exhibit $D_{U}(X)$ as a recollement of $D^{\wedge}(U)$ and $D(Z)$. If
$U=\coprod_{i}U_{i}$ is the decomposition of $U$ into its irreducible
components, then

$D^{\wedge}(U)\cong\prod_{i}D^{\wedge}(U_{i})$

and we define the perverse $t$-structure on $D^{\wedge}(U)$ as the product
perverse $t$-structure, where the perverse $t$-structure on each
$D^{\wedge}(U_{i})$ is defined by the property that its heart consists of
those objects of the form $F[-\dim(U_{i})]$ with $F$ in the heart of the
ordinary $t$-structure on $D(U_{i})$. We have a perverse $t$-structure on
$D(Z)$ by induction, and thus a perverse $t$-structure on $D_{U}(X)$, by
[BBDG18, Théorème 1.4.10]. If $Z^{\prime}$ is a nowhere dense open subscheme
of $X$ containing $Z$ and if $U^{\prime}$ is the complement of $Z^{\prime}$,
since the extension by zero functor $D(Z)\to D(Z^{\prime})$ is $t$-exact (by
induction on dimension, property (ii) holds true), and since the restriction
functor $D^{\wedge}(U)\to D^{\wedge}(U^{\prime})$ is obviously $t$-exact, we
see that the inclusion functor

$D_{U}(X)\hookrightarrow D_{U^{\prime}}(X)$

is $t$-exact for the perverse $t$-structures. Writing

$\varinjlim_{U}D_{U}(X)=D_{c}(X)$

where the filtered colimit is indexed by dense open subschemes $U\subset X$
such that $U_{\mathit{red}}$ is regular thus defines a $t$-structure on
$D_{c}(X)$. It is clear that properties (i), (ii) and (iii) are true by
construction.

We denote by ${}^{p}D^{\leq i}_{c}(X)$ (by ${}^{p}D^{\geq i}_{c}(X)$
respectively) the full subcategory of $D_{c}(X)$ that consists of
constructible objects with cohomology concentrated in degrees $\leq i$ (in
degrees $\geq i$, respectively) with respect to the perverse $t$-structure
defined above.

An object of $D(X)$ is called _perverse_ if it is in the heart of the perverse
$t$-structure on $D_{c}(X)$.

######  Lemma 2.4.

For any closed immersion $i\colon Z\to X$, the functor $i^{*}\colon D(X)\to
D(Z)$ is right $t$-exact with respect to the perverse $t$-structure.

###### Proof.

This is obvious since its right adjoint $i_{*}$ is (left) $t$-exact with
respect to the perverse $t$-structure. ∎

######  2.5.

Observe that, by generic base change, the functors $f_{*}$ preserve
constructible objects for any morphism $f$ between $k$-schemes of finite type.
This implies that, for any separated morphism of finite type $f\colon X\to Y$,
the functor $f^{!}$ preserves constructible objects in $D$: since this is a
Zariski-local property, it suffices to check this property for $f$ quasi-
projective. By relative purity, for $f$ smooth, we have
$f^{!}(\mathbf{1})\otimes f^{*}\cong f^{!}$ with the object
$f^{!}(\mathbf{1})$ $\otimes$-invertible (hence dualizable and therefore
constructible), so that it suffices to check that $i^{!}$ preserves
constructible objects for $i\colon Z\to X$ a closed immersion. If $j\colon
U\to X$ is the open complement, the fiber sequence

$i^{!}\to i^{*}\to i^{*}\,j_{*}\,j^{*}$

shows that $i^{!}$ preserves constructible objects. We also see that the
internal Hom preserve constructible objects: For $F$ and $G$ constructible in
$D(X)$, we choose an open immersion $j\colon U\to X$ so that $j^{*}(F)$ is
dualizable. The internal Hom $\mathit{Hom}(F,G)$ then fits into a fiber
sequence

$i_{*}\,i^{!}\mathit{Hom}(F,G)\to\mathit{Hom}(F,G)\to
j_{*}\,j^{*}\mathit{Hom}(F,G)$

that is isomorphic to

$i_{*}\mathit{Hom}(i^{*}(F),i^{!}(G))\to\mathit{Hom}(F,G)\to
j_{*}\mathit{Hom}(j^{*}(F),j^{*}(G))\,.$

We see that $\mathit{Hom}(i^{*}(F),i^{!}(G))$ by induction on the Krull
dimension whereas the object $\mathit{Hom}(j^{*}(F),j^{*}(G))$ is the tensor
product of the dual of the dualizable object $j^{*}(F)$ with the constructible
object $j^{*}(G)$, hence is constructible. We can thus define Verdier duality
in $D$ as the functor

$D_{X}\colon D_{c}(X)^{\mathit{op}}\to D_{c}(X)\ ,\quad
F\mapsto\mathit{Hom}(F,a^{!}(\mathbf{1}))$

where $a\colon X\to\mathrm{Spec}(k)$ is the structural map. Observe that,
since

$f^{!}\mathit{Hom}(A,B)\cong\mathit{Hom}(f^{*}(A),f^{!}(B))$

and

$\mathit{Hom}(f_{!}(A),B)\cong f_{*}\mathit{Hom}(A,f^{!}(B))\,,$

we see that Verdier duality commutes with $f_{*}$ for $f$ proper.

######  Remark 2.6.

On regular schemes, Verdier duality preserves dualizable objects: For $F$
dualizable in $D(X)$ with $X_{\mathit{red}}$ regular, we have
$D_{X}(\mathbf{1}_{X})=\mathbf{1}_{X}(d)[2d]$ with $d=\dim X$ (it suffices to
prove this for $D=DM$ since $r_{D}\colon DM\to D$ preserves the six
operations) so that

$D_{X}(F)\cong F^{\wedge}\otimes\mathbf{1}_{X}(d)[2d]=F^{\wedge}(d)[2d]$

with $F^{\wedge}$ the ordinary dual of $F$. In particular, we have

$D_{X}(F[d]])\cong D_{X}(F)[-d]\cong F^{\wedge}[d](d)\,.$

######  Definition 2.7.

An _Artin-Verdier six-functors formalism_ is a six-functors formalism $D$
defined for schemes of finite type over a field $k$ as in 1.1 equipped with an
ordinary $t$-structure in the sense of 2.1 with the following two extra
properties:

  * (i)

Verdier duality holds in $D$: for any scheme of finite type $X$ over $k$,
Verdier duality induces an equivalence

$D_{X}\colon D_{c}(X)^{\mathit{op}}\cong D_{c}(X)\,.$

  * (ii)

Artin vanishing holds in $D$: for any affine morphism of $k$-schemes of finite
type $f\colon X\to Y$ the induced functor $f_{*}\colon D_{c}(X)\to D_{c}(Y)$
is right $t$-exact with respect to the perverse $t$-structure.

######  Example 2.8.

$D(-,\mathbf{F})$ with $\mathbf{F}$ a finite field, $D(-,\mathbf{Q}_{\ell})$
with $\ell$ any prime number, and $D(-,\mathbf{F}_{\mathcal{U}})$ with
$\mathcal{U}$ any ultrafilter on the set of prime numbers, as defined in 1.3
and 1.4 are Artin-Verdier six-functors formalisms – properties (i) and (ii)
are even known to hold over any quasi-excellent base, after the work of Gabber
[ILO14, Exposé XVII, Théorème 0.2 and Exposé XV, Théorème 1.1.2]. Other
examples are the de Rham six-functors formalism, the $\mathbf{F}$-linear
version of Betti six-functors formalism (for any field of coefficient
$\mathbf{F}$), and the rigid six-functors formalism.

######  2.9.

From now on, we fix an Artin-Verdier six-functors formalism $D$. We consider
the associated perverse $t$-structure on $D$ (see 2.3).

######  Remark 2.10.

We have formulas of the form

$f^{!}\,D_{X}\cong D_{Y}\,f^{*}\quad\text{and}\quad D_{Y}\,f_{!}\cong
f_{*}\,D_{X}$

for any morphism of $k$-schemes of finite type $f\colon X\to Y$. Since Verdier
duality is an equivalence, this implies dually that

$f^{*}\,D_{X}\cong D_{Y}\,f^{!}\quad\text{and}\quad D_{Y}\,f_{*}\cong
f_{!}\,D_{X}$

holds as well. Given an immersion $\imath\colon S\to X$ with
$S_{\mathit{red}}$ regular, and $F$ constructible in $D(X)$ such that
$\imath^{*}(F)$ (with $\imath^{!}(F)$, respectively) is dualizable, we observe
that

$D_{S}\,\imath^{*}(F)\cong\imath^{!}\,D_{S}(F)\quad\text{($D_{S}\,\imath^{!}(F)\cong\imath^{*}\,D_{S}(F)$,
respectively)}$

is dualizable as well: it is the tensor product of the $\otimes$-invertible
object $D_{S}(\mathbf{1}_{S})\cong\mathbf{1}_{S}(\dim S)[2\dim S]$ with the
ordinary dual of $i^{*}(F)$ (of $i^{!}(F)$, respectively); see 2.6.

######  Proposition 2.11.

Verdier duality is $t$-exact with respect to the perverse $t$-structure.

###### Proof.

We proceed by induction on the Krull dimension $d$ of $X$. For $d\leq 0$, all
constructible objects are dualizable and Verdier duality coincides with
ordinary duality, and the bidual of a dualizable object $F$ is canonically
isomorphic to $F$. Let us assume that $d>0$. We choose a nowhere dense closed
subscheme $i\colon Z\to X$ with open complement $j\colon U\to X$ with
$U_{\mathit{red}}$ regular. We consider the category $D_{U}(X)$ of
constructible objects $F$ in $D(X)$ such that $j^{*}(F)$ is dualizable. We
observe that Verdier duality induces a $t$-exact functor with respect to the
perverse $t$-structure

$D_{c}(Z)^{\mathit{op}}\to D_{c}(Z)$

by induction on the Krull dimension, and an equivalence

$D_{U}(U)^{\mathit{op}}\cong D_{U}(U)$

because, up to a Tate twist, Verdier duality on $D_{U}(U)$ simply is ordinary
duality restricted to dualizable objects; see Remark 2.6. In fact, the
perverse $t$-structure on $D_{U}(U)$ is the ordinary $t$-structure shifted by
$\dim U$: the $t$-structure whose heart is

$D^{\leq\dim U}_{U}(U)\cap D^{\geq\dim U}_{U}(U)\,.$

Verdier duality induces an equivalence $D_{U}(U)^{\mathit{op}}\cong D_{U}(U)$
that is $t$-exact with respect to the perverse $t$-structure because of
formula $D_{X}(F[d]])\cong F^{\wedge}[d](d)$ for any $F$ in the ordinary heart
of $D_{U}(U)$. Verdier duality is $t$-exact with respect to the perverse
$t$-structure on $D(Z)$ by induction on $d$. Verdier duality restricts to
$D_{U}(X)$, and since the inclusion $D_{U}(X)\hookrightarrow D(X)$ is
$t$-exact with respect to the perverse $t$-structure by construction, it
suffices to prove that Verdier duality is $t$-exact on $D_{U}(X)$. In other
words, we want to prove that an object $F$ of $D_{U}(X)$ is in ${}^{p}D^{\leq
0}_{U}(X)$ if and only if is Verdier dual $D_{X}(F)$ is in ${}^{p}D^{\geq
0}_{U}(X)$. By definition of glued $t$-structures, an object $F$ is in
${}^{p}D^{\leq 0}_{U}(X)$ if only if $i^{*}(F)$ lies in ${}^{p}D^{\leq 0}(Z)$
and $j^{*}(F)$ lies in ${}^{p}D^{\leq 0}_{U}(U)$. Dually, an object $F$ is in
${}^{p}D^{\leq 0}_{U}(X)$ if only if $i^{!}(F)$ lies in ${}^{p}D^{\geq 0}(Z)$
and $j^{*}(F)$ lies in ${}^{p}D^{\leq 0}_{U}(U)$. Since Verdier duality
exchanges $i^{*}$ and $i^{!}$ and commutes with $j^{*}$, this achieves the
proof. ∎

######  Corollary 2.12.

For any affine morphism of $k$-schemes of finite type $f\colon X\to Y$ the
induced functor $f_{!}\colon D_{c}(X)\to D_{c}(Y)$ is left $t$-exact with
respect to the perverse $t$-structure.

######  Corollary 2.13.

For any finite morphism $f\colon X\to Y$, the functor $f_{!}=f_{*}\colon
D(X)\to D(Y)$ is $t$-exact with respect to the perverse $t$-structure.

######  Lemma 2.14.

Let $X$ be of pure dimension $d$, with $X_{\mathit{red}}$ regular and $F$
dualizable in $D(X)$. Then $F$ is perverse (is in ${}^{p}D^{\leq 0}(X)$) if
and only if $F[-d]$ is in the heart of the ordinary $t$-structure (is in
$D^{\leq 0}(X)$, respectively).

###### Proof.

We will prove the respective case first. Let us assume that $F$ is in
${}^{p}D^{\leq 0}(X)$ but not in ${}^{p}D^{<0}(X)$. There is smallest integer
$n$ such that $F$ lies in $D^{\leq n}_{c}(X)$. Then $F[-n]$ is in $D^{\leq
0}_{c}(X)$ and thus $F[d-n]$ is in ${}^{p}D^{\leq 0}(X)$, which yields $n\leq
d$.

Let us assume now that $F$ is perverse. Since $F$ is dualizable and
$X_{\mathit{red}}$ regular of dimension $d$, we have $D_{X}(F)\cong
F^{\wedge}(d)[2d]$ (Remark 2.6). By virtue of Proposition 2.11, this means
that

$F^{\wedge}[2d]=(F[-d])^{\wedge}[d]$

is dualizable and perverse as well. This means that both $F[-d]$ and
$(F[-d])^{\wedge}$ are dualizable and in $D^{\leq 0}(X)$. Since ordinary
duality of dualizable objects is $t$-exact with respect to the ordinary
$t$-structure, this implies that $F[-d]$ lies in the heart of the ordinary
$t$-structure. ∎

######  Proposition 2.15.

Let $F$ be a constructible object of $D(X)$ and $n\in\mathbf{Z}$ an integer.
The following conditions are equivalent.

  * (i)

$F$ belongs to ${}^{p}D^{\leq 0}_{c}(X)$ (to ${}^{p}D^{\geq 0}_{c}(X)$,
respectively);

  * (ii)

For any irreducible closed subscheme $T\subset X$, there is a dense open
subscheme $S$ in $T$ inducing an embedding $\imath_{S}\colon S\to X$ such that
$\imath^{*}_{S}(F)$ belongs to $D^{\leq\dim S}_{c}(S)$ (such that
$\imath^{!}_{S}(F)$ belongs to $D^{\geq\dim S}_{c}(S)$, respectively).

  * (iii)

For any finite stratification $\mathcal{S}$ of $X$ by strata that are
irreducible with regular reduction, and for any embedding $\imath_{S}\colon
S\to X$ with $S\in\mathcal{S}$, assuming that both $i^{*}_{S}(F)$ and
$i^{!}_{S}(F)$ are dualizable, we have $\imath^{*}_{S}(F)$ in $D^{\leq\dim
S}_{c}(S)$ (we have $\imath^{!}_{S}(F)$ in $D^{\geq\dim S}_{c}(S)$,
respectively).

  * (iv)

There exists a finite stratification $\mathcal{S}$ of $X$ by strata that are
irreducible with regular reduction, such that, for any embedding
$\imath_{S}\colon S\to X$ with $S\in\mathcal{S}$, both $i^{*}_{S}(F)$ and
$i^{!}_{S}(F)$ are dualizable and we have $\imath^{*}_{S}(F)$ in $D^{\leq\dim
S}_{c}(S)$ (we have $\imath^{!}_{S}(F)$ in $D^{\geq\dim S}_{c}(S)$,
respectively).

###### Proof.

Let $Y$ be a scheme of finite type over $k$. For any constructible object $K$
in $D(Y)$, there is a dense open subscheme $U$ of $Y$ such that the
restriction of $K$ on $U$ is dualizable in $D(U)$. Since any scheme of finite
over scheme is excellent, there is a dense subscheme $U$ of $Y$ that is a
finite disjoint uinion of irreducible schemes whose reductions are regular.
Therefore, there exists at least one finite stratification $\mathcal{S}$ of
$X$ by strata that are irreducible with regular reduction, such that, for any
embedding $\imath_{S}\colon S\to X$ with $S\in\mathcal{S}$, both
$i^{*}_{S}(F)$ and $i^{!}(S)$ are dualizable. The fact that we have
equivalences (i)–(iv) in the case of ${}^{p}D^{\leq 0}_{c}(X)$ follows right
away from Lemmata 2.4 and 2.14. Since Verdier duality exchanges ${}^{p}D^{\leq
0}_{c}(X)$ and ${}^{p}D^{\geq 0}_{c}(X)$ (because it is $t$-exact), the
respective case follows now from the last part of Remark 2.10. ∎

######  Corollary 2.16.

We have the following exactness properties with respect to the perverse
$t$-structure.

  * (a)

For any quasi-finite morphism $f\colon X\to Y$, the functor $f^{*}\colon
D_{c}(Y)\to D_{c}(X)$ is right $t$-exact and the exceptional pullback functor
$f^{!}\colon D_{c}(Y)\to D_{c}(X)$ is left $t$-exact.

  * (b)

For any quasi-finite morphism $f\colon X\to Y$, the functor $f_{!}\colon
D_{c}(Y)\to D_{c}(X)$ is right $t$-exact and $f_{*}\colon D_{c}(Y)\to
D_{c}(X)$ is left $t$-exact.

###### Proof.

To prove (a), we use criterium (ii) of Proposition 2.15. Let $T$ be a closed
subscheme of $X$. There is a dense open subscheme $U$ of the closure of the
image of $T$ in $Y$ such that $\imath_{U}^{*}(F)$ belongs to $D^{\leq\dim
U}_{c}(U)$. Therefore, if $S$ denotes the pullback of $U$ in $T$, we have
$\imath_{U}^{*}\,f^{*}(F)$ in $D^{\leq\dim U}_{c}(S)$. Since $\dim S=\dim U$
(because $f$ is quasi-finite and $S$ dominates $U$ by construction), this
proves that $f^{*}(F)$ is in ${}^{p}D^{\leq 0}(X)$. By Verdier duality, this
proves that $f^{!}$ is left $t$-exact. Property (b) follows straight away from
property (a). ∎

######  Corollary 2.17.

If $f\colon X\to Y$ is étale, then $f^{*}$ is $t$-exact with respect to the
perverse $t$-structure.

######  Corollary 2.18.

If $f\colon X\to Y$ is both affine and quasi-finite, then both functors
$f_{!}$ and $f_{*}$ are $t$-exact with respect to the perverse $t$-structure.

######  Proposition 2.19.

If $f\colon X\to Y$ is a finite flat morphism with both $X_{\mathit{red}}$ and
$Y_{\mathit{red}}$ regular, then, for any $F$ that is both perverse and
dualizable in $D(Y)$, there is an isomorphism $f^{!}(F)\cong f^{*}(F)$ and
$f^{*}(F)$ is both perverse and dualizable in $D(X)$.

###### Proof.

There is always a canonical morphism

$f^{!}(\mathbf{1}_{Y})\otimes f^{*}(F)\to f^{!}(F)\,.$

The latter is an isomorphism for $F$ dualizable (this is a nice exercise that
does not use any property of $f$) and
$f^{!}(\mathbf{1}_{Y})\cong\mathbf{1}_{X}$ by relative purity (this times
using that both $X$ and $Y$ are regular up to reduction and of same
dimension). If $F$ is dualizable in $D(Y)$, since $Y_{\mathit{red}}$ is
regular, it is perverse if and only if $F[-\dim Y]$ is in the heart of the
ordinary $t$-structure (Lemma 2.14). The functor $f^{*}$ preserves dualizable
because it is symmetric monoidal. Since $f^{*}$ is $t$-exact for the ordinary
$t$-structure, this proves the proposition. ∎

######  Proposition 2.20.

Let $a\colon X\to Y$ be a flat affine morphism between $k$-schemes of finite
type, of relative dimension $d$.

  1. (a)

For any $F$ in $D^{\leq 0}_{c}(X)$, there is a dense open subscheme $V$ of $Y$
such that the restriction $a_{*}(F)_{|_{V}}$ belongs to $D^{\leq d}_{c}(V)$.

  2. (b)

For any $F$ in $D^{\geq 0}_{c}(X)$, such that $F[\dim X]$ belongs to
${}^{p}D^{\geq 0}_{c}(X)$, there is a dense open subscheme $V$ of $Y$ such
that the restriction $a_{!}(F)_{|_{V}}$ belongs to $D^{\geq d}_{c}(V)$.

Moreover, in both cases, if $F$ is strictly of geometric origin, then the open
subscheme $V$ may be chosen independently of $D$.

###### Proof.

For (a), We proceed by induction on $d$. Since we are allowed to shrink $Y$ at
will, we may assume $Y_{\mathit{red}}$ to be regular. Since $D$ is insensitive
to nil-immersions, we may as well assume that $Y$ is regular. For $d\leq 0$,
this is obvious: since $X$ is affine and quasi-finite over $Y$, we know that
$a_{*}$ is $t$-exact by Corollary 2.18. For $d>0$, given $F$ constructible in
$D^{\leq 0}(X)$, we choose $i\colon Z\to X$ a nowhere dense closed immersion
with affine complement $j\colon U\to X$ such that $U_{\mathit{red}}$ is
regular and everywhere of dimension $\dim X$, and $j^{*}F$ is dualizable in
$D(U)$ (if $F$ is strictly of geometric origin we make this choice in
$\mathit{DM}$ to make this independent of $D$, by Proposition 1.9). We then
have $j^{*}F[\dim U]$ in ${}^{p}D^{\leq 0}(U)$ hence $j_{!}\,j^{*}F[\dim U]$
is in ${}^{p}D^{\leq 0}(X)$. Therefore, Artin vanishing tells us that
$a_{*}\,j_{!}\,j^{*}F[\dim U]$ belongs to ${}^{p}D^{\leq 0}(Y)$. Shrinking
$Y$, we may impose that $a_{*}\,j_{!}\,j^{*}F$ is dualizable (if $F$ is
strictly of geometric origin, so is $a_{*}\,j_{!}\,j^{*}F$ and we make this
shrinking at the motivic level, as in the proof of 1.9). Since $d=\dim U-\dim
Y$, by virtue of Lemma 2.14, this means that $a_{*}\,j_{!}\,j^{*}F[d]$ belongs
to $D^{\leq 0}(Y)$, hence $a_{*}\,j_{!}\,j^{*}F$ to $D^{\leq d}(Y)$. By virtue
of Grothendieck’s generic flatness theorem, shrinking $Y$, we may assume that
$Z$ is flat over $Y$ as well (this choice is purely geometric and does not
depend on $F$ nor on $D$). By induction, since the relative dimension of $Z$
over $Y$ is $<d$, possibly after shrinking $Y$ one more time, we have
$(ai)_{*}\,i^{*}(F)$ in $D^{\leq d-1}(Y)$. The cofiber sequence

$a_{*}\,j_{!}\,j^{*}F\to a_{*}F\to a_{*}\,i_{*}\,i^{*}F$

thus implies that $a_{*}(F)$ belongs to $D^{\leq d}(Y)$.

In order to prove (b), this is more direct. Let $F$ be in $D^{\geq 0}_{c}(X)$,
such that $F[\dim X]$ belongs to ${}^{p}D^{\geq 0}_{c}(X)$. Since $a_{!}$ is
left $t$-exact with respect to the perverse $t$-structure (Cor. 2.12), we have
$a_{!}(F)[\dim X]$ in ${}^{p}D^{\geq 0}_{c}(Y)$. Shrinking $Y$, we may assume
that $a_{!}(F)$ is dualizable and that $Y_{\mathit{red}}$ is regular, which
implies by Lemma 2.14 that

$a_{!}(F)[\dim X-\dim Y]=a_{!}(F)[d]$

belongs to $D^{\geq 0}_{c}(Y)$. ∎

######  Remark 2.21.

Let $a\colon X\to\mathrm{Spec}(k)$ be a separated morphism of finite type. For
any $F$ in $D(X)$, we can define $\Gamma(X,F)=a_{*}(F)$ and
$\Gamma_{c}(X,F)=a_{!}(F)$. Taking cohomology in the sense of the ordinary
$t$-structure, we get objects

$H^{i}(X,F)=H^{i}(\Gamma(X,f))\quad\text{and}\quad
H^{i}_{c}(X,F)=H^{i}(\Gamma_{c}(X,F))$

in the heart $D^{\heartsuit}(\mathrm{Spec}(k))=D^{\leq
0}(\mathrm{Spec}(k))\cap D^{\geq 0}(\mathrm{Spec}(k))$.

The following statement was observed by Beilinson [Bei87]. Recall the notion
of hyperplane section transverse to an object of $D$, as defined in 1.18.

######  Theorem 2.22.

Let $X$ be an affine scheme of finite type over $k$ and $e\colon X\to\bar{X}$
an affine open immersion into a projective scheme. We consider given an affine
open immersion $u\colon U\to X$ with $U_{\mathit{red}}$ regular. We assume
that $F$ is dualizable and in the heart of the ordinary $t$-structure on
$D_{c}(U)$. If $j\colon X-H\hookrightarrow X$ denotes the embedding of the
complement of a hyperplane section that is transverse to $u_{!}F$, then
$H^{q}(X,j_{!}\,j^{*}u_{!}F)=0$ for any $q\neq\dim X$.

###### Proof.

Let $d=\dim X$. Keeping track of the notations of 1.18, the transversality
assumption on $H$ implies that we have canonical isomorphisms

$H^{q}_{c}(\bar{X}-\bar{H},e_{*}\,u_{!}F)\cong H^{q}(X,j_{!}\,j^{*}\,u_{!}F)$

with both $X$ and $\bar{X}-\bar{H}$ affine of dimension $d$. By virtue of
Proposition 2.20 (a), since all functors $j_{!}$, $j^{*}$ and $u_{!}$ are
$t$-exact with respect to the ordinary $t$-structure,
$H^{q}(X,j_{!}\,j^{*}\,u_{!}F)$ vanishes for $q>d$. On the other hand, by
virtue of Lemma 2.14, we see that $F[d]$ is perverse on $U$, and since both
$u$ and $e$ are affine immersions, Corollaries 2.17 and 2.18 imply that
$e_{*}\,u_{!}F[d]$ is perverse on $\bar{X}-\bar{H}$. Therefore, Proposition
2.20 (b) implies that $H^{q}_{c}(\bar{X}-\bar{H},e_{*}u_{!}F)$ vanishes for
$q<d$. ∎

##  3\. Uniformly perverse motivic generators

######  3.1.

Let $X$ be a scheme of finite type over a field $k$. We denote by $M(X)$ the
motive of $X$, i.e. the object of $\mathit{DM}(\mathrm{Spec}(k))$ defined by

$M(X)=a_{!}\,a^{!}(\mathbf{Z})$

with $a\colon X\to\mathrm{Spec}(k)$ the structural morphism.555If we define
$\mathit{DM}(X)$ through $h$-motives, its is a
$(\mathbf{P}^{1},\infty)$-stabilization of the $\mathbf{A}^{1}$-localization
of the derived $\infty$-category of sheaves of abelian groups on the big site
of $k$ with respect to the $h$-topology. The object $M(X)$ is then what
remains of the sheaf represented by $X$ through all this process. This
construction defines a functor from the category of schemes of finite type
over $k$ to the stable $\infty$-category $\mathit{DM}(\mathrm{Spec}(k))$.
Given a closed subscheme $Z$ of $X$, we define $M(X,Z)$ through the cofiber
sequence

$M(Z)\to M(X)\to M(X,Z)$

For any object $A$ in $\mathit{DM}(\mathrm{Spec}(k))$, we thus have a fiber
sequence in the derived category of abelian groups

$\mathrm{RHom}(M(X,Z),A)\to\mathrm{RHom}(M(X),A)\to\mathrm{RHom}(M(Z),A)$

with $\mathrm{RHom}$ the cochain complex of maps in
$\mathit{DM}(\mathrm{Spec}(k))$ (the latter is $\mathbf{Z}$-linear by
construction). Since $(a_{!}a^{!}(\mathbf{Z}))^{\wedge}\cong
a_{*}a^{*}(\mathbf{Z})$ by Verdier duality, we observe that

$\mathrm{RHom}(M(X),A)\cong\mathrm{RHom}(\mathbf{Z},a_{*}a^{*}(\mathbf{Z}))\cong\mathrm{RHom}(\mathbf{Z}_{X},A_{X})$

where third instance of $\mathrm{RHom}$ is the cochain complex of maps in
$\mathit{DM}(X)$, $\mathbf{Z}_{X}=\mathbf{1}_{X}$ is the $\otimes$-unit of
$\mathit{DM}(X)$, and $A_{X}=a^{*}(A)$ with $a$ the structural morphism of $X$
as above. Therefore, if $j\colon X-Z\hookrightarrow X$ is the open immersion
of the complement of $Z$ in $X$, and if $i\colon Z\hookrightarrow X$ is the
embedding of $Z$ into $X$, the cofiber sequence

$j_{!}\,j^{*}(A_{X})\to A_{X}\to i_{*}\,A_{Z}$

thus shows that we have a canonical isomorphism

$\mathrm{RHom}(M(X,Z),A)\cong\mathrm{RHom}(\mathbf{Z}_{X},j_{!}\,j^{*}A_{X})\,.$

Let $h:H\hookrightarrow X$ be another closed embedding. We have a cocartesian
square

${M(Z\cap H)}$${M(Z)}$${M(H)}$${M(H\cup Z)}$

hence cartesian squares of the form below.

${\mathrm{RHom}(M(X,Z\cup
H),A)}$${\mathrm{RHom}(M(X,Z),A)}$${\mathrm{RHom}(M(X,H),A)}$${\mathrm{RHom}(M(X,H\cap
Z),A)}$

Those correspond to cartesian squares

${\mathrm{RHom}(\mathbf{Z}_{X},u_{!}\,u^{*}\,j_{!}\,j^{*}A_{X})}$${\mathrm{RHom}(\mathbf{Z}_{X},j_{!}\,j^{*}A_{X})}$${\mathrm{RHom}(\mathbf{Z}_{X},u_{!}\,u^{*}A_{X})}$${\mathrm{RHom}(\mathbf{Z}_{X},v_{!}\,v^{*}A_{X})}$

where $u\colon X-H\hookrightarrow X$ is the complement of $H$ and $v\colon
X-(H\cap Z)\hookrightarrow X$ is the complement of $H\cap Z$ (observe that
$u_{!}\,u^{*}A_{X}\cong u_{!}\,u^{*}\,v_{!}\,v^{*}A_{X})$ and
$j_{!}\,j^{*}A_{X}\cong j_{!}\,j^{*}\,v_{!}\,v^{*}A_{X})$).

######  Definition 3.2.

Let $X$ be an affine scheme of finite type over a field $k$ of dimension $d$.
A _Lefschetz subscheme of $X$_ is a closed subscheme $Z$ of $X$ of the form
$Z=W\cup H$, where $w\colon W\hookrightarrow X$ is a closed subscheme of
dimension $<d$, with affine complement, that contains all irreducible
components of $X$ of dimension $<d$, such that $X-W$ is regular, and $H$ is an
hyperplane section that is transverse to the motivic sheaf
$j_{!}\,j^{*}\mathbf{Z}_{X}$ in the sense of 1.18, where $j\colon X-W\to X$
denotes the complement of $W$ in $X$. The _rank of the pair $(X,Z)$_ is the
trace of the identity of $M(X,Z)[-d]$ in $\mathit{DM}(\mathrm{Spec}(k))$; we
will denote it

$\mathit{rk}(X,Z)=(-1)^{d}\mathit{Tr}\big{(}1_{M(X,Z)}\big{)}\in\mathbf{Z}$

(this is always an integer because it coincides with the Euler characteristic
of its $\ell$-adic realizations [Cis21, Cor. 2.2.7]). A _Lefschetz pair_ is a
pair $(X,Z)$ that consists of an affine scheme $X$ and of a Lefschetz
subscheme $Z\subset X$.

A _Lefchetz filtration_ of $X$ is a finite sequence of closed subschemes of
the form

$\varnothing=X_{-1}\subset X_{0}\subset\ldots\subset X_{d-1}\subset X_{d}=X$

where, for $0\leq i\leq d$, each $X_{i-1}$ is a Lefschetz subscheme of
$X_{i}$.

Lefschetz filtrations exist in practice thanks to the following proposition.

######  Proposition 3.3.

Let $k$ be an infinite field. Then, for any affine $k$-scheme $X$ and any
closed subscheme $Y$ in $X$ of dimension $<\dim X$, there is a Lefschetz
subscheme $Z$ of $X$ containing $Y$. If $k$ is a finite field, this is true
only after pulling back $X$ along a finite extension of $k$.

###### Proof.

Define $W$ to be the union of $Y$ with the singular locus of $X$ as well as
with the irreducible components of $X$ of dimension $<d$. We may enlarge $W$
so that its complement if affine, and then apply Corollary 1.17 to
$F=j_{!}\,\mathbf{Z}_{X-W}$, where $j\colon X-W\to X$ denotes the complement
of $W$ in $X$, in order to find a section $H$ that is transverse to $F$, and
put $Z=W\cup H$. The case of a finite field is proved similarly, applying
Theorem 1.16 instead of its corollary: the dense open space $U$ of hyperplane
sections will only have rational points after a finite extension of $k$. ∎

######  Lemma 3.4.

Let $X$ be an affine scheme of dimension $d\geq 0$, $Z$ a Lefschetz subscheme
of $X$, and $j\colon X-Z\hookrightarrow X$ the corresponding open immersion.
Then $\mathit{rk}(X,Z)$ is a non-negative integer. Furthermore, for any Artin-
Verdier six-functors formalism $D$ defined on $k$-schemes of finite type, the
induced realization functor

$r_{D}\colon\mathit{DM}(\mathrm{Spec}(k))\to D(\mathrm{Spec}(k))$

yields

$H^{q}(X,j_{!}\mathbf{1}_{X-Z})\cong\begin{cases}0&\text{if $q\neq d$,}\\\
r_{D}(M(X,Z)[-d])^{\wedge}&\text{if $q=d$.}\end{cases}$

(where the cohomology is defined using the ordinary $t$-structure as in Remark
2.21). The rank of $H^{d}(X,j_{!}\mathbf{1}_{X-Z})$ (defined as the trace of
its identity) is equal to the class of $\mathit{rk}(X,Z)$ in
$\pi_{0}\operatorname{Map}_{D(\mathrm{Spec}(k))}(\mathbf{1},\mathbf{1})$.

In particular, if $k$ is separably closed, for any prime $\ell$ distinct from
the characteristic of $k$, we can compute the $\ell$-adic cohomology of the
pair $(X,Z)$ motivically:

$H^{q}_{\mathrm{\acute{e}t}}(X,j_{!}\mathbf{Z}_{\ell})\cong\begin{cases}0&\text{if
$q\neq d$,}\\\ \mathbf{Z}_{\ell}^{\mathit{rk}(X,Z)}&\text{if
$q=d$.}\end{cases}$

###### Proof.

Let $D$ be an Artin-Verdier six-functors formalism defined on $k$-schemes of
finite type. We then have colimit preserving functors

$r_{D}\colon\mathit{DM}(X)\to D(X)$

that commute with the six operations. With the notations of Remark 2.21, if
$M(X,Z)^{\wedge}\cong a_{*}j_{!}j^{*}\mathbf{Z}_{X}$ denotes the dual of
$M(X,Z)$ in $\mathit{DM}(\mathrm{Spec}(k))$, we thus have

$r_{D}(M(X,Z)^{\wedge})\cong\Gamma(X,j_{!}\,j^{*}\mathbf{1}_{X})\,.$

Therefore,

$r_{D}(M(X,Z)[-d])\cong\Gamma(X,j_{!}\,j^{*}\mathbf{1}_{X}[d])^{\wedge}$

is concentrated in degree $0$ by virtue of Theorem 2.22 applied to
$M=\mathbf{Z}_{W}$ and $U=W$ (with the notations of Definition 3.2). Since
$r_{D}$ is symmetric monoidal, it preserves the formation of traces of
endomorphisms of dualizable objects. For any $D$ such that the constructible
heart $D^{\heartsuit}_{c}(\mathrm{Spec}(k))$ is tannakian over $\mathbf{Q}$,
we get that $\mathit{rk}(X,Z)$ is the Euler characteristic of the image of
$a_{*}j_{!}j^{*}\mathbf{Z}_{X}[d]$ through $r_{D}$, and thus a non-negative
integer. For instance, if $\ell$ is a prime not equal to the characteristic of
$k$, and if $\bar{k}$ is a separable closure of $k$, we can associate to each
$k$-scheme $X$ the $\bar{k}$-scheme $\bar{X}$ obtained by pulling back $X$ to
$\mathrm{Spec}(\bar{k})$. The assignment

$X\mapsto D(\bar{X},\mathbf{Q}_{\ell})$

then defines an Artin-Verdier six functors formalism $D$ such that
$D^{\heartsuit}_{c}(\mathrm{Spec}(k))$ is the category of finite dimensional
$\mathbf{Q}_{\ell}$-vector spaces. This proves that $\mathit{rk}(X,Z)$ is a
non-negative integer.

Let us assume that $k=\bar{k}$ is separably closed (in order to simplify the
notations). We can analyze the situation a little bit further as follows. The
morphisms of rings
$\mathbf{Q}_{\ell}\leftarrow\mathbf{Z}_{\ell}\to\mathbf{F}_{\ell}$ induce
isomorphisms on their Grothendieck rings

$\mathbf{Z}=K_{0}(\mathbf{Q}_{\ell})\cong K_{0}(\mathbf{Z}_{\ell})\cong
K_{0}(\mathbf{F}_{\ell})=\mathbf{Z}\,.$

Therefore, for any perfect complex of $\mathbf{Z}_{\ell}$-modules $C$, we
always have:

$\sum_{q}(-1)^{q}\dim_{\mathbf{Q}_{\ell}}H^{q}(C)\otimes\mathbf{Q}=\sum_{q}(-1)^{q}\dim_{\mathbf{F}_{\ell}}H^{q}(C\otimes^{L}_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell})\,.$

If ever both $C\otimes\mathbf{Q}$ and
$C\otimes^{L}_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell}$ are concentrated in
cohomological degree $d$, this means that the dimensions of
$H^{d}(C\otimes^{L}_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell})$ and of
$H^{d}(C)\otimes\mathbf{Q}$ are the same. A direct consequence can be seen in
the exact sequence

$0\to H^{q}(C)\otimes_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell}\to
H^{q}(C\otimes^{L}_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell})\to\mathrm{Tor}_{1}(H^{q+1}(C),\mathbf{F}_{\ell})\to
0\,.$

This yields an isomorphism (an injective linear map from a space that has at
least the dimension of the codomain)

$H^{q}(C)\otimes_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell}\cong
H^{q}(C\otimes^{L}_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell})$

for all $q$ (for $q\neq d$ this is even easier) and tells us that $H^{q+1}(C)$
is free for all $q$. Finally, we must have $H^{q}(C)\cong 0$ for any $q\neq d$
and $H^{d}(C)\cong\mathbf{Z}^{r}_{\ell}$ with $r$ the dimension of
$H^{d}(C)\otimes\mathbf{Q}$ as a $\mathbf{Q}_{\ell}$-vector space.

On constructible objects, the $\ell$-adic realization functor

$r_{\ell}\colon\mathit{DM}_{c}(Y)\to D^{b}_{c}(Y,\mathbf{Z}_{\ell})$

induces the realization functors

$\mathit{DM}_{c}(Y)\to
D^{b}_{c}(Y,\mathbf{F}_{\ell})\quad\text{and}\quad\mathit{DM}_{c}(Y)\to
D^{b}_{c}(Y,\mathbf{Z}_{\ell})\otimes\mathbf{Q}=D^{b}_{c}(Y,\mathbf{Q}_{\ell})$

by the formulæ

$M\mapsto
r_{\ell}(M)\otimes^{L}_{\mathbf{Z}_{\ell}}\mathbf{F}_{\ell}\quad\text{and}\quad
M\mapsto r_{\ell}(M)\otimes\mathbf{Q}$

respectively. Therefore, by virtue of Theorem 2.22 for
$D^{b}_{c}(-,\mathbf{Q}_{\ell})$ and $D^{b}_{c}(-,\mathbf{F}_{\ell})$, we see
that we can apply the discussion above with $C$ be the $\ell$-adic realization
of the motive $a_{*}j_{!}j^{*}\mathbf{Z}_{X}[d]$ in the stable
$\infty$-category
$D^{b}_{c}(\mathrm{Spec}(k),\mathbf{Z}_{\ell})\cong\mathit{Perf}(\mathbf{Z}_{\ell})$.
This proves the last part of the lemma. ∎

######  Remark 3.5.

Let $(X,Z)$ be a Lefschetz pair, with $X$ of dimension $d$ over a separably
closed field. Let $j\colon X-Z\to X$ be the corresponding embedding. Given any
prime number $\ell$ distinct from the characteristic of the ground field, we
have a long exact sequence

$\cdots\to H^{q}_{\mathrm{\acute{e}t}}(X,j_{!}\mathbf{Z}_{\ell})\to
H^{q}_{\mathrm{\acute{e}t}}(X,\mathbf{Z}_{\ell})\to
H^{q}_{\mathrm{\acute{e}t}}(Z,\mathbf{Z}_{\ell})\to
H^{q+1}_{\mathrm{\acute{e}t}}(X,j_{!}\mathbf{Z}_{\ell})\to\cdots$

By virtue of the preceding lemma, this yields isomorphisms

$H^{q}_{\mathrm{\acute{e}t}}(X,\mathbf{Z}_{\ell})\cong
H^{q}_{\mathrm{\acute{e}t}}(Z,\mathbf{Z}_{\ell})\quad\text{for $q\neq
d,d-1$,}$

and an exact sequence of the form below.

$0\to H^{d-1}_{\mathrm{\acute{e}t}}(X,\mathbf{Z}_{\ell})\to
H^{d-1}_{\mathrm{\acute{e}t}}(Z,\mathbf{Z}_{\ell})\to\mathbf{Z}_{\ell}^{\mathit{rk}(X,Z)}\to
H^{d}_{\mathrm{\acute{e}t}}(X,\mathbf{Z}_{\ell})\to 0$

The map $H^{d-1}_{\mathrm{\acute{e}t}}(X,\mathbf{Z}_{\ell})\to
H^{d-1}_{\mathrm{\acute{e}t}}(Z,\mathbf{Z}_{\ell})$ is thus a split
monomorphism inducing an isomorphism on torsion subgroups
$H^{d-1}_{\mathrm{\acute{e}t}}(X,\mathbf{Z}_{\ell})_{\mathit{tors}}\cong
H^{d-1}_{\mathrm{\acute{e}t}}(Z,\mathbf{Z}_{\ell})_{\mathit{tors}}$.

######  Theorem 3.6.

Let $X$ be an affine scheme of Krull dimension $d$, of finite type over a
separably closed field $k$, and equipped with a Lefschetz filtration

$\varnothing=X_{-1}\subset X_{0}\subset\ldots\subset X_{d-1}\subset
X_{d}=X\,.$

For any prime $\ell$ distinct from the characteristic of $k$, and $0\leq q\leq
d$, we have:

$\mathit{rk}\,H^{q}(X,\mathbf{Z}_{\ell})\leq\mathit{rk}\,H^{q}(X_{q},\mathbf{Z}_{\ell})\leq\mathit{rk}(X_{q},X_{q-1})\,.$

###### Proof.

This follows right away from Lemma 3.4 and from Remark 3.5 by induction on
$d$. ∎

######  Remark 3.7.

Together with Proposition 3.3, this gives uniform bounds of Betti numbers for
all quasi-projective schemes over a field, using Jouanolou’s trick [Jou73,
Lemme 1.5]: the latter proves that any quasi-projective scheme is
$\mathbf{A}^{1}$-homotopy equivalent to an affine one.

Lefschetz pairs may be used to prove the existence of uniform bounds of Betti
numbers of any constructible motive; see Corollary 3.10 below.

######  3.8.

We can interpret the existence of Lefschetz filtrations as an approximation of
the motivic $t$-structure on $\mathit{DM}(X)$ (whose existence would imply
Grothendieck’s standard conjectures [Bei10]). In order to simplify our task,
we will work with $\mathbf{Q}$-linear coefficients (it is straightforward to
produce an integral version from any $\mathbf{Q}$-linear one using the
rigidity theorem).

We observe that objects of the form $M(X)(n)$, with $X$ affine of finite type
over the base field $k$ and $n\in\mathbf{Z}$ are compact and generate
$\mathit{DM}(\mathrm{Spec}(k))$ as a compactly generated stable
$\infty$-category. If $k$ is infinite, this implies that the objects of the
form $M(X,Z)(n)$ with $(X,Z)$ a Lefschetz pair and $n\in\mathbf{Z}$ also form
a generating family. In order to allow any field, we proceed as follows.

######  Proposition 3.9.

Let $k$ be any field. We define the family $\mathcal{G}_{k}$ as follows: this
is the family of motives of the form $f_{*}M(X,Z)[-\dim X](n)$, where
$f\colon\mathrm{Spec}(K)\to\mathrm{Spec}(k)$ is a Galois extension of the base
field, with $(X,Z)$ a Lefschetz pair defined over $K$, and $n\in\mathbf{Z}$.
Then $\mathcal{G}_{k}$ is a family of compact generators of
$\mathit{DM}(k,\mathbf{Q})$ as a cocomplete stable $\infty$-category.

###### Proof.

The case where $k$ is infinite is obvious from Proposition 3.3: any affine
scheme has a Lefschetz filtration and given a Lefschetz filtration by
subschemes $X_{i}$, one can reconstruct the motive of $X$ from the
$M(X_{i},X_{i-1})$’s. Since the motives of the form $M(X)(n)$ with $X$ affine
form a generating family, this proves the claim if $k$ is infinite, in
particular for $k$ separably closed. In general, we know that, if $\bar{k}$ is
a separable closure of $k$, then the pullback functor

$\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})\to\mathit{DM}(\mathrm{Spec}(\bar{k}),\mathbf{Q})$

is conservative and preserve the formation of internal Hom’s of the form
$\mathit{Hom}(A,B)$ as long as $A$ is compact; see [CD19, Lemma 4.3.7]. By a
continuity argument (as in [CD19, Prop. 4.3.1]), it is now easy to see that it
suffices to prove that $\mathcal{G}_{k}$ generates
$\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$ if $\mathcal{G}_{\bar{k}}$
generates $\mathit{DM}(\mathrm{Spec}(\bar{k}),\mathbf{Q})$, which we already
know. ∎

######  Corollary 3.10.

Let $M$ be any constructible object in
$\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$. There exists a constant $C>0$ and
integer $a\leq b$ with the following properties. For any $\mathbf{Q}$-linear
Artin-Verdier six-functors formalism $D$ defined over $k$-schemes of finite
type with tannakian heart $D^{\heartsuit}_{c}(\mathrm{Spec}(k))$, the induced
functor

$r_{D}\colon\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})\to D(\mathrm{Spec}(k))$

sends $M$ to an object $r_{D}(M)$ of cohomological amplitude $[a,b]$ with
respect to the ordinary $t$-structure and so that the rank of each cohomology
objects $H^{q}(r_{D}(M))$ (i.e. the traces of their identity) is bounded by
$C$.

###### Proof.

The preceding proposition means that such an $M$ is obtained by finitely many
operations (such as extracting a direct factor in a finite coproducts, taking
cofibers, shifting by some integer) from finitely many objects of the form
$f_{*}M(X,Z)[-\dim X](n)$, where $f\colon\mathrm{Spec}(K)\to\mathrm{Spec}(k)$
is a Galois extension of the base field, with $(X,Z)$ a Lefschetz pair defined
over $K$, and $n\in\mathbf{Z}$. Therefore, it suffices to prove the result for
$M=f_{*}M(X,Z)[-\dim X](n)$. Since $f_{*}$ is an exact functor for $f$ finite,
we conclude with Lemma 3.4. ∎

######  3.11.

We call _standard generators_ objects of the form

$A^{\wedge}_{1}\otimes\cdots\otimes A^{\wedge}_{m}\otimes
B_{1}\otimes\cdots\otimes B_{n}$

where each $A_{i}$ and $B_{j}$ are in $\mathcal{G}_{k}$ and $(-)^{\wedge}$ is
the duality operator on dualizable objects in
$\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$. We define $\mathit{DM}^{\leq
0}(\mathrm{Spec}(k),\mathbf{Q})$ as the smallest full subcategory of
$\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$ stable under small colimits and
stable under extensions spanned by standard generators. This is a compactly
generated presentable $\infty$-category. Therefore, the inclusion functor

$\mathit{DM}^{\leq
0}(\mathrm{Spec}(k),\mathbf{Q})\hookrightarrow\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$

has a right adjoint

$\tau^{\leq
0}\colon\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})\to\mathit{DM}^{\leq
0}(\mathrm{Spec}(k),\mathbf{Q})$

We define $\mathit{DM}^{\geq 0}(\mathrm{Spec}(k),\mathbf{Q})$ as the full
subcategory of objects $B$ such that

$\operatorname{Map}(A[1],B)=0$

for any $A$ in $\mathit{DM}^{\leq 0}(\mathrm{Spec}(k),\mathbf{Q})$. This
defines a $t$-structure on $\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$. By
construction (and thanks to Lemma 3.4), for any $\mathbf{Q}$-linear Artin-
Verdier six-functors formalism $D$ defined over $k$-schemes of finite type,
the induced functor

$r_{D}\colon\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})\to D(\mathrm{Spec}(k))$

is right $t$-exact and induces a symmetric monoidal colimit preserving functor

$r_{D}\colon\mathit{DM}^{\leq 0}(\mathrm{Spec}(k),\mathbf{Q})\to D^{\leq
0}(\mathrm{Spec}(k))\,.$

This $t$-structure on $\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$ is an
approximation of the conjectural motivic $t$-structure - if we are optimistic,
we could hope it is the right one. We would like to do something similar in
$\mathit{DM}(Y,\mathbf{Q})$ for any $Y$ of finite type over a field:
approximating the perverse $t$-structure. For this, we need to define a
relative notion of Lefschetz pairs.

######  3.12.

Let $Y$ be a regular scheme of finite type over $k$. An affine $Y$-scheme of
finite type $X$ is _admissible_ if the structural map $a\colon X\to Y$ is flat
and if the motivic sheaf $a_{*}(\mathbf{Q}_{X})$ is dualizable in
$\mathit{DM}(Y,\mathbf{Q})$ (up to replacing $Y$ by a dense open subscheme,
any flat scheme over $Y$ is admissible; by Grothendieck’s generic flatness
theorem, this means that, up to replacing $Y$ by a dense open subscheme, any
$Y$-scheme of finite type can be made admissible).

Given an admissible $Y$-scheme $X$ with structural morphism $a\colon X\to Y$,
we define

$M_{Y}(X)=a_{*}(\mathbf{Z}_{X})^{\wedge}\,.$

Given a closed subscheme $Z$ of $X$ that is admissble over $Y$ as well, we
define $M_{Y}(X,Z)$ by a cofiber sequence

$M_{Y}(Z)\to M_{Y}(X)\to M_{Y}(X,Z)$

######  Definition 3.13.

Let $Y$ be a regular scheme of finite type over $k$. Let $a\colon X\to Y$ be
flat morphism between $k$-schemes of finite type of relative dimension $d$. A
_Lefschetz subscheme of $X$ over $Y$_ is a closed subscheme $Z$ of $X$ that is
flat over $Y$, of the form $W\cup H$, where $W$ is a closed subscheme of $X$,
with affine complement, and that if flat over $Y$, with $\dim(W)=\dim(X)-1$,
which contains all irreducible components of $X$ of dimension $\dim(X)$, such
that $X-W$ is regular, and $H$ is an hyperplane section that is transverse to
the motivic sheaf $j_{!}\,j^{*}\mathbf{Z}_{X}$ in the sense of 1.18, where
$j\colon X-W\to X$ is the embedding of the complement of $W$. We will then say
that $(X,Z)$ is a Lefschetz pair over $Y$. A Lefschetz pair $(X,Z)$ is
_admissible_ over $Y$ if both $X$ and $Z$ are admissible.

######  3.14.

Let $a\colon X\to Y$ be the structural morphism of the admissible affine
$Y$-scheme $X$. We choose an affine open embedding $e\colon X\to\bar{X}$
followed by a projective morphism $\bar{a}\colon\bar{X}\to Y$ with
$a=\bar{a}\,e$. Possibly after shrinking $Y$, we may assume that $\bar{X}$ is
flat over $Y$. Let us consider an admissible Lefschetz subscheme $Z$ in $X$,
and write $p\colon\bar{X}-\bar{H}\to Y$ for the structural map of the
complement of the chosen hyperplane section $H$. We have

$a_{*}\,j_{!}\,\mathbf{Q}_{X-Z}\cong p_{!}\,e_{*}\,j_{!}\mathbf{Q}_{X-W}\,.$

We will say that $(X,Z)$ is _tight_ if the conclusion of Proposition 2.20
holds without shrinking $Y$ any further for the realizations of
$a_{*}\,j_{!}\,\mathbf{Q}_{X-Z}$ and of $p_{!}\,e_{*}\,j_{!}\mathbf{Q}_{X-W}$
through any $\mathbf{Q}$-linear Artin-Verdier six-functors formalism $D$. The
aforementioned proposition says that, given any admissible Lefschetz pair
$(X,Z)$ over $Y$, there always exists a dense open subscheme of $Y$ over which
$(X,Z)$ becomes tight.

######  Proposition 3.15.

If $(X,Z)$ is a tight admissible Lefschetz pair over a regular scheme $Y$,
with $X$ of relative dimension $d$ over $Y$, then, for any $\mathbf{Q}$-linear
Artin-Verdier six-functors formalism $D$, the induced functor

$r_{D}\colon\mathit{DM}(Y,\mathbf{Q})\to D(Y)$

sends

$M_{Y}(X,Z)^{\wedge}[d]\cong a_{*}\,j_{!}\,\mathbf{Q}_{X-Z}[d]$

to a dualizable object that is concentrated in degree $0$ for the ordinary
$t$-structure in $D(Y)$.

###### Proof.

This is a direct consequence of the definition, exactly as in the proof of
Theorem 2.22 and of Lemma 3.4. ∎

######  3.16.

Let $Y$ be a scheme of finite type over $k$ with $Y_{\mathit{red}}$ regular,
and let $\imath\colon Y_{\mathit{red}}\to Y$ be the canonical embedding. We
denote by $\mathcal{G}_{Y}$ the family of objects of the form

$\imath_{*}\,f_{*}M_{Y^{\prime}}(X^{\prime},Z^{\prime})(n)[-d]$

with $f\colon Y^{\prime}\to Y_{\mathit{red}}$ a finite flat morphism,
$Y^{\prime}$ regular, $(X^{\prime},Z^{\prime})$ a tight admissible Lefschetz
pair over $Y^{\prime}$, $d$ the relative dimension of $X$ over $Y^{\prime}$
(seen as a locally constant function), and $n$ any integer. As we did
previously we define _standard generators_ over $Y$ objects of the form

$A^{\wedge}_{1}\otimes\cdots\otimes A^{\wedge}_{m}\otimes
B_{1}\otimes\cdots\otimes B_{n}$

where each $A_{i}$ and $B_{j}$ are in $\mathcal{G}_{Y}$ and $(-)^{\wedge}$ is
the duality operator on dualizable objects in $\mathit{DM}(Y,\mathbf{Q})$.

For a scheme of finite type $X$ over $k$, we define the _perverse generating
family_ $P(X)$ by noetherian induction on dimension as follows. For $\dim
X\leq 0$, $P(X)=\mathcal{G}_{X}$. For $\dim X>0$, $P(X)$ is the collections of
objects of the following two possible forms:

  * •

$i_{*}(M)$ for $i\colon Z\to X$ a nowhere dense closed immersion and $M$ in
$P(Z)$;

  * •

$j_{!}M[\dim Y]$ for $j\colon Y\to X$ an affine dense open immersion with
$Y_{\mathit{red}}$ affine and regular, and $M$ is a standard generator over
$Y$.

######  Proposition 3.17.

The family $P(X)$ forms a family of compact generators in
$\mathit{DM}(X,\mathbf{Q})$.

###### Proof.

We already know this is true in the case where $X$ is the spectrum of a field.
The case where $X$ is of dimension $\leq 0$ follows right away. Theorem 1.16
together with Proposition 2.20 and Grothendieck’s generic flatness theorem
ensure that tight admissible Lefschetz pairs exist in abundance generically.
The proposition follows easily from [CD19, Prop. 4.3.17] applied to
$\mathit{DM}$ by continuity arguments. ∎

######  3.18.

We define ${}^{p}\mathit{DM}^{\leq 0}(X,\mathbf{Q})$ as the smallest full
subcategory of $\mathit{DM}(X,\mathbf{Q})$ stable under small colimits and
stable under extensions spanned by $P(X)$. This is a compactly generated
presentable $\infty$-category. Therefore, the inclusion functor

${}^{p}\mathit{DM}^{\leq
0}(X,\mathbf{Q})\hookrightarrow\mathit{DM}(\mathrm{Spec}(k),\mathbf{Q})$

has a right adjoint

$\tau^{\leq 0}\colon\mathit{DM}(X,\mathbf{Q})\to{{}^{p}\mathit{DM}^{\leq
0}(X,\mathbf{Q})}$

We define ${}^{p}\mathit{DM}^{\geq 0}(X,\mathbf{Q})$ by orthogonality as the
full subcategory of objects $B$ such that

$\operatorname{Map}(A[1],B)=0$

for any $A$ in ${}^{p}\mathit{DM}^{\leq 0}(X,\mathbf{Q})$. This defines a
$t$-structure on $\mathit{DM}(X,\mathbf{Q})$ that would be a candidate for the
perverse motivic $t$-structure.

If $D$ is any $\mathbf{Q}$-linear Artin-Verdier six-functors formalism defined
over $k$-schemes of finite type, the perverse $t$-structure on $D_{c}$ extends
to a perverse $t$-structure on $D$ such that the embedding

$D_{c}(X)\hookrightarrow D(X)$

is $t$-exact for all $X$: simply define ${}^{p}D^{\leq 0}(X)$ as the smallest
full subcategory of $D(X)$ stable under small colimits and stable under
extensions spanned by perverse sheaves in $D_{c}(X)$. For any $k$-scheme of
finite type $X$, Proposition 3.15 imply that the induced functor

$r_{D}\colon\mathit{DM}(X,\mathbf{Q})\to D(X)$

is right $t$-exact and induces a colimit preserving functor

$r_{D}\colon{{}^{p}\mathit{DM}^{\leq 0}(X,\mathbf{Q})}\to{{}^{p}D^{\leq
0}(X)}\,.$

There is an analogous construction for a candidate of the ordinary
$t$-structure on $\mathit{DM}(X,\mathbf{Q})$. Define $G(X)$ as follows:
$G(X)=P(X)$ for $X$ of dimension $\leq 0$, and for $X$ of dimension $d>0$,
define $G(X)$ as the collection of objecrs of the form

  * •

$i_{*}(M)$ for $i\colon Z\to X$ a nowhere dense closed immersion and $M$ in
$G(Z)$;

  * •

$j_{!}M$ for $j\colon Y\to X$ an affine dense open immersion with
$Y_{\mathit{red}}$ affine and regular, and $M$ is a standard generator over
$Y$.

We define $\mathit{DM}^{\leq 0}(X,\mathbf{Q})$ as the smallest full
subcategory of $\mathit{DM}(X,\mathbf{Q})$ stable under small colimits and
stable under extensions spanned by $P(X)$. This defines as above a
$t$-structure such that $r_{D}$ is right $t$-exact for any $D$ (with respect
to the ordinary $t$-structure on $D$). By construction, $\mathbf{Q}_{X}$
belongs to $\mathit{DM}^{\leq 0}(X,\mathbf{Q})$ and the tensor product defines
a functor that preserves colimit in each variables

$(-)\otimes(-)\colon\mathit{DM}^{\leq
0}(X,\mathbf{Q})^{\mathit{op}}\times\mathit{DM}^{\leq
0}(X,\mathbf{Q})\to\mathit{DM}^{\leq 0}(X,\mathbf{Q})\,.$

A wild optimistic guess would be that any element of $G(X)$ is in the heart of
this $t$-structure.

## References

  * [Ara05] D. Arapura, _The Leray spectral sequence is motivic_ , Invent. Math. 160 (2005), no. 3, 567–589.
  * [Ara13] by same author, _An abelian category of motivic sheaves_ , Adv. Math. 233 (2013), 135–195.
  * [Ayo07] J. Ayoub, _Six opérations de Grothendieck et cycles évanescents dans le monde motivique_ , Astérisque, vol. 314–315, Soc. Math. France, 2007.
  * [Ayo14] by same author, _La réalisation étale et les opérations de Grothendieck_ , Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 1, 1–145.
  * [Ayo23] by same author, _Weil cohomology theories and their motivic Hopf algebroids_ , arXiv:2312.11906, 2023.
  * [Bac21] T. Bachmann, _Rigidity in étale motivic stable homotopy theory_ , Alg. & Geom. Topol. 21 (2021), 173–209.
  * [BBDG18] A. Beilinson, J. Bernstein, P. Deligne, and O. Gabber, _Faisceaux pervers_ , Astérisque, vol. 100, Soc. Math. France, 2018, 2nd edition.
  * [Bei87] A. Beilinson, _On the derived category of perverse sheaves_ , $K$-theory, arithmetic and geometry (Moscow, 1984–1986), Lecture Notes in Math., vol. 1289, Springer, 1987, pp. 27–41.
  * [Bei10] by same author, _Remarks on Grothendieck’s standard conjectures_ , arXiv:arXiv:1006.1116, 2010.
  * [BS15] B. Bhatt and P. Scholze, _The pro-étale topology for schemes_ , De la Géométrie algébrique aux formes automorphes (I) - (Une collection d’articles en l’honneur du soixantième anniversaire de Gérard Laumon), Astérisque, vol. 369, Soc. Math. France, 2015, pp. 99–201.
  * [CD16] D.-C. Cisinski and F. Déglise, _Étales motives_ , Compositio Math. 152 (2016), 556–666.
  * [CD19] by same author, _Triangulated categories of mixed motives_ , Springer Monographs in Mathematics, Springer, 2019.
  * [Cis21] D.-C. Cisinski, _Cohomological methods in intersection theory_ , Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects (F. Neumann and A. Pál, eds.), Lecture Notes in Math., vol. 2292, Springer, 2021, pp. 49–105.
  * [Del77] P. Deligne, _Cohomologie étale – SGA 4 1/2_ , Lecture Notes in Mathematics, vol. 569, Springer, Berlin, 1977.
  * [DG22] B. Drew and M. Gallauer, _The universal six-functors formalism_ , Annals of $K$-theory 7 (2022), no. 4, 599–649.
  * [HMS17] A. Huber and S. Müller-Stach, _Periods and Nori motives_ , Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 65, Springer, Cham, 2017, with contributions by Benjamin Friedrich and Jonas von Wangenheim.
  * [HRS23] T. Hemo, T. Richarz, and J. Scholbach, _Constructible sheaves on schemes_ , arXiv:2305.18131, 2023.
  * [ILO14] L. Illusie, Y. Lazslo, and F. Orgogozo, _Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents_ , Astérisque, vol. 361–362, Soc. Math. France, 2014, Séminaire à l’École Polytechnique 2006–2008.
  * [IM19] F. Ivorra and S. Morel, _The four operations on perverse motives_ , arXiv:1901.02096, to appear in J. Eur. Math. Soc., 2019.
  * [Jou73] J. P. Jouanolou, _Une suite exate de Mayer-Vietoris en $K$-théorie algébrique_, Algebraic $K$-theory, I: Higher $K$-theories (Pro. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math., vol. 341, Springer, Berlin, 1973, pp. 293–316.
  * [Kat93] N. Katz, _Affine cohomological transforms, perversity, and monodromy_ , J. Amer. Math. Soc. 6 (1993), no. 1, 149–222.
  * [Kat01] by same author, _Sums of Betti numbers in arbitrary characteristic_ , Finite Fields and Their Applications 7 (2001), no. 1, 29–44.
  * [KL85] N. Katz and G. Laumon, _Transformation de Fourier et majoration de sommes exponentielles_ , Publ. Math. IHES 62 (1985), 145–202.
  * [Lur17] J. Lurie, _Higher Algebra_ , available at https://www.math.ias.edu/~lurie/, 2017.
  * [Mil64] J. Milnor, _On the Betti numbers of real varieties_ , Proc. Amer. Math. Soc. 15 (1964), 275–280.
  * [Org19] F. Orgogozo, _Constructibilité et modération uniformes en cohomologie étale_ , Compositio Math. 155 (2019), 711–757.
  * [Tho65] R. Thom, _Sur l’homologie des varétés algébriques réelles_ , Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, NJ, 1965, pp. 255–265.