# On the Nori and Hodge realisations of Voevodsky étale motives

Swann Tubach

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Swann Tubach. On the Nori and Hodge realisations of Voevodsky étale motives. 2024. hal-04740242

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# On the Nori and Hodge realisations of Voevodsky motives

Swann Tubach

## Abstract

We show that the derived category of perverse Nori motives and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations and therefore to obtain realisation functors from the category of Voevodsky étale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We also prove that if a motivic t-structure exists then Voevodsky étale motives and the derived category of perverse Nori motives are equivalent. Finally we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky étale motives.

## Contents

<table><tr><td>1</td><td>Recollections.</td><td>7</td></tr><tr><td>1.1</td><td>Categories of étale motives.</td><td>7</td></tr><tr><td>1.2</td><td>Perverse Nori motives.</td><td>8</td></tr><tr><td>1.3</td><td>Mixed Hodge modules</td><td>9</td></tr><tr><td>2</td><td>Adapting Nori's argument to perverse Nori motives.</td><td>10</td></tr><tr><td>2.1</td><td>Cohomology of motives over an affine variety.</td><td>11</td></tr><tr><td>2.2</td><td>Admissibility of constant motives on any variety.</td><td>14</td></tr><tr><td>2.3</td><td>Lisse motives enter the party.</td><td>16</td></tr><tr><td>3</td><td>Higher categorical enhancements.</td><td>17</td></tr><tr><td>3.1</td><td>Categorical preliminaries.</td><td>17</td></tr><tr><td>3.2</td><td>Enhancement of the 6 functors formalism.</td><td>23</td></tr></table>

2020 Mathematics subject classification. 14F42, 14F25, 18G80, 14C15

Key words and phrases. Voevodsky motives, Nori motives, mixed Hodge modules, motivic realisation, six functor formalism.
<table>
  <tbody>
    <tr>
      <td><b>4</b></td>
      <td><b>The realisation functor and applications.</b></td>
      <td><b>29</b></td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Construction of the realisation.</td>
      <td>29</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Relation with the t-structure conjecture.</td>
      <td>34</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Nori motives as modules in étale motives.</td>
      <td>37</td>
    </tr>
    <tr>
      <td><b>A</b></td>
      <td><b>Coefficient systems.</b></td>
      <td><b>43</b></td>
    </tr>
  </tbody>
</table>

Introduction

Let $k$ be a field of characteristic zero. Following the vision of Beilinson, Deligne, Grothendieck and others there should exist an abelian category of mixed motives $\mathcal{M}\mathcal{M}(k)$, target of the universal cohomology theory

$$
M^* : \mathrm{Var}_k^{\mathrm{op}} \to \mathcal{M}\mathcal{M}(k)
$$

on $k$-varieties in the sense that any other reasonable cohomology theory on $k$-varieties
would factor uniquely through $M^*$. This is out of reach of the current technology. How-
ever two constructions have almost all the required properties to provide a category of
mixed motives. The first one is Voevodsky, Levine, Hanamura and others' triangulated
category of geometric motives $DM_{gm}(k, \mathbb{Q})$ ([VSF00]), a candidate for the bounded de-
rived category of mixed motives. The second is the abelian category of Nori motives
$\mathcal{M}_{\text{Nori}}(k)$ ([Fak00]), a candidate for $\mathcal{M}\mathcal{M}(k)$. These categories have realisation functors
factoring the known cohomology theories of varieties. Indeed, Huber ([Hub00]), Levine
([Lev05]) and Nori have constructed a Hodge realisation functor

$$
\mathrm{DM}_{\mathrm{gm}}(k, \mathbb{Q}) \to \mathrm{D}^b(\mathrm{MHS}_{\mathbb{Q}}^p)
$$

to the derived category of polarisable rational mixed Hodge structures that realises the
Hodge cohomology of varieties and factors through the derived category of the $\mathbb{Q}$-linear
abelian category of motives constructed by Nori thanks to the work of Harrer ([Har16])
and Choudhury–Gallauer ([CGAdS17]). When $k$ is not a necessarily a subfield of $\mathbb{C}$
they also constructed a $\ell$-adic realisation with values in the derived category of Galois
representations.

There exist relative versions of the categories of Voevodsky and Nori. The triangulated category of constructible rational étale motives $DM_c(-)$ constructed by Ayoub ([Ayo14a]) and Cisinski and Déglise ([CD16]) gives a triangulated category of étale motivic sheaves generalising Voevodsky’s category, and the abelian category of perverse Nori motives $M_{\text{perv}}(X)$ constructed by Ivorra and S. Morel ([IM22]) gives a category of motivic sheaves generalising the abelian category of Nori. Both settings afford a 6 functors formalism, the tensor product on perverse Nori motives being constructed by Terenzi ([Ter24]). Note that another approach based on constructible sheaves has been considered by Arapura in [Ara13] and [Ara23] but his construction does not include all 6 operations. If $X$ is a quasi-projective smooth $k$-variety, a realisation functor

$$
\mathrm{DM}_c(X) \to \mathrm{D}^b(\mathrm{MHM}(X))
$$
to Saito's derived category of mixed Hodge modules ([Sai90]) has been constructed by Ivorra ([Ivo16]). It also factors through the derived category of perverse Nori motives and computes the relative Hodge homology of X-schemes. Unfortunately, Ivorra's functor has no obvious compatibility with the 6 operations that exist on $DM_c(-)$ and $D^b(MHM(-))$. The reason behind this difficulty is that the functor is built by constructing an explicit complex $K^\bullet \in \mathrm{Ch}^b(MHM(X))$ that computes the relative homology sheaf $f!f!\mathbb{Q}_X \in \mathrm{D}^b(MHM(X))$ of a smooth affine X-scheme $f:Y \to X$, and that we do not know if one can construct the 6 operations explicitly on complexes only in terms of objects living in the abelian category $MHM(X)$.

On the other hand, we know since Robalo's thesis [Rob15] that the $\infty$-category $DM_c(X)$ (which is a stable $\infty$-category lifting $DM_c(X)$) affords an universal property that enables one to construct realisations functors to any stable $\infty$-category $\mathcal{C}$ that have reasonable properties. Indeed, he shows that $DM(X) = \mathrm{Ind}DM_c(X)$ is the target of the universal symmetric monoidal functor $\mathrm{Sm}_X \to DM(X)$ to a stable $\mathbb{Q}$-linear presentably symmetric monoidal $\infty$-category, satisfying $\mathbb{A}^1$-invariance, étale hyperdescent and $\mathbb{P}^1$-stability. The work of Drew and Gallauer ([DG22]) shows that this universal property works very well in families, so that the $\infty$-functor

$$DM : \mathrm{Sch}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Pr}_{\mathrm{st}}^L)$$

is the target of the universal natural transformation $\mathrm{Sm}_{(-)} \Rightarrow DM$ of functors $\mathrm{Sch}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty)$ sending $f:Y \to X$ in $\mathrm{Sch}_k$, smooth, to a right adjoint and such that $DM$ is $\mathbb{Q}$-linear, satisfy non-effective étale descent and $\mathbb{A}^1$-invariance, together with $\mathbb{P}^1$-stability. Moreover Cisinski and Déglise (see also Ayoub's [Ayo07, Scholie 1.4.2]) proved in [CD19, Theorem 4.4.25] that the full subcategory of geometric objects $\mathcal{C}_{\mathrm{gm}}$ of any such functor $\mathcal{C}$ affords the six operations, and that any natural transformation $DM \to \mathcal{C}$ between such functors would induce a natural transformation between the categories of geometric objects that commutes with the six operations. In other words, to construct a family of realisation functors $\mathrm{Hdg}^* : DM_c \to D^b(MHM(-))$ compatible with the 6 operations, it suffices to lift the 6 operations on mixed Hodge modules to the $\infty$-categorical setting. This is what we do.

There is another approach to the Hodge realisation of étale motives considered by Brad Drew in [Dre18]. For each finite type $k$-scheme $X$, he constructs a new category $\mathcal{DH}_c(X)$ of motivic Hodge modules that has a tautological functor $DM_c(X) \to \mathcal{DH}_c(X)$ commuting with all the operations. If $X = \mathrm{Spec} k$ is the spectrum of a field, then $\mathcal{DH}_c(\mathrm{Spec} k) \hookrightarrow D^b(MHS_k^p)$ embeds in the derived category of mixed Hodge structures. Moreover one can compute Deligne cohomology as a Hom-group in $\mathcal{DH}_c(X)$. However it was unclear how to relate $\mathcal{DH}_c(X)$ with $D^b(MHM(X))$ as it is hard to construct an adequate t-structure on $\mathcal{DH}_c(X)$. Using our result on the existence of the Hodge realisation and a description of geometric mixed Hodge modules as modules in $DM(X)$ (see Proposition 4.20 for the case of Nori motives), it should not be too hard to prove that Drew's category $\mathcal{DH}_c(X)$ embeds fully faithfully in the derived category of mixed Hodge modules. We plan to pursue this in a sequel to this paper.

Our main result is the following:
**Theorem 0.1** (Theorem 4.2, Theorem 4.4 and Proposition 4.8). The 6 operations on the categories $D^b(MHM(-))$ and $D^b(M_{\text{perv}}(-))$ constructed by Saito, Ivorra, S. Morel and Terenzi admit $\infty$-categorical lifts and are defined over any finite type k-scheme. For every finite type k-scheme there exist $\infty$-functors

$$ \mathrm{Nor}^* : \mathcal{DM}_c(X) \to \mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X)), $$

$$ R_H : \mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X)) \to \mathcal{D}^b(\mathrm{MHM}(X)), $$

and

$$ \mathrm{Hdg}^* : \mathcal{DM}_c(X) \to \mathcal{D}^b(\mathrm{MHM}(X)) $$

that commute with the 6 operations and such that $R_H \circ \mathrm{Nor}^* \simeq \mathrm{Hdg}^*$. Moreover, $\mathrm{Nor}^*$ is compatible with the Betti and the $\ell$-adic realisations, $R_H$ is t-exact and all functors are weight-exact.

How to give $\infty$-categorical lifts of triangulated functors? In general this is a hard question. One general solution would be to use the formalism developed by Liu and Zheng in [LZ15]. In our particular case we have a simpler method. Indeed, we use that it is very easy to construct $\infty$-categorical lifts of derived functors. Most of the 6 operations are not t-exact (even on one side) for the perverse t-structure. However, they are t-exact (at least on one side) for the constructible t-structure, that is the t-structure whose heart behaves like the abelian category of constructible sheaves. In [Nor02], Nori proves that the triangulated category of cohomologically constructible sheaves $D_c^b(X(\mathbb{C}), \mathbb{Q})$ on the $\mathbb{C}$-points of an algebraic variety $X$ is the derived category of its constructible heart. We show that one can adapt his argument to make things work for mixed Hodge modules and perverse Nori motives, and obtain the following result:

**Theorem 0.2** (Corollary 2.19). Let $X$ be a quasi-projective $k$-variety. Denote by $\mathrm{MHM}_c(X)$ (resp. by $\mathcal{M}_{\mathrm{ct}}(X)$) the heart of the constructible t-structure on $D^b(\mathrm{MHM}(X))$ (resp. on $D^b(\mathcal{M}_{\mathrm{perv}}(X))$). The canonical $\infty$-functors

$$ \mathrm{real}_{\mathrm{MHM}_c(X)} : \mathcal{D}^b(\mathrm{MHM}_c(X)) \to \mathcal{D}^b(\mathrm{MHM}(X)) $$

and

$$ \mathrm{real}_{\mathcal{M}_{\mathrm{ct}}(X)} : \mathcal{D}^b(\mathcal{M}_{\mathrm{ct}}(X)) \to \mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X)) $$

are equivalences of $\infty$-categories.

This theorem enables us to make the handy change of variables $\mathcal{D}^b(\mathcal{M}_{\mathrm{ct}}(X)) \simeq \mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X))$ and see, for example, pullback functors $f^*$ as derived functors of their restriction to the constructible heart, so that they are $\infty$-functors for a simple reason! This consideration was the starting idea of this paper. Each of the 6 operations can be obtained as a right derived functor on the constructible or the perverse heart. This is how we obtain the $\infty$-categorical lifts of the operations.
Let $\mathcal{DN}(X) = \text{Ind}\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ be the indization of the bounded derived $\infty$-category of $\mathcal{M}_{\text{perv}}(X)$. This is a compactly generated stable presentably symmetric monoidal $\infty$-category whose compact objects consist of $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$. The functor $\text{Nor}^* : \mathcal{DM}_c(X) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ extends to a functor

$$ \text{Nor}^* : \mathcal{DM}(X) \to \mathcal{DN}(X) $$

which preserves colimits, hence has a right adjoint $\text{Nor}_*$. For each finite type $k$-scheme, denote by $\mathcal{N}_X := \text{Nor}_*\text{Nor}^*\mathbb{Q}_X \in \mathcal{DM}(X)$. As $\text{Nor}_*$ is naturally lax symmetric monoidal, we have that $\mathcal{N}_X \in \text{CAlg}(\mathcal{DM}(X))$ is an $\mathbb{E}_\infty$-algebra of $\mathcal{DM}(X)$. Using the proof of the same result for the Betti realisation in [Ayo22, Theorem 1.93], we prove

**Proposition 0.3 (Proposition 4.20).** For each scheme $X$ of finite type over some field $k$ of characteristic zero, denote by $p_X : X \to \text{Spec} \mathbb{Q}$ the unique morphism. Then $p_X^*\mathcal{N}_X \simeq \mathcal{N}_X$ and the natural functor

$$ \text{Mod}_{\mathcal{N}_X}(\mathcal{DM}(X)) \to \mathcal{DN}(X) $$

is an equivalence of categories that commutes with pullback functors.

Although we do not go in the details, the same arguments would prove that the $\infty$-category $\mathcal{D}\mathcal{H}^{\text{geo}}(X) \subset \text{Ind}\mathcal{D}^b(\text{MHM}(X))$ of geometric origin objects in the indization of the derived category of mixed Hodge modules is also the category of modules over some algebra $\mathcal{H}_X$ in $\mathcal{DM}(X)$ (See [Tub24] for more details). This shows that Drew's constructions in [Dre18] indeed provide an module presentation of geometric origin mixed Hodge modules (and even, by considering motives enriched in mixed Hodge structures, a presentation of all mixed Hodge modules as modules in a motivic category).

As the functor $(\mathcal{A}, \mathcal{C}) \mapsto \text{Mod}_{\mathcal{A}}(\mathcal{C})$ that sends a symmetric monoidal $\infty$-category pointed at an algebra object $\mathcal{A} \in \text{CAlg}(\mathcal{C})$ to the $\infty$-category of modules over $\mathcal{A}$ preserves colimits, we obtain the following corollary:

**Corollary 0.4 (Corollary 4.22).** The $\infty$-functor $X \mapsto \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ has an unique extension $\mathcal{DN}_c(X)$ to quasi-compact and quasi-separated schemes of characteristic zero such that for all limit $X = \lim_i X_i$ of such schemes with affine transitions the natural functor

$$ \text{colim}_i \mathcal{DN}_c(X_i) \to \mathcal{DN}_c(X) $$

is an equivalence of $\infty$-categories.

Now that a well behaved comparison functor $\text{Nor}^* : \mathcal{DM}_c(X) \to \mathcal{DN}_c(X)$ is available, one wants to see to what extend we can compare both categories. We have a result in that direction, conditional to the existence of a motivic t-structure on $\text{DM}_{\text{gm}}(k)$, an out of reach conjecture that we know imply all standard conjectures in characteristic zero ([Bei12]). Moreover, by the work of Bondarko in [Bon15, Theorem 3.1.4], if such a t-structure exists, then for all finite type $k$-schemes, we have a perverse and a constructible t-structure on $\mathcal{DM}_c(X)$.
**Theorem 0.5** (Theorem 4.12). Assume that a motivic t-structure exists for all fields of characteristic 0. Let $X$ be a finite type $k$ scheme for such a field $k$. Then the heart of the perverse t-structure of $\mathcal{DM}_c(X)$ is canonically equivalent to $\mathcal{M}_{\text{perv}}(X)$ the category of perverse Nori motives. Moreover, the functor $\text{Nor}^* : \mathcal{DM}_c(X) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ is an equivalence of stable $\infty$-categories. This implies that $\mathcal{DM}_c(X)$ is both the derived category of its perverse and constructible hearts.

In particular over a field we would have $\text{DM}_{\text{gm}}(k, \mathbb{Q}) \simeq \text{D}^b(\mathcal{MM}(k))$, showing that the "even more optimistic" expectation of [And04, 21.1.8] is in fact no more optimistic than expecting the standard conjectures to be proven.

# Organisation of the paper.

After some recollections on motivic constructions in Section 1, in our Section 2 we review Nori's proof in [Nor02] that the category of cohomologically constructible complexes of sheaves on the $C$-points of a quasi-projective variety $X$ is equivalent to the derived category of constructible sheaves and show that it gives a proof that the derived categories of the perverse and constructible hearts on the categories of mixed Hodge modules (or of perverse Nori motives) are equivalent. The only idea here was to replace local systems with dualisable objects and cohomology of the global sections with $\text{Hom}_{\text{P}}^{b}(\mathcal{M}_{\text{perv}}(X))(\mathbb{Q}X, -[q])$. The main point of the proof goes by showing effaceability of cohomology of the constant object on affine $n$-spaces, then deduce it for any affine variety thanks to Noether normalisation.

We divided Section 3 in three parts. In the first one we prove general results about the functoriality of the functor $\mathcal{D}^b(-)$ sending an abelian category to its bounded derived $\infty$-category. In the second part we prove that the 6 operations on mixed Hodge modules and on perverse Nori motives can be lifted to $\infty$-categorical functors of stable $\infty$-categories. The third part is devoted to prove étale descent for the derived category of perverse Nori motives.

In Section 4, we use what have been proven on $\infty$-categorical enhancements to construct the realisation functors. We then prove the result related to the t-structure conjecture. The main argument is the repeated use of the universal properties of $\mathcal{DM}_c(X)$ and $\mathcal{M}_{\text{perv}}(X)$ that force the Nori realisation functor to be an equivalence when $\mathcal{DM}_c(X)$ has a perverse t-structure. Finally we prove that Nori motives are modules over an algebra in $\mathcal{DM}(X)$. The arguments follow more or less Ayoub's proof of the similar statement for the Betti realisation in [Ayo22].

# Forthcoming and future work

Using the $\infty$-categorical enhancement of mixed Hodge modules we will be able to extend them to Artin stacks, together with the operations, t-structures and weights. In joint work with Raphaël Ruimy we will construct an integral version of the derived category
of perverse Nori motives. This will give an Abelian category of motivic sheaves with
integral coefficients over any finite dimensional scheme of characteristic zero.

**Notations and conventions**

For the whole paper, $k$ is a field of characteristic $chark = 0$. When $k$ is a subfield of $\mathbb{C}$, if $X$ is a finite type $k$-scheme, we denote by $D_c^b(X, \mathbb{Q})$ the category of constructible complexes of sheaves on $X(\mathbb{C})$. By [Bei87], it is also the derived category of the abelian category of perverse sheaves $Perv(X, \mathbb{Q})$ defined in [BBD82]. For a general field $k$, we denote by $D_c^b(X, \mathbb{Q}_\ell)$ the category of $\mathbb{Q}_\ell$-adic étale sheaves on $X$.

Starting with Section 3, we will use the language of $\infty$-categories as developed in [Lura] by Lurie, after Joyal’s work on quasi-categories. We fix an universe and call small an $\infty$-category whose mapping spaces and core are in this universe. We denote by $\mathrm{Cat}_\infty$ the $\infty$-category of non necessarily small $\infty$-categories. If $\mathcal{C}$ is a symmetric monoidal $\infty$-category, and $A \in \mathcal{C}$ is a commutative algebra, we denote by $\mathrm{Mod}_A(\mathcal{C})$ the category of modules over $A$ in $\mathcal{C}$. For example, the category of $\mathbb{H}\mathbb{Q}$-modules in spectra $\mathrm{Mod}_{\mathbb{Q}} := \mathrm{Mod}_{\mathbb{H}\mathbb{Q}}(\mathrm{Sp})$ is the unbounded derived category of $\mathbb{Q}$-vector spaces. We will also have to consider the $\infty$-category of $\mathrm{Mod}_{\mathbb{Q}}$-modules in the $\infty$-category of stable $\infty$-categories, that we denote by $\mathrm{Mod}_{\mathbb{Q}}(\mathrm{St})$ and which is equivalent to the category of stable $\mathbb{Q}$-linear $\infty$-categories. We will also denote by $\mathrm{Gpd}_\infty$ the $\infty$-category of $\infty$-groupoids, which is also the $\infty$-category of Kan complexes.

We try to always put the index $\mathcal{C}$ when mentioning $\mathrm{Hom}$, as in $\mathrm{Hom}_{\mathcal{C}}(A, B)$. In the case where $\mathcal{C}$ is an ordinary category, this is the $\mathrm{Hom}$-set, and in the case $\mathcal{C}$ is an $\infty$-category this is the $\pi_0$ of the mapping space $\mathrm{map}_{\mathcal{C}}(A, B)$. In stable categories, the mapping spectra will be denoted by $\mathrm{Map}_{\mathcal{C}}(-, -)$.

**Acknowledgements**

This work was written during my PhD supervised by Sophie Morel at the UMPA in
Lyon. I would like to thank her for the faith and constant attention she has for my work
and for the immense amount of time she is spending to accompany me in this thesis.
I am grateful to the anonymous referee for their valuable comments and suggestions
that helped making this paper more readable. I am much obliged to Joseph Ayoub
for finding a mistake during a talk I gave, whose correction simplified the third part
of this paper. I thank also Frédéric Déglise for sharing some ideas with me and Robin
Carlier for taking the time of explaining to me so much facts about $\infty$-categories. I
had interesting email exchanges with Marc Hoyois. I also had useful conversations with
Georg Lehner, Raphaël Ruimy and Luca Terenzi.
# 1 Recollections.

## 1.1 Categories of étale motives.

Ayoub ([Ayo10] and [Ayo14a]), Cisinski-Déglise ([CD19] and [CD16]) and Voevodsky ([Voe02]) have constructed triangulated categories of motivic sheaves, that are the relative versions of DM<sub>gm</sub>(k). In this paper, we are interested in the étale versions. We will use the stable ∞-categorical construction due to Robalo ([Rob15, Section 2.4]) as this is the one we will be using. Also, we only consider Q-linear categories.

Let S be a scheme. One starts with the category Sm<sub>S</sub> of smooth S-schemes, and consider presheaves of spectra on it : PSh(Sm<sub>S</sub>, Sp). The category of rational effective étale motivic sheaves is the full subcategory of PSh(Sm<sub>S</sub>, Sp) whose objects are the presheaves F that are Q-linear (i.e. whose image lands in D(Q) ≃ Mod<sub>HQ</sub> ⊂ Sp), A<sup>1</sup>-invariant (i.e. the natural map F(X) → F(A<sub>X</sub><sup>1</sup>) is an equivalence for any smooth S-scheme X), and satisfy étale hyper-descent (i.e. for any étale hyper cover U<sub>•</sub> → X of a smooth S-scheme X, the natural map F(X) → lim<sub>Δ</sub> F(U<sub>n</sub>) is an equivalence.) It is a symmetric monoidal stable ∞-category, the tensor product being inherited from the Cartesian structure of Sm<sub>S</sub>. Inverting the object cofib(S <sup>1</sup> → Gm<sub>S</sub>) for the tensor product gives the category that we will denote by DM(S) and call the category of Voevodsky (étale) motives.

As a formula, we have

$$DM(S) := (\mathrm{L}_{\mathbb{A}^1} \mathrm{Shv}_{\hat{\mathcal{E}}t}^{\wedge}(\mathrm{Sm}_S, \mathcal{D}(\mathbb{Q})))[(\mathbb{G}_{m_S}/1_S)^{-1}].$$

This is a stable presentably symmetric monoidal ∞-category.

We will denote by DM<sub>c</sub>(S) the full subcategory of étale motives whose objects are compact objects (i.e. M ∈ DM(S) such that Hom<sub>DM</sub>(M, −) commutes with small filtered colimits.) As we work rationally, one can show ([Ayo14a, Proposition 8.3]) that the subcategory of compact objects coincides with the idempotent complete stable subcategory generated by motives of the form M(X)(−n) for X a smooth S-scheme and n ∈ N, where we have denoted by M(X) the image of X ∈ Sm<sub>S</sub> by the Yoneda functor. As DM(S) is compactly generated, we have IndDM<sub>c</sub>(S) = DM(S). Moreover, if S = Spec k is the spectrum of a field, then there is a natural equivalence ho(DM<sub>c</sub>(Spec k)) ≃ DM<sub>gm</sub>(k, Q) between the homotopy category of étale motives and the triangulated now classical category of Voevodsky motives over a field.

The categories DM(X) can be put together to give a functor Sch<sub>S</sub><sup>op</sup> → CAlg(Pr<sup>L</sup>) from the category of finite type S-schemes (with S a fixed scheme) to the ∞-category of presentable symmetric monoidal ∞-categories (see [Rob14, Section 9.1]). Moreover, this functor DM is a coefficient system in the sense of Drew and Gallauer (see Definition A.1), and the ∞-category of constructible objects DM<sub>c</sub>(−) is part of a six functors formalism (see [CD19, Section A.5] for a detailed definition), which includes that all pullbacks f* have right adjoints f*, that there exists exceptional functoriality consisting of a f<sub>↓</sub> left adjoint to f<sup>↓</sup>, such that f<sup>↓</sup> ≃ f* for f étale and p* ≃ p<sub>↓</sub> for p proper.
Ayoub constructed in [Ayo10] the Betti realisation of $DM(S)$ with values in the derived category of sheaves on the analytification $S^{an}$ on $S$ when $S$ is of finite type over a subfield of the complex numbers, which restricts to $DM_c(S)$ and then lands into $D_c^b(S(C), \mathbb{Q})$ the stable category of bounded complexes of sheaves with constructible cohomology. This category is both the derived category of perverse sheaves ([Bei87]) and of the abelian category of constructible sheaves ([Nor02]). One can find in [Ayo14a] and [CD16] the construction of the $\ell$-adic realisation functors with values in the derived category of $\ell$-adic sheaves. An $\infty$-categorical version of this functor can be found in [RS20, 2.1.2]. This is a functor $DM_c(S) \to D_c^b(S, \mathbb{Q}_\ell)$, the latter category being enhanced using condensed coefficients in [HRS23] where it is denoted by $D_{\text{cons}}(S, \mathbb{Q}_\ell)$. It also has a version on the big category $DM(S)$ by taking indization. All the realisations commute with the 6 operations.

## 1.2 Perverse Nori motives.

Let $X$ be a quasi-projective $k$-scheme with $k$ a field of characteristic zero. In [IM22], Ivorra and S. Morel introduced an abelian category of perverse Nori motives $M_{\text{perv}}(X)$. The construction of the category $M_{\text{perv}}(X)$ goes as follows:

Let $X$ be a quasi-projective $k$-scheme. Pick your favourite prime number $\ell$. We can compose the $\ell$-adic realisation

$$ \rho_{\ell} : DM_c(X) \to D^b(X_{\acute{e}t}, \mathbb{Q}_{\ell}) $$

with the perverse truncation functor

$$ {}^{\text{P}}H^0 : D^b(X_{\acute{e}t}, \mathbb{Q}_{\ell}) \to \text{Perv}(X, \mathbb{Q}_{\ell}) $$

to obtain a homological functor ${}^{\text{P}}H_{\ell}^0$ that will factor through $\mathcal{R}(DM_c(X))$. This latter category is the abelian category of coherent modules over $DM_c(X)$: it is the full subcategory of additive presheaves $\mathcal{F} : DM_c(X)^{\text{op}} \to \text{Ab}$ that fit in exact sequences

$$ y_M \to y_N \to \mathcal{F} \to 0 $$

where $M \to N$ is a morphism in $DM_c(X)$. Denote by $Z_\ell$ the kernel of ${}^{\text{P}}H_\ell^0 : \mathcal{R}(DM_c(X)) \to \text{Perv}(X, \mathbb{Q}_\ell)$, that is the objects mapping to zero. The quotient category $\mathcal{R}(DM_c(X))/Z_\ell$ is the category of perverse Nori motives $M_{\text{perv}}(X)$.

By construction, $M_{\text{perv}}(X)$ has the following universal property: Any homological functor $H : DM_c(X) \to \mathcal{A}$ to an abelian category $\mathcal{A}$ such that $H(M) = 0$ whenever ${}^{\text{P}}H_{\ell}^0(M) = 0$ for all $M \in DM_c(X)$ will factor uniquely through the universal functor

$$ DM_c(X) \xrightarrow{H_{\text{univ}}} M_{\text{perv}}(X). $$

As we will see in Lemma 4.10 this construction is the best approximation possible of the heart of the conjectural perverse t-structure on $DM_c(X)$. A priori this category depends on the chosen $\ell$, but one can show, using a continuity argument that this is
not the case because for finite type fields (or more generally, for fields embeddable into $\mathbb{C}$), the Betti realisation can also be used to define $\mathcal{M}_{\text{perv}}(X)$.

In fact, every realisation functor existing on $\mathcal{DM}_c(X)$ gives us a realisation functor on $\mathcal{M}_{\text{perv}}(X)$: we have the $\ell$-adic realisation $R_\ell$ and the Betti realisation $R_B$. If $S = \text{Spec } k$, then Ivorra and S. Morel have proven that $\mathcal{M}_{\text{perv}}(\text{Spec } k)$ coincide with the category of Nori constructed by Nori, and thus also have an Hodge realisation.

Ivorra and S. Morel show in [IM22, Theorem 5.1] that the derived category of perverse motives $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(-))$ is part of a homotopical 2-functor in the sense of Ayoub [Ayo07, Definition 1.4.1]. This means that they constructed four of the six operations, namely for $f$ a morphism of quasi-projective varieties we have $f^*, f_*, f_1, f^!$, together with Verdier duality $\mathbb{D}$, verifying $f^! \simeq \mathbb{D} \circ f^*\mathbb{D}$ and $f_1 \simeq \mathbb{D} \circ f_* \circ \mathbb{D}$. In his PhD thesis Terenzi [Ter24] constructed the remaining two operations: tensor product and internal $\mathcal{Hom}$. This proves that $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(-))$ is part of a full 6 functor formalism. The 6 operations commute with the realisation $R_\ell$ and $R_B$ that become morphism of symmetric monoidal stable homotopical 2-functors. The construction of the operations of perverse Nori motives is very similar to Saito's construction of mixed Hodge modules.

## 1.3 Mixed Hodge modules

Let $X$ be a separated finite type $\mathbb{C}$-scheme. In [Sai90], Saito constructed an abelian category $\text{MHM}(X)$ of mixed Hodge modules, together with the six operations on $\mathcal{D}^b(\text{MHM}(X))$. It has a faithful exact functor $\text{MHM}(X) \to \text{Perv}(X^{\text{an}}, \mathbb{Q})$ to the abelian category of perverse sheaves on the analytification of $X$, such that the induced functor on the derived categories commutes with the operations. If $X = \text{Spec } \mathbb{C}$, then there is a natural equivalence $\text{MHM}(\text{Spec } \mathbb{C}) \simeq \text{MHS}_{\mathbb{Q}}^p$ between polarisable mixed Hodge structures and mixed Hodge modules. One of the most interesting property of mixed Hodge modules is that its *lisse* objects, that is the objects such that the underlying perverse sheaf is a shift of a locally constant sheaf are exactly polarisable variations of mixed Hodge structures. We will not go in details in the construction of Saito's category, and we will mostly need to know that the functor taking a complex of mixed Hodge modules to its underlying complex of perverse sheaves is conservative and commutes with the operations.

# 2 Adapting Nori's argument to perverse Nori motives.

## Hypothesis on the coefficient system.

$k$ is a field of characteristic zero. For this section we will call *variety* an element of a fixed subcategory $\text{Var}_k$ of finite type separated $k$-schemes. For example $\text{Var}_k$ could be the full subcategory of quasi-projective $k$-varieties or of separated reduced $k$-schemes of finite type. Choose $D : \text{Var}_k \to \text{Triang}$ your favourite $\mathbb{Q}$-linear triangulated category
of coefficients (e.g. the derived category of perverse Nori motives [IM22, Theorem 5.1] and [Ter24, Main theorem] or of mixed Hodge modules [Sai90, Theorem 0.1]) with a 6 functors formalism and a conservative Betti realisation $R_B : \mathbf{D} \to \mathbf{D}_c^b$ compatible with the operations. Note that in the mixed Hodge module situation the Betti realisation is nothing more that the functor forgetting the structure of a mixed Hodge module, and is called ‘taking the $\mathbb{Q}$-structure” by Saito. It has a constructible t-structure for which $R_B$ is t-exact when $\mathbf{D}_c^b(X(\mathbb{C}), \mathbb{Q})$ is endowed with its natural t-structure of heart the abelian category of constructible sheaves $\text{Cons}(X)$. For $X \in \text{Var}_k$ we will call motives the elements of $\mathbf{D}(X)$.

Recall the following definition:

**Definition 2.1.** The constructible t-structure on $\mathbf{D}(X)$ is the unique t-structure on $\mathbf{D}(X)$ that makes all pullback functors t-exact. Explicitly it is the t-structure given by

$$ {}^{\text{ct}}\mathbf{D}^{\le 0}(X) := \{M \in \mathbf{D}(X) \mid \forall x \in X, x^*M \in \mathbf{D}^{\le 0}(\text{Spec } \kappa(x))\} $$

and

$$ {}^{\text{ct}}\mathbf{D}^{\ge 0}(X) := \{M \in \mathbf{D}(X) \mid \forall x \in X, x^*M \in \mathbf{D}^{\ge 0}(\text{Spec } \kappa(x))\}. $$

That this indeed gives a t-structure can be proven by gluing, using that a lisse object (see Definition 2.2) is positive (resp. negative) for the constructible t-structure if and only it is positive (resp. negative) for the perverse t-structure when shifted by the dimension.

Denote by $\mathcal{M}_{\text{ct}}(X)$ the heart of the constructible t-structure on $\mathbf{D}(X)$. As $R_B$ is t-exact and conservative, all known t-exactness results about the 6 functors are true in $\mathbf{D}$. We will call elements of $\mathcal{M}_{\text{ct}}(X)$ constructible motives. Denote by $\mathcal{Hem}(-, -)$ and $\mathbb{Q}_X$ the internal Hom and unit object of $\mathbf{D}(X)$. If $k$ is too big to be embedded in $\mathbb{C}$, then one needs an $\ell$-adic realisation instead of a Betti realisation. Except for Lemma 2.6 which is different in both cases, in this section we will say Betti realisation even for big fields $k$.

We assume that $\mathbf{D}(X)$ has enough structure so that there exists a natural functor $\mathbf{D}^b(\mathcal{M}_{\text{ct}}(X)) \to \mathbf{D}(X)$ (for example if $\mathbf{D}(X)$ is a derived category as shown by Beilinson, Bernstein, Deligne and Gabber in [BBD82, section 3.1] or more generally if $\mathbf{D}(X)$ is the homotopy category of some stable $\infty$-category $\mathcal{D}(X)$ as proven in [BCKW24, Remark 7.4.13]), and we assume that the natural functor $\mathbf{D}^b(\mathcal{M}_{\text{ct}}(\text{Spec } k)) \to \mathbf{D}(\text{Spec } k)$ is an equivalence over the base field. This is true for perverse Nori motives and mixed Hodge modules.

**Definition 2.2.** Let $X$ be a $k$-variety. A motive $L$ is a *lisse object* if its Betti realisation is a local system, that is a locally constant bounded complex of finite dimensional $\mathbb{Q}$-vector spaces. We denote $\mathbf{D}^{\text{liss}}(X)$ the full subcategory of lisse objects in $\mathbf{D}(X)$.

*Remark 2.3.* Note that as $\mathbf{D}(X)$ is closed as a tensor category, given an object $M \in \mathbf{D}(X)$, we have a natural candidate for the strong dual of $M$: it is $\mathcal{Hem}(M, \mathbb{Q}_X)$. Therefore, an object $M$ is dualisable if and only if the map

$$ \mathcal{Hem}(M, \mathbb{Q}_X) \otimes M \to \mathcal{Hem}(M, M) \qquad (2.3.1) $$
obtained by the $\otimes$-Hom adjunction from the map $ev_M \otimes \text{Id}_M : \text{Hom}(M, \mathbb{Q}_X) \otimes M \otimes M \to M$, is an isomorphism. We will denote by $M^\vee = \text{Hom}(M, \mathbb{Q}_X)$ the dual of $M$ when $M$ is dualisable.

**Proposition 2.4.** Let $X$ be a $k$-variety. An object $L \in \text{D}(X)$ is lisse if and only if it is dualisable and in that case $L \in \mathcal{M}_{\text{ct}}(X)$ if and only if $L^\vee \in \mathcal{M}_{\text{ct}}(X)$. In particular, for any variety $Y$ and for every object $N \in \text{D}(Y)$, there exists a dense open on which $N$ is dualisable.

*Proof.* For the first point, dualisable objects in $\text{D}_c^b(X, \mathbb{Q})$ are exactly local systems by Ayoub's [Ayo22, Lemma 1.24], and the Betti realisation is conservative, so we can test if the map (2.3.1) is an isomorphism after applying Betti realisation. If $L \in \mathcal{M}_{\text{ct}}(X)$ is dualisable, the functor $\text{Hom}(L, -) = - \otimes L^\vee$ is t-exact because it is left and right adjoint to the exact functor $- \otimes L$. Therefore, $L^\vee = \text{Hom}(L, \mathbb{Q}) \in \mathcal{M}_{\text{ct}}(X)$ lies in the heart.

The second point follows from the fact that for $N \in \text{D}(Y)$, the complex of sheaves $R_B(N)$ is constructible so there exists a dense open $U$ of $Y$ on which $R_B(N)$ is locally constant. $\square$

## 2.1 Cohomology of motives over an affine variety.

**Definition 2.5.** 1. An additive functor $F : \mathcal{A} \to \mathcal{B}$ between abelian categories is called *effaceable* if for any $M \in \mathcal{A}$ there is an injection $M \to N$ in $\mathcal{A}$ such that the induced map $F(M) \to F(N)$ is the zero map.

2. An object $M \in \text{D}(X)$ is called *admissible* if the functor $\text{Hom}_{\text{D}(X)}(M, -[q]) : \mathcal{M}_{\text{ct}}(X) \to \text{Vect}_{\mathbb{Q}}$ is effaceable for every $q > 0$.

**Lemma 2.6.** Let $X$ be a variety and $M$ be a constructible motive on $\mathbb{A}_X^1$. Assume that there is $U \subset \mathbb{A}_X^1$ a smooth open subset on which $M$ is lisse and such that the restriction of $\pi : \mathbb{A}_X^1 \to X$ to $Z = \mathbb{A}_X^1 \setminus U$ is finite. Assume that $M|_Z = 0$. Then there exists $N \in \mathcal{M}_{\text{ct}}(\mathbb{A}_X^1)$ such that $\pi_*N = 0$ and $M$ injects into $N$.

*Proof.* Assume that $k$ is a subfield of $\mathbb{C}$. Then the following holds: Denote by $p_i : \mathbb{A}_X^2 \to \mathbb{A}_X^1$ the projections, and by $\Delta : \mathbb{A}_X^1 \to \mathbb{A}_X^2$ the diagonal. The map $\alpha : p_1^*M \to \Delta_*M$ in $\mathcal{M}_{\text{ct}}(X)$ (both functors are t-exact) obtained by adjunction from $\text{id} : \Delta^*p_1^*M \simeq M \to M$ is surjective because it is surjective after realisation. We therefore have an exact sequence

$$0 \to \ker \alpha \to p_1^*M \to \Delta_*M \to 0.$$

Denote $N := \text{H}^1(p_2)_* \ker \alpha$. Then the long exact sequence of cohomology obtained after applying $(p_2)_*$ to the above sequence gives a map $M = (p_2)_* \Delta_* M = \text{H}^0((p_2)_* \Delta_* M) \to N$, which is injective by [Nor02, Proposition 2.2].

Moreover, Nori proves in *loc cit.* that for all $p \ge 0$ we have $\text{H}^p(\pi_*N) = 0$, which is equivalent to $\pi_*N = 0$.

In the case $k$ is too big to be embedded in $\mathbb{C}$, one may use [Bar21, Proposition 3.2] and the $\ell$-adic realisation, noting that the construction of $N$ is the same. $\square$
Nori's method gives in our setting a slightly less good result than in the case of constructible sheaves. It will thankfully be sufficient for us.

**Theorem 2.7** (cf. [Nor02] Theorem 1). For every $n \in \mathbb{N}$, the motive $\mathbb{Q}_{\mathbb{A}_k^n} \in \mathcal{M}_{\text{ct}}(\mathbb{A}_k^n)$ is admissible.

*Proof.* We argue as in Nori's paper. We proceed by induction on $n$ to prove the theorem. First the case of a point. If $n = 0$ and $M \in \mathcal{M}_{\text{ct}}(\text{Spec } k)$, as $\text{D}(\text{Spec } k) = \text{D}^b(\mathcal{M}_{\text{ct}}(\text{Spec } k))$, the functors $\text{Hom}_{\text{D}(\text{Spec } k)}(M, -[q])$ are effaceable by [BBD82, Proposition 3.1.16].

Suppose that the result is known for $n-1$. Let $M \in \mathcal{M}_{\text{ct}}(\mathbb{A}_k^n)$ and $q > 0$. Let $\pi: \mathbb{A}_k^n \to \mathbb{A}_k^{n-1}$ be the projection on the $n-1$ first coordinates. Up to change of coordinates, one can choose a smooth open of the form $D(f)$ in $\mathbb{A}_k^n$ such that $M$ is lisse on that open and the natural projection $\pi|_{V(f)}: V(f) \to \mathbb{A}_k^{n-1}$ is finite. Denote by $j$ and $i$ the inclusions of $D(f) = U$ and $V(f) = Z$ in $\mathbb{A}_k^n$.

By Lemma 2.6, $j_*M|_U$ is a submotive of a constructible motive $N'$ such that $\pi_*N' = 0$. Taking $N$ to be the push-out of the maps $j_*M|_U \to M$ and $j_*M|_U \to N'$ we get a morphism of exact sequences:

$$ \begin{array}{ccccl} 0 & \longrightarrow & j_*M|_U & \longrightarrow & M \longrightarrow i_*M|_Z \longrightarrow 0 \\ & \Big\downarrow & \delta & & \Big\downarrow & \gamma \\ & & & & \Big\downarrow & \text{id} \\ 0 & \longrightarrow & N' & \longrightarrow & N & \longrightarrow i_*M|_Z \longrightarrow 0 \end{array} $$

with $\gamma$ also injective.

Also, as $\pi_*N' = 0$, we see that $\pi_*N \to \pi_*i_*M|_Z = (\pi|_Z)_*M|_Z$ is an isomorphism because of the distinguished triangle $N' \to N \to i_*M_Z \xrightarrow{+1}$. As $\pi|_Z$ is finite, its pushforward is t-exact, hence $\pi_*N \in \mathcal{M}_{\text{ct}}(\mathbb{A}_k^{n-1})$.

By the induction hypothesis, we can find an injection $g: \pi_*N \to K$ with $K$ a constructible motive such that $\text{Hom}_{\text{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, g[q]) = 0$. Take $L$ the pushout of the map $\pi^*\pi_*N \to N$ and the injection $(\pi^* \text{ is t-exact}) \pi^*g: \pi^*\pi_*N \to \pi^*K$. We get a morphism of exact sequences :

$$ \begin{array}{ccccl} 0 & \longrightarrow & \pi^*\pi_*N & \xrightarrow{\pi^*g} & \pi^*K \longrightarrow C \longrightarrow 0 \\ & \Big\downarrow & & & \Big\downarrow & \text{id}_C \\ 0 & \longrightarrow & N & \xrightarrow{h} & L & \longrightarrow C \longrightarrow 0 \end{array} \qquad (2.7.1) $$

The unit map $\pi_*N \to \pi_*\pi^*\pi_*N$ is an isomorphism by $\mathbb{A}^1$-invariance. Therefore in the diagram above the left and the right vertical maps become isomorphisms after applying $\pi_*$, thus this is also the case for the middle map $\iota: \pi^*K \to L$.

Using the adjunction $(\pi^*, \pi^*)$ and the fact that $\pi_*N \to \pi_*\pi^*\pi_*N$ is an isomorphism one gets that the map

$$ \text{Hom}_{\text{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, \pi^* \pi_* N[q]) \to \text{Hom}_{\text{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, N[q]) \qquad (2.7.2) $$
is an isomorphism. Moreover the map

$$ \mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, \pi^*\pi_* N[q]) \to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, \pi^*K[q]) \quad (2.7.3) $$

induced by $\pi^*g$ vanishes as it was already the case before applying $\pi^*$.

Therefore when applying $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, -[q])$ to the left square of (2.7.1), we see that the map

$$ \mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, N[q]) \to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, L[q]) \quad (2.7.4) $$

induced by $h$ factors through the zero map.

Hence we see that the injection $M \xrightarrow{\gamma} N \xrightarrow{h} L$ induces the zero map after applying the functor $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, -[q])$, finishing the induction. $\square$

**Corollary 2.8.** If $X$ is affine, the object $\mathbb{Q}_X$ is admissible.

*Proof.* Let $q > 0$. Take $i : X \to \mathbb{A}_k^n$ a closed immersion and $M \in \mathcal{M}_{\mathrm{ct}}(X)$. By Theorem 2.7, there is an injection $i_*M \to K$ with $K$ a constructible motive such that $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{|\mathbb{A}_k^n}, i_*M[q]) \to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{|\mathbb{A}_k^n}, K[q])$ is the zero map. We have an inclusion $f : M \to N := i^*K$, and the image of the map in the Hom-sets is

$$ \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X, i^*i_*M[q]) \simeq \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X, M[q]) \to \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X, i^*K[q]) $$

which fits in a commutative diagram :

$$ \begin{tikzcd} \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X, M[q]) \arrow[r, "f \circ -"'] & \mathrm{Hom}_{\mathrm{D}(X)}(\mathbb{Q}_X, i^*K[q]) \arrow[d, "adj \uparrow"'] \\ \mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, i_*M[q]) \arrow[r, "0"'] & \mathrm{Hom}_{\mathrm{D}(\mathbb{A}_k^n)}(\mathbb{Q}_{\mathbb{A}_k^n}, K[q]) \arrow[d, "adj \uparrow"'] \end{tikzcd} $$

giving the result. $\square$

## 2.2 Admissibility of constant motives on any variety.

**Notation 2.9.** If $j : U \hookrightarrow X$ is an open subset of a variety $X$, and $M \in \mathcal{M}_{\mathrm{ct}}(X)$ is a constructible motive, we will use the notation $M[U] := j_*M|_U$. Note that by localisation we have $M/M[U] = i_*M|_Z$ with $i : Z = X \setminus U \to X$ the closed complement.

**Lemma 2.10** (Stability of admissibility). Let

$$ 0 \to M' \to M \to M'' \to 0 $$

be an exact sequence of constructible motives on $X$.

1. Assume that $M''$ is admissible and at least one of $M'$, $M$ is admissible. Then all three are admissible.
2. Assume that $M'$ and $M$ are admissible. If the functor

$$ \operatorname{coker}(\operatorname{Hom}_{\mathrm{D}(X)}(M, -) \to \operatorname{Hom}_{\mathrm{D}(X)}(M', -)) $$

is effaceable. Then $M''$ is admissible.

*Proof.* These are formal properties of abelian categories, proven in [Nor02, Lemma 3.2]. □

**Corollary 2.11** ([Nor02] Lemma 3.5). Let $M \in \mathcal{M}_{\mathrm{ct}}(X)$

1. If $U \subset X$ is open, and if $M$ and $M[U]$ are admissible, then so is $M/M[U]$.

2. If $V, W$ are open of $X$, and if $M[V], M[W], M[V \cap W]$ are admissible, then the same is true for $M[V \cup W]$.

*Proof.* We use part 2. of Lemma 2.10. For 1., it suffices to show that any $P \in \mathcal{M}_{\mathrm{ct}}(X)$ is a subsheaf of a $Q \in \mathcal{M}_{\mathrm{ct}}(X)$ such that the map $\operatorname{Hom}_{\mathrm{D}(X)}(M, Q) \to \operatorname{Hom}_{\mathrm{D}(X)}(M[U], Q)$ is surjective. Denote by $j: U \to X$ the open immersion and by $i: Z \to X$ its closed complement. Then define $Q = i_*i^*P \oplus j_*j^*P$. Localisation ensures that the map $P \to Q$ is injective, and the induced map between Hom's is

$$ \begin{array}{ccc} \operatorname{Hom}_{\mathrm{D}(X)}(M, i_*P|_Z) \oplus \operatorname{Hom}_{\mathrm{D}(U)}(M|_U, P|_U) & \simeq & \operatorname{Hom}_{\mathrm{D}(X)}(M, Q) \\ \downarrow & & \\ \operatorname{Hom}_{\mathrm{D}(U)}(M|_U, P|_U) & \simeq & 0 \oplus \operatorname{Hom}_{\mathrm{D}(X)}(j!M|_U, j_*P|_U) \simeq \operatorname{Hom}_{\mathrm{D}(X)}(M[U], Q) \end{array} $$

is surjective as it is just the second projection.

For 2. first one has the Mayer-Vietoris exact sequence

$$ 0 \to M[V \cap W] \to M[V] \oplus M[W] \to M[V \cup W] \to 0. \qquad (2.11.1) $$

For a given $P$ we take the same $Q$ as above for $U = V \cap W$ that gives surjectivity of $\operatorname{Hom}_{\mathrm{D}(X)}(M, Q) \to \operatorname{Hom}_{\mathrm{D}(X)}(M[U], Q)$. Now the map $a: M[U] \to M$ factors through $M[V]$ thus the map

$$ \operatorname{Hom}_{\mathrm{D}(X)}(M[V] \oplus M[W], Q) \xrightarrow{a^* \oplus 0} \operatorname{Hom}_{\mathrm{D}(X)}(M[U], Q) \qquad (2.11.2) $$

is surjective. Finally the map induced by $M[U] \to M[V] \oplus M[W]$ is also surjective as one of its direct summands is. □

**Proposition 2.12** (Avoiding the use of injective objects, after Nori [Nor02] Remark 3.8). Let $F: \mathcal{A} \to \mathcal{B}$ be an exact functor between abelian categories, and let $G: \mathcal{B} \to \mathcal{A}$ be a right adjoint to $F$. Let $H: \mathcal{A} \to \mathcal{C}$ be an effaceable functor. Then $HG$ is also effaceable.
**Proof.** Take $B$ an object of $\mathcal{B}$. By assumption on $H$, there is an injection $u : G(B) \to A$ such that the induced map $H(u) : HG(B) \to H(A)$ vanishes. By exactness of $F$, the morphism $F(u) : FG(B) \to F(A)$ is a monomorphism. We also have the counit of the adjunction $\varepsilon : FG(B) \to B$. Take $B'$ the pushout of these two maps :

$$ \begin{tikzcd} FG(B) \arrow[r, "F(u)"'] & F(A) \arrow[d, "\varepsilon"] \\ B \arrow[r, "\tau"'] & B' \arrow[d, "t"'] \end{tikzcd} $$

The map $v : B \to B'$ is a monomorphism and we have a commutative diagram (with $\eta$ the unit of the adjunction) :

$$ \mathrm{Id} \begin{tikzcd} G(B) \arrow[r, "u"] & A \arrow[d, "\eta_{G(B)}"] \\ GFG(B) \arrow[r, "GF(u)"'] & GF(A) \arrow[d, "\eta_A"] \\ G(\varepsilon) \arrow[d, "G(v)"'] & G(t) \arrow[d, "G(t)"'] \\ G(B) \arrow[r, "G(v)"'] & G(B') \arrow[d, "G(v)"'] \end{tikzcd} $$

The composition $G(\varepsilon) \circ \eta$ is the identity, therefore, the map $G(v)$ factors as $w \circ u$ with $w = G(t) \circ \eta$. Now, $HG(v) = H(w) \circ H(u) = 0$ hence $HG$ is effaceable. $\square$

**Corollary 2.13.** Let $F : \mathrm{D}(X) \to \mathrm{D}(Y)$ be a t-exact exact functor which is left adjoint to a t-exact functor. If $M \in \mathcal{M}_{\mathrm{ct}}(X)$ is admissible, then so is $F(M) \in \mathcal{M}_{\mathrm{ct}}(Y)$.

**Proof.** Let $q > 0$ and let $G$ be a right adjoint of $F$. The functor $H = \mathrm{Hom}_{\mathrm{D}(X)}(M, -[q])$ is effaceable by definition. By Proposition 2.12, $H \circ G$ is therefore effaceable. We have a natural isomorphism $H \circ G \simeq \mathrm{Hom}_{\mathrm{D}(Y)}(F(M), -[q])$ which gives the claim. $\square$

**Corollary 2.14.** Let $M \in \mathcal{M}_{\mathrm{ct}}(X)$ be an admissible motive.

1. Let $j : X \to Y$ be an étale morphism. Then $j!M$ is admissible.

2. Let $L \in \mathcal{M}_{\mathrm{ct}}(X)$ be a lisse motive. Then $M \otimes L$ is admissible.

**Proof.** We use Corollary 2.13. For the first point, we have the adjunction $(j_!, j^*)$ $j_!$ and $j^*$ are t-exact. Let $L \in \mathcal{M}_{\mathrm{ct}}(X)$ be a lisse motive. The functor $- \otimes L$ is left adjoint to $\mathcal{Hem}(L, -)$ and right adjoint to $\mathcal{Hem}(L^\vee, -)$ hence is exact. The same can be said about $\mathcal{Hem}(L, -) = - \otimes L^\vee$. $\square$

**Proposition 2.15** ([Nor02] Proposition 3.6). Let $U \subset X$ an open of a variety $X$. Then the constructible motive $\mathbb{Q}_X[U]$ is admissible.
*Proof.* Let $j: U \to X$ be an open immersion. We prove the theorem by induction on the number $n$ of affines needed to cover $U$. If $U$ is affine, Corollary 2.8 ensures that $Q_U$ is admissible hence by Corollary 2.14, $j_!Q_U = Q_X[U]$ is admissible. For the induction, one can write $U = V \cup W$ with $V$ affine and $W$ covered by $n-1$ affines, and separateness of $X$ gives that $V \cap W$ is covered by $n-1$ affines. Therefore by induction $Q_X[V], Q_X[V \cap W]$ and $Q_X[W]$ are admissible. Then, Corollary 2.11 ensures that $Q_X[U]$ is admissible. □

**Corollary 2.16** ([Nor02] Theorem 2). Let $X$ be a variety over $k$. Then the constant motive $Q_X$ is admissible, that is, for every $q > 0$, every constructible motive $M \in \mathcal{M}_{\text{ct}}(X)$ can be embedded in a constructible motive $N \in \mathcal{M}_{\text{ct}}(X)$ such that the map

$$ \mathrm{Hom}_{\mathrm{D}(X)}(Q_X, M[q]) \to \mathrm{Hom}_{\mathrm{D}(X)}(Q_X, N[q]) $$

vanishes.

*Proof.* One takes $U = X$ in the previous proposition. □

## 2.3 Lisse motives enter the party.

**Proposition 2.17.** Let $X$ be a variety. Any lisse motive in $\mathcal{M}_{\text{ct}}(X)$ is admissible.

*Proof.* By Corollary 2.16 the unit object $Q_X$ on $X$ is admissible. By Corollary 2.14 $L \simeq Q_X \otimes L$ is admissible. □

**Theorem 2.18.** Let $X$ be a variety. Then any $M \in \mathcal{M}_{\text{ct}}(X)$ is admissible, that is, for every $q > 0$, the functor $\mathrm{Hom}_{\mathrm{D}(X)}(M, -[q]): \mathcal{M}_{\text{ct}}(X) \to \mathrm{Vect}_Q$ is effaceable.

*Proof.* We prove the result by Noetherian induction on the support $S$ of $M$ in $X$, that is the complement of the largest open subset $V$ of $X$ such that $M|_V = 0$. Nothing is added to the proof of [Nor02, Proposition 3.10].

Let $S^d$ be the union of the irreducible components of maximal dimension of $S$. Let $U_{\text{liss}}$ be the largest open subset of $S^d$ on which $M$ is lisse. Then there is an affine open subset $U_1$ of $X$ such that $X \cap S^d \subset U_{\text{liss}}$ and is nonempty. Let $U_2 = U_1 \cup_i C_i$ be the open subset of $X$ obtained by removing from $U_1$ the irreducible components of $S$ that are not of maximal dimension. Then $U_2$ is an affine open subset of $X$, such that $U_2 \cap S$ is nonempty and contained in $U_{\text{liss}}$. Choose a Noether normalisation $U_2 \cap S \to \mathbb{A}^d$ that extends (because $U_2 \cap S$ is closed in $U_2$) to $f_2: U_2 \to \mathbb{A}^d$. Let $W \subset A^d$ be a nonempty affine open subset over which $f_2$ is étale. We set $U := f_2^{-1}(W)$. This is an nonempty open subset of $X$, that has a finite étale map $f$ to $\mathbb{A}^d$.

Let $Z = U \cap S$ and $g: Z \to A^d$ be the restriction of $f$ to $Z$. Note that because the restriction of $M$ to $Z$ is lisse and $g$ is finite étale, the object $g_*M|_Z$ is still lisse, so that $L := f^*g_*M|_Z$ is lisse. The restriction to $Z$ of $L$ is $g^*g_*M|_Z$ which has $M|_Z$ as a direct factor because $g$ is étale.

Let $i: Z \to U$ be the closed immersion. Because the support of $M|_U$ is contained in $Z$ we have that $i_*i^*M|_U = M|_U$. By the above, this shows that $M|_U$ is a direct factor
of $i_*L_{|Z}$. Let $j$ be the inclusion of the open complement of $Z$ in $U$. By localisation we have that $i_*L_{|Z} = L/j!j^*L$, so that, because $L$ and $j^*L$ are admissible by Corollary 2.14 and then by Lemma 2.10 1. the object $i_*L_{|Z}$ is admissible, thus the restriction $M_{|U}$ of $M$ to $U$ is admissible. If $\iota: U \to X$ is the open immersion, $\iota!M_{|U}$ is admissible by Corollary 2.14 and we have an exact sequence

$$0 \to \iota!M_{|U} \to M \to M/\iota!M_{|U} \to 0$$

in which the support of $M/\iota!M_{|U}$ is strictly smaller than the support of $M$, so that by Noetherian induction and Lemma 2.10 1. again, we finally obtain that $M$ is admissible. $\square$

**Corollary 2.19.** *The natural functor $D^b(M_{\mathrm{ct}}(X)) \to D(X)$ is an equivalence.*

*Proof.* We apply [BBD82, Lemme 3.1.16]. The conditions are verified because of the previous Theorem 2.18. $\square$

# 3 Higher categorical enhancements.

## 3.1 Categorical preliminaries.

Recall the following fundamental definition of Lurie:

**Definition 3.1.** An $\infty$-category $\mathcal{C}$ is stable if it has all finite limits and colimits, if the initial objects coincides with the final object and if any commutative square in $\mathcal{C}$ is cocartesian if and only if it is Cartesian. An exact functor between stable $\infty$-categories is an $\infty$-functor preserving finite limits and colimits.

The homotopy category $\mathrm{ho}(\mathcal{C})$ is always a triangulated category and exact functors induce triangulated functors.

Recall that by [Lura, Tag 01F1], we have the following proposition :

**Proposition 3.2.** Let $F: \mathcal{C} \to \mathcal{D}$ be a functor of $\infty$-categories. Assume that for a collection $A \subset \mathcal{C}$ of objects of $\mathcal{C}$ is given the data, for each $a \in A$ of an isomorphism $f_a: F(a) \to d_a$ with $d_a \in \mathcal{D}$. Then there exist a functor $G: \mathcal{C} \to \mathcal{D}$ and a natural equivalence $g: F \Rightarrow G$ such that for every $a \in A$, we have $g_a = f_a$ (and this is really an equality).

*Proof.* Indeed, denote by $\iota: A \to \mathcal{C}$ the canonical monomorphism of simplicial sets. Then by [Lura, Tag 01F1], the restriction

$$\iota^*: \mathrm{Fun}(\mathcal{C}, \mathcal{D}) \to \mathrm{Fun}(A, \mathcal{D}) \qquad (3.2.1)$$

is an isofibration of simplicial sets (cf. [Cis19, 3.3.15].) The data of the $f_a$ defines an isomorphism $\gamma$ in $\mathrm{Fun}(A, \mathcal{D})$ between $F|_A$ and the map of simplicial sets $A \to \mathcal{D}$ defined by $a \mapsto d_a$. By the lifting property of isofibrations, there is a functor $G: \mathcal{C} \to \mathcal{D}$ and an isomorphism $F \simeq G$, that is a natural equivalence $g: F \Rightarrow G$ such that $\iota^*g = \gamma$, which is exactly the statement of the proposition. $\square$
**Definition 3.3.** The bounded derived $\infty$-category of an abelian category $\mathcal{A}$ is the $\infty$ category $\mathcal{D}^b(\mathcal{A})$ obtained by inverting quasi-isomorphisms in the category $\mathrm{Ch}^b(\mathcal{A})$. It is a stable $\infty$-category with a t-structure, and its homotopy category is the usual derived category $\mathrm{D}^b(\mathcal{A})$.

We will use the following result of Bunke, Cisinski, Kasprowski, and Winges:

**Theorem 3.4.** Let $\mathcal{C}$ be a stable $\infty$-category and let $\mathcal{A}_1, \dots, \mathcal{A}_n$ be abelian categories. Denote by $\mathrm{Fun}^{\mathrm{nex}}(\prod_i \mathcal{D}^b(\mathcal{A}_i) \to \mathcal{C})$ the category of $n$-multi-exact functors $\prod_i \mathcal{D}^b(\mathcal{A}_i) \to \mathcal{C}$ that are exact in each variables (meaning that if one fixes one variable then we obtain an $(n-1)$-multi-exact functor). Denote also by $\mathrm{Fun}^{\mathrm{II,nex}}(\prod_i \mathcal{A}_i \to \mathcal{C})$ the category of finite coproduct preserving $n$-multi-exact functors, that is functors $\prod_i \mathcal{A}_i \to \mathcal{C}$ that are finite coproduct preserving and that in each variable $i$ send short exact sequences $0 \to x \to y \to z \to 0$ to cocartesian squares

$$ \begin{array}{ccc} f(x) & \longrightarrow & f(y) \\ \downarrow & & \downarrow \\ 0 & \longrightarrow & f(z) \end{array} $$

(so fixing one variable gives a coproduct preserving $(n-1)$-multi-exact functor). Then the restriction to the hearts functor

$$ \mathrm{Fun}^{\mathrm{nex}}(\prod_i \mathcal{D}^b(\mathcal{A}_i) \to \mathcal{C}) \to \mathrm{Fun}^{\mathrm{II,nex}}(\prod_i \mathcal{A}_i \to \mathcal{C}) \qquad (3.4.1) $$

is an equivalence of $\infty$-categories.

*Proof.* We give the proof for $n=2$, the general case being strictly the same except for a too large number of indices. Recall that we have canonical equivalence of $\infty$-categories $\mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_1) \times \mathcal{D}^b(\mathcal{A}_2), \mathcal{C}) \simeq \mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_1), \mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_2), \mathcal{C}))$. By definition, the $\infty$-category $\mathrm{Fun}^{2\mathrm{ex}}(\mathcal{D}^b(\mathcal{A}_1) \times \mathcal{D}^b(\mathcal{A}_2) \to \mathcal{C})$ consists exactly of functors that are sent in $\mathrm{Fun}^{\mathrm{ex}}(\mathcal{D}(\mathcal{A}_1), \mathrm{Fun}^{\mathrm{ex}}(\mathcal{D}^b(\mathcal{A}_2), \mathcal{C}))$ by that equivalence. By [BCKW24, Corollary 7.4.12] the latter category is exactly $\mathrm{Fun}^{\mathrm{II,ex}}(\mathcal{A}_1, \mathrm{Fun}^{\mathrm{II,ex}}(\mathcal{A}_2, \mathcal{C}))$, which again is equivalent to $\mathrm{Fun}^{\mathrm{II,2ex}}(\mathcal{A}_1 \times \mathcal{A}_2 \to \mathcal{C})$. This finishes the proof. $\square$

We will mostly use the particular case of $n=1$:

**Corollary 3.5 ([BCKW24]).** Let $\mathcal{C}$ be a stable $\infty$-category and let $\mathcal{A}$ be an abelian categories. Then restriction to the heart gives an equivalence of $\infty$-categories

$$ \mathrm{Fun}^{\mathrm{ex}}(\mathcal{D}^b(\mathcal{A}), \mathcal{C}) \to \mathrm{Fun}^{\mathrm{II,ex}}(\mathcal{A}, \mathcal{C}). $$

We will also need a result on the monoidal structure of the bounded derived category.
**Proposition 3.6.** Let $\mathcal{A}$ be an abelian category endowed with a tensor product which is exact in each variable. Then there is a canonical symmetric monoidal structure on the stable $\infty$-category $\mathcal{D}^b(\mathcal{A})$ such that the inclusion functor $\mathcal{A} \to \mathcal{D}^b(\mathcal{A})$ is symmetric monoidal. Moreover, for any category $I$ and any functor $\mathcal{A} : I \to \text{SymMono}_1$ of such abelian categories with exact symmetric monoidal transitions, there is a canonical lift of the functor $\mathcal{D}^b \circ \mathcal{A} : I \to \text{Cat}_\infty$ to a functor $I \to \text{CAlg}(\text{Cat}_\infty)$.

*Proof.* Let $\mathcal{A} : I \to \text{SymMono}_1$ be a diagram of symmetric monoidal abelian categories and symmetric monoidal exact functors. We can compose this functor with $\text{Ch}^b(-)$ to obtain a diagram of symmetric monoidal additive categories (the monoidal structure on $\text{Ch}^b(\mathcal{A}(i))$ is induced by that of $\mathcal{A}(i)$.) By Barkan's [Bar22, Corollary A] or Lurie's [Lur22, Corollary 5.1.1.7], the forgetful functor induces an equivalence of 2-categories

$$ \text{CAlg}(\text{Cat}_1) \to \text{SymMono}_1 $$

thus we can see $\text{Ch}^b \circ \mathcal{A}$ as a functor

$$ I \to \text{CAlg}(\text{Cat}_1) \subset \text{CAlg}(\text{Cat}_\infty). $$

Now by [Lur22, Theorem 2.4.3.18 and Proposition 2.4.2.5] the data of this functor is equivalent to the data of a $I^{\text{II}}$-monoid, which is classified by a cocartesian fibration $p: \mathcal{C} \to I^{\text{II}}$ (we denote by $I^{\text{II}}$ the $\infty$-operad constructed in [Lur22, Construction 2.4.3.1]). By [Hin16, Proposition 2.1.4] we can fiberwise invert quasi-isomorphisms to obtain a cocartesian fibration $q: \mathcal{D} \to I^{\text{II}}$ which classifies (using unstraightening<fnref n="1" /> and [Lur22, Theorem 2.4.3.18, Proposition 2.4.2.5] again) a functor $I \to \text{CAlg}(\text{Cat}_\infty)$ sending an object $i \in I$ to $\mathcal{D}^b(\mathcal{A}(i))$. The fact that the functor $\mathcal{A}(i) \to \mathcal{D}^b(\mathcal{A}(i))$ is symmetric monoidal comes from the exactness of the tensor product on $\mathcal{A}(i)$. $\square$

We also have a version of the universal property of $\mathcal{D}^b(\mathcal{A})$ (see Corollary 3.5) that works in families.

**Proposition 3.7.** Let $\mathcal{C} : I \to \text{CAlg}(\text{Cat}_\infty)$ be a diagram of stably symmetric monoidal $\infty$-categories such that for each $i \in I$ the $\infty$-category $\mathcal{C}(i)$ has a $t$-structure such that the tensor product is $t$-exact in each variable and every arrow $i \to j$ in $I$ induces a $t$-exact functor. Then the canonical functors $\mathcal{D}^b(\mathcal{C}(i)^\diamond) \to \mathcal{C}(i)$ assemble to give a natural transformation $\mathcal{D}^b(\mathcal{C}^\diamond) \Rightarrow \mathcal{C}$ of functors $I \to \text{CAlg}(\text{Cat}_\infty)$.

*Proof.* For $\mathcal{B}$ an $\infty$-category, denote by $\mathcal{K}^b(\mathcal{B})$ the $\infty$-category of bounded complexes in $\mathcal{B}$, that is the $\infty$-categorical localisation of the additive category of bounded chain complexes $\text{Ch}^b(\mathcal{B})$ with respect to chain homotopies. By the work of Bunke, Cisinski, Kasprowski and Winges, we know ([BCKW24, Theorem 7.4.0]) that $\mathcal{K}^b(\mathcal{B})$ is the value at $\mathcal{B}$ of a functor

$$ \mathcal{K}^b(-): \text{Cat}_\infty^{\text{add}} \to \text{Cat}_\infty^{\text{ex}} $$

<footnote n="1">¹One has to verify that the functor $I^{\text{II}} \to \text{Cat}_\infty$ obtained by unstraightening is still a $I^{\text{II}}$-monoid, but this is clear because $\mathcal{D}^b(-)$ commutes with finite products: this is the case for $\text{Ch}^b(-) \subset \text{Fun}(\mathbb{Z}, -)$ and then we invert products of maps.</footnote>
factors through

$$ \mathcal{K} \to \mathcal{K}[W^{-1}]. $$

By [Hin16, Proposition 2.1.4], the map $\mathcal{K}[W^{-1}] \to I^{\Pi}$ is a cocartesian fibration, and moreover, the fiber above a $\tilde{i} = (i_1, \dots, i_n)$ is the localisation of $\prod_{j=1}^n \mathcal{K}^b(\mathcal{A}(i_j))$ with respect to products of quasi-isomorphisms, which is $\prod_{j=1}^n \mathcal{D}^b(\mathcal{A}(i_j))$. Because $I^{\Pi}$ is a 1-category, to check that the diagram

<figure><img src="image_2.png" /></figure>

commutes it suffices by adjunction to check that $\mathrm{Ho}(-)$ of it commutes, which is clear because the horizontal map preserves fibers. Thus after unstraightening we obtain a map of $I^{\Pi}$-monoids

$$ \mathcal{D}^b \circ \mathfrak{A} \to \mathfrak{C} $$

which express exactly the symmetric monoidality of a natural transformation $\mathcal{D}^b \circ \mathfrak{A} \to \mathfrak{C}$, finishing the proof. $\square$

*Remark 3.8.* The proof of Proposition 3.7 also give the same result with $\mathcal{K}^b(-)$ instead of $\mathcal{D}^b(-)$, and the fact that $\mathcal{K}^b(-) \Rightarrow \mathcal{D}^b(-)$ is symmetric monoidal on symmetric monoidal abelian categories and symmetric monoidal exact functors.

We will use the following two theorems :

**Theorem 3.9 (Theorem 3.3.1 of [NRS20])**. Let $\mathcal{D}$ be an $\infty$-category which admits finite limits and let $\mathcal{C}$ be an $\infty$-category. Let $G : \mathcal{D} \to \mathcal{C}$ be a functor which preserves finite limits. Then $G$ admits a left adjoint if and only if $\mathrm{ho}(G) : \mathrm{ho}(\mathcal{D}) \to \mathrm{ho}(\mathcal{C})$ does.

**Corollary 3.10.** Let $\mathcal{C}$ and $\mathcal{D}$ be a stable $\infty$-categories. Let $G : \mathcal{D} \to \mathcal{C}$ be an exact functor. Then $G$ admits a left (resp. a right) adjoint if and only if $\mathrm{ho}(G) : \mathrm{ho}(\mathcal{D}) \to \mathrm{ho}(\mathcal{C})$ does.

*Proof.* The case of the left adjoint is the above theorem, and the case of the right adjoint follows by passing to the opposite $\infty$-categories. $\square$

**Theorem 3.11 (Theorem 7.6.10 of [Cis19])**. Let $F : \mathcal{D} \to \mathcal{C}$ be a functor between $\infty$-categories having finite limits. Assume that $F$ preserves finite limits. Then $F$ is an equivalence of $\infty$-categories if and only if $\mathrm{ho}(F) : \mathrm{ho}(\mathcal{D}) \to \mathrm{ho}(\mathcal{C})$ is an equivalence of categories.

To prove descent statement for categories of perverse Nori motives and mixed Hodge modules we will use the following proposition and its convenient corollary. Robin Carlier really helped the author in the writing of this proposition.
**Proposition 3.12.** Let $\chi : (\mathcal{C}^{\text{op}})^{\triangleleft} \to \text{Cat}_{\infty}$ be a functor. For any $f : c \to c'$ in $(\mathcal{C}^{\text{op}})^{\triangleleft}$, denote $f^* = \chi(f)$, and assume that any such $f^*$ has a right adjoint $f_*$. For each $c \in \mathcal{C}$, denote $f_c : c \to v$ the unique map from $c$ to the end point $v$ (we have that $(\mathcal{C}^{\text{op}})^{\triangleleft} \simeq (\mathcal{C}^{\triangleleft})^{\text{op}}$.) Let $\pi : \overline{\mathcal{D}} \to (\mathcal{C}^{\text{op}})^{\triangleleft}$ be the cocartesian fibration that classifies $\chi$, and let $\pi : \mathcal{D} \to \mathcal{C}^{\text{op}}$ be its pullback by the inclusion $\mathcal{C}^{\text{op}} \to (\mathcal{C}^{\text{op}})^{\triangleleft}$.

Then $\chi$ is a limit diagram if and only if the following two conditions are verified :

1. The family $(f_c^*)_c \in \mathcal{C}$ is conservative.

2. For any cocartesian section $X : \mathcal{C}^{\text{op}} \to \mathcal{D}$ of $\pi$, the limit $X(v) := \lim_{c \in \mathcal{C}} (f_c)_* X(c)$ exists in $\chi(v)$ and the map $f_c^* X(v) \to X(c)$ adjoint to the canonical map

$$ \lim_{c} (f_c)_* X(c) \to (f_c)_* X(c) $$

is an equivalence.

*Proof.* By [Lura, Construction 02T0], there is a diffraction functor

$$ \mathrm{Df} : \overline{\mathcal{D}}_v \to \mathrm{Fun}_{\mathcal{C}^{\text{op}}}^{\mathrm{Cocart}}(\mathcal{C}^{\text{op}}, \mathcal{D}) \qquad (3.12.1) $$

that fits in a commutative triangle :

$$ \begin{tikzcd} \overline{\mathcal{D}}_v \arrow[r, "\mathrm{Df}"] \arrow[d, "\mathrm{ev}_v"] & \mathrm{Fun}_{\mathcal{C}^{\text{op}}}^{\mathrm{Cocart}}(\mathcal{C}^{\text{op}}, \mathcal{D}) \arrow[r, "\mathrm{res}", "(3.12.2)"'] & \\ \mathrm{Fun}_{(\mathcal{C}^{\text{op}})^{\triangleleft}}^{\mathrm{Cocart}}((\mathcal{C}^{\text{op}})^{\triangleleft}, \overline{\mathcal{D}}) & & \end{tikzcd} $$

By the diffraction criterion [Lura, Theorem 02T8], $\chi$ is a limit diagram if and only if the restriction morphism in (3.12.2) is an equivalence. In that case by [Lura, Remark 02TF], the functor Df is an equivalence.

Assume that for any cocartesian section $X : \mathcal{C}^{\text{op}} \to \mathcal{D}$ of $\pi$, the functor $c \mapsto (f_c)_* X(c)$ admits a limit in $\overline{\mathcal{D}}_v$. By the preservation of limits of any right adjoint in $(\mathcal{C}^{\text{op}})^{\triangleleft}$ and [Lura, Corollary 0311 and Corollary 02KY], this is equivalent to saying that for any cocartesian section $X : \mathcal{C}^{\text{op}} \to \mathcal{D}$ of $\pi$, the lifting problem

$$ \begin{tikzcd} \mathcal{C}^{\text{op}} \arrow[r, "X"] \arrow[d, "i"] & \overline{\mathcal{D}} \arrow[d, "\pi"] \\ (\mathcal{C}^{\text{op}})^{\triangleleft} \arrow[d, "\sim"] & (\mathcal{C}^{\text{op}})^{\triangleleft} \end{tikzcd} \qquad (3.12.3) $$

admits a solution $\overline{X}$ which is a $\pi$-limit ([Lura, Definition 02KG].) By [Lura, Example 02ZA], in that case $\overline{X}$ is the right Kan extension functor along the inclusion $i : \mathcal{C}^{\text{op}} \to (\mathcal{C}^{\text{op}})^{\triangleleft}$. This easily implies that the restriction functor $\mathrm{res} : \mathrm{Fun}_{(\mathcal{C}^{\text{op}})^{\triangleleft}}^{\mathrm{Cocart}}((\mathcal{C}^{\text{op}})^{\triangleleft}, \overline{\mathcal{D}}) \to \mathrm{Fun}_{\mathcal{C}^{\text{op}}}^{\mathrm{Cocart}}(\mathcal{C}^{\text{op}}, \mathcal{D})$ is fully faithful as the left adjoint of the right Kan extension along $i : \mathcal{C}^{\text{op}} \to (\mathcal{C}^{\text{op}})^{\triangleleft}$.
Therefore, assuming point 2. of the proposition, $\chi$ is a limit diagram if and only if the restriction functor is conservative by [Lura, Corollary 03UZ]. But as the family of evaluations $ev_c : \text{Fun}(\mathcal{C}^{\text{op}}, \mathcal{D}) \to \mathcal{D}$ is conservative, this is equivalent to the family of functors $ev_c \circ \text{res} = ev_c \circ \text{Df} \circ ev_v = f_c^*$ being conservative, which is point 1. of the proposition.

By (3.12.2), if $\chi$ is a limit diagram, $\text{Df}$ is an equivalence which implies that 1. holds. To finish the proof of the proposition, we therefore only have to show that if $\chi$ is a limit diagram, then 2. holds. Let $\bar{X}$ be a cocartesian section of $\bar{\pi}$. Assume first that $\bar{X}$ is a $\bar{\pi}$-limit diagram. By [Lura, Example 03ZA] this is equivalent to suppose that $\bar{X}$ is the $\bar{\pi}$-right Kan extension of its restriction $X$ to $\mathcal{C}^{\text{op}}$. Then by [Lura, Corollary 0307], the limit of $c \mapsto (f_c)_*X(c)$ exists and the fact that $\bar{X}$ is cocartesian is equivalent to the fact that for every $c \in \mathcal{C}$, the canonical map

$$f_c^*(\lim_{c'} (f_{c'})_* X(c)) \to X(c) \qquad (3.12.4)$$

is an equivalence.

Therefore, to show that 2. holds, we only have to show that any cocartesian section $\bar{X}$ of $\bar{\pi}$ is a $\bar{\pi}$-limit. This is exactly what is proved in the second paragraph of the proof of [Lur11, Theorem 5.17]. □

**Corollary 3.13** (See also [Lur22, Corollary 5.2.2.37]). Let $\chi, \chi' : (\mathcal{C}^{\text{op}})^{\triangleleft} \to \text{Cat}_{\infty}$ be two functors as in Proposition 3.12, and assume one has a natural transformation $B : \chi \to \chi'$ which commutes with the rights adjoints. Suppose also that the limits of point 2. in Proposition 3.12 exist for $\chi$, that $B_v : \chi(v) \to \chi'(v)$ preserves them and that each $B_c$ for $c \in \mathcal{C}$ is conservative. Then if $\chi'$ is a limit diagram, so is $\chi$.

## 3.2 Enhancement of the 6 functors formalism.

In this subsection, we will work with perverse Nori motives or mixed Hodge modules identically. Therefore we denote by $\text{Var}_k$ the category of quasi-projective $k$-schemes when dealing with perverse Nori motives or the category of separated and reduced finite type $\mathbb{C}$-schemes when dealing with mixed Hodge structures. We will call elements of $\text{Var}_k$ varieties or $k$-varieties. For $X \in \text{Var}_k$, we will denote by $\mathcal{M}_{\text{perv}}(X)$ the category of perverse Nori motives constructed in [IM22] or the category of Saito's mixed Hodge modules constructed in [Sai90]. We will have to use realisation functors hence we denote by $\text{rat} : \text{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \text{D}_c^b(X)$ either the Betti realisation of perverse Nori motives when $k \subset \mathbb{C}$, the underlying $\mathbb{Q}$-structure functor of mixed Hodge modules, or the $\ell$-adic realisation of perverse Nori motives. Here $\text{D}_c^b(X)$ is the derived category of bounded cohomologically constructible sheaves on $X^{\text{an}}$ in the two first cases, and the derived category of constructible $\ell$-adic étale sheaves on $X$. The triangulated category $\text{D}_c^b(X)$ is endowed with a perverse and a constructible t-structure for which $\text{rat}$ is t-exact and conservative.
We begin with a recollection on the $\infty$-categorical enhancement of derived categories of constructible sheaves.

**Proposition 3.14.** *There exists a functor*

$$ \mathcal{D}_c^b: \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty) $$

such that for each map $f: Y \to X$ in $\mathrm{Var}_k$, the symmetric monoidal induced by $\mathcal{D}_c^b(f)$ on the homotopy categories is the classical pullback of constructible sheaves.

*Proof.* In both analytic sheaves an étale sheaves, for $X \in \mathrm{Var}_k$, the category $\mathcal{D}_c^b(X)$ is embedded in a bigger category $\mathcal{D}(X)$ which is of the form $\mathcal{D}(\mathrm{Shv}(\mathcal{X}, \mathcal{O}))$ with $\mathcal{X}$ a topos and $\mathcal{O}$ a sheaf of rings on $\mathcal{X}$ and is naturally a symmetric monoidal stable $\infty$-category. Indeed, in the case of analytic sheaves one can take $\mathcal{X}$ to be the categories of analytic opens of the complex points of $X$ and $\mathcal{O}$ is the constant sheaf $\mathbb{Q}_X$. In the case of $\ell$-adic sheaves, following [HRS23] one can take $\mathcal{X}$ to be the proétale topos of $X$, and $\mathcal{O}$ to be the sheaf of ring induced by the condensed ring $\mathbb{Q}_\ell = \mathbb{Z}_\ell[1/\ell]$. This construction has a canonical lift to the world of symmetric monoidal $\infty$-categories thanks to [Lurb, Corollary 2.1.2.3]. As the constructions $X \mapsto \mathcal{X}$ and $\mathcal{X} \mapsto \mathcal{D}(\mathrm{Shv}(\mathcal{X}, \mathcal{O}))$ are functorial, we obtain a functor

$$ \mathcal{D}: \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty). $$

Now as for each $f$ in $\mathrm{Var}_k$, the functor $f^*: \mathcal{D}(Y) \to \mathcal{D}(X)$ preserves constructible objects, this induces a functor

$$ \mathcal{D}_c^b: \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty) $$

which gives the usual pullback on homotopy categories by definition in the case of analytic sheaves, and by [HRS23, Theorem 7.7] for $\ell$-adic sheaves. $\square$

**Construction 3.15.** We have a symmetric monoidal homotopy 2-functor

$$ \mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(-)) : \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_1). $$

By restricting to the constructible hearts, this induces a functor

$$ \mathcal{M}_{\mathrm{ct}}(-) : \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_1) $$

such that the tensor product is exact in each variable and the pullback functors are exact (this can be checked after realisation.) By applying Proposition 3.6 we obtain a functor

$$ \mathcal{D}^b(\mathcal{M}_{\mathrm{ct}}(-)) : \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty). $$

Moreover, the equivalence Corollary 2.19 lifts to an equivalence of $\infty$-categories thanks to Theorem 3.11. The Proposition 3.2 then gives a functor

$$ \mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(-)) : \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty), $$

canonically equivalent to $\mathcal{D}^b(\mathcal{M}_{\mathrm{ct}}(-))$.
**Proposition 3.16.** Let $f: Y \to X$ be a map in $\mathrm{Var}_k$. Then the functor induced by $\mathcal{D}^b(\mathcal{M}_{\mathrm{ct}}(f))$ on the homotopy categories is the pullback functor constructed by Ivorra and Morel or by Saito, depending on which case we are.

*Proof.* We claim that if we know that the pullback functors constructed by Ivorra, Morel and Saito admit $\infty$-categorical lifts $f^*$, then they will be canonically equivalent to $f_{\mathrm{new}}^* := \mathcal{D}^b(\mathcal{M}_{\mathrm{ct}}(f))$. Indeed in that case the $\infty$-functors $f^*$ and $f_{\mathrm{new}}^*$ coincide on the constructible heart by definition, hence they are equivalent by Corollary 3.5.

We prove this first for mixed Hodge modules: For any map $f: Y \to X$ of varieties, there is the factorisation

$$Y \xrightarrow{i} X \times Y \xrightarrow{p} X$$

where $i$ is the closed immersion given by the inclusion of the graph of $f$ (our varieties are separated) and $p$ is the projection. By definition, we have $f^* \simeq p^* \circ i^*$. The functor $p^*$ is given by the external product with the constant object: $p^*(M) = M \boxtimes_{\mathbb{Q}_Y} M$, hence is an $\infty$-functor because the external product is defined as the derived functor on the perverse hearts. The functor $i^*$ is the left adjoint of the functor $i_*$, which is obtained ([Sai90, (4.2.4) and (4.2.10)]) by deriving the functor $i_*$ on the perverse heart and is therefore an $\infty$-functor. Thus by Corollary 3.10 the functor $i^*$ is an $\infty$-functor. Finally $f^*$ admits an $\infty$-categorical lift.

Now for perverse Nori motives, any map $f: Y \to X$ being a map of quasi-projective varieties, factors as

$$Y \xrightarrow{i} Z \xrightarrow{g} X$$

with $i$ a closed immersion and $g$ a smooth map. Hence $f^* \simeq i^*g^*$. The case of $i^*$ is the same as for mixed Hodge modules, and thus we have an $\infty$-functor. For the smooth map $g$, we can assume that $X$ is connected (as a direct sum of $\infty$-functors is an $\infty$-functor), so that $g$ is smooth of pure relative dimension $d$ for some $d \in \mathbb{N}$. Now by definition $f^* = (f^*[d])[-d]$ where $f^*[d]$ is the derived functor of the exact functor $f^*[d]$ on the perverse heart, hence we have an $\infty$-functor. $\square$

**Corollary 3.17.** Let $f: Y \to X$ be a map in $\mathrm{Var}_k^{\mathrm{op}}$, then the $\infty$-functor

$$f^*: \mathcal{D}(\mathcal{M}_{\mathrm{perv}}(X)) \to \mathcal{D}(\mathcal{M}_{\mathrm{perv}}(Y))$$

admits a right adjoint $f_*$. If $f$ is moreover smooth, $f^*$ admits a left adjoint $f_*$. □

*Proof.* These adjoints exist on the homotopy categories, thus they have $\infty$-categorical lifts by Corollary 3.10. $\square$

**Proposition 3.18.** For each $X \in \mathrm{Var}_k$, the monoidal structure obtained in Construction 3.15 induces on the homotopy categories the same monoidal structure as the one constructed by M. Saito for mixed Hodge modules and by Terenzi ([Ter24]) for perverse Nori motives.

Moreover, the projection formula for a smooth morphism holds in $\mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X))$.
Proof. Note that in both [Ter24] and [Sai90] the tensor product is built in the following way : there is an external tensor product $\boxtimes : \mathcal{M}_{\text{perv}}(X) \times \mathcal{M}_{\text{perv}}(X) \to \mathcal{M}_{\text{perv}}(X \times X)$ that is exact in both variable hence induces by Theorem 3.4 an $\infty$-functor $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \times \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X \times X))$ which we compose with the pullback $\Delta^*$ under the diagonal $\Delta : X \to X \times X$ to obtain the tensor product, thus given by $M \otimes N = \Delta^*(M \boxtimes N)$. The other structure morphism of the monoidal structure are obtained similarly. Denote by $\otimes^T : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \times \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ Saito's or Terenzi's tensor product and denote by $\otimes^\infty : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \times \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ the one we just constructed. We want to check that $\text{ho}(\otimes^\infty) = \text{ho}(\otimes^T)$. For this there are several isomorphisms to check (see [ML98, Section XI.1]) :

1. A functorial isomorphism $\otimes^T \Rightarrow \otimes^\infty$.

2. Modulo point 1. isomorphism, an identification of the unit isomorphism $\rho : M \otimes 1 \to M$ and $\lambda : 1 \otimes M \to M$.

3. Modulo point 1. isomorphism, an identification of the associativity isomorphism $c : M \otimes (N \otimes P) \to (M \otimes N) \otimes P$.

4. Modulo point 1. isomorphism, an identification of the commutativity isomorphism $\gamma : M \otimes N \to N \otimes M$.

By definition, $\otimes^\infty$ coincides with $\otimes^T$ when restricted to $\mathcal{M}_{\text{ct}}(X) \times \mathcal{M}_{\text{ct}}(X)$. By Theorem 3.4 this gives 1. We explain how to obtain 3. and the other checks will be left to the reader as they are very similar. Denote by $t_1^\infty : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \times \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \times \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ the functor sending $M, N, P$ to $M \otimes^\infty (N \otimes^\infty P)$ and by $t_2^\infty : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \times \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \times \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ the functor sending $M, N, P$ to $(M \otimes^\infty N) \otimes^\infty P$. The associativity isomorphism is a natural equivalence $c^\infty : t_1^\infty \Rightarrow t_2^\infty$. We give the same definition for $t_1^T, t_2^T$ and $c^T$. Again by definition, the image by the restriction functor of the morphisms $c^\infty$ and $c^T$ are the same, hence up to the equivalence of 1., we obtain 3.

For the projection formula, the arrow exists by a play of adjunctions, and is an equivalence on the homotopy categories. $\square$

**Corollary 3.19.** Let $X$ be a variety. The tensor structure on $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ is closed.

*Proof.* Again this is true on the homotopy categories and we can apply Corollary 3.10. $\square$

**Proposition 3.20.** Let $X$ be a variety. The realisation functor

$$R : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}_c^b(X)$$

admits a canonical lift to a map in $\text{CAlg}(\text{Cat}_\infty)$.
Moreover, it induces a natural transformation

$$ \mathcal{D}^b(\mathcal{M}_{\text{perv}}(-)) \Rightarrow \mathcal{D}_c^b(-) $$

on functors

$$ \text{Var}_k^{\text{op}} \rightarrow \text{CAlg}(\text{Cat}_{\infty}). $$

*Proof.* In the case of the Betti realisation this is easy as we have an equivalence

$$ \mathcal{D}^b(\text{Cons}(X)) \simeq \mathcal{D}_c^b(X) $$

thanks to Nori's theorem. This enables us to consider the functor $\Delta^1 \times \text{Var}_k^{\text{op}} \rightarrow \text{CAlg}(\text{Cat}_1)$ that send $(0, f)$ to $f^* : \mathcal{M}_{\text{ct}}(Y) \rightarrow \mathcal{M}_{\text{ct}}(X)$, the edge $(1, f)$ to $f^* : \text{Cons}(Y) \rightarrow \text{Cons}(X)$ and $(0 \rightarrow 1, X)$ to the realisation. Therefore by Proposition 3.6 we obtain the result.

For the $\ell$-adic realisation it is not true that the canonical functor

$$ \text{real}_X : \mathcal{D}^b(\text{Cons}(X, \mathbb{Q}_\ell)) \rightarrow \mathcal{D}_c^b(X, \mathbb{Q}_\ell) $$

is an equivalence. What we did in the case of the Betti realisation gives a symmetric monoidal natural transformation

$$ \mathcal{D}^b(\mathcal{M}_{\text{ct}}(-)) \Rightarrow \mathcal{D}^b(\text{Cons}(-, \mathbb{Q}_\ell)). $$

Hence we have to check that the realis ensemble to give a symmetric monoidal natural transformation

$$ \mathcal{D}^b(\text{Cons}(-, \mathbb{Q}_\ell)) \Rightarrow \mathcal{D}_c^b(-). $$

This is Proposition 3.7. $\square$

**Corollary 3.21.** *The realisation functor*

$$ \mathcal{R}: \mathcal{D}^b(\mathcal{M}_{\text{perv}}(-)) \rightarrow \mathcal{D}_c^b(-) $$

commutes with all the operations constructed above.

*Proof.* Indeed, this is true on the homotopy categories, and the exchange morphisms exist because $\mathcal{R}$ commutes with pullbacks and tensor products by definition. $\square$

For free, we also obtain the exceptional functoriality:

**Corollary 3.22.** *The exceptional functors $f^!$ and $f_!$ constructed by Ivorra, Morel and Saito have $\infty$-categorical lifts that assemble to give functors*

$$ \text{Sch}_k^{(\text{op})} \rightarrow \text{Cat}_{\infty}. $$

*Moreover Verdier duality is a natural transformation between the star functoriality ($f_*$ or $f^*$) and the shriek functoriality of the opposite category ($f_!$ or $f^!$).*
Proof. The last statement of this corollary gives a construction. Indeed the formula $\mathbb{D} \circ f_* \simeq f_*\mathbb{D}$ tells us that if we apply Proposition 3.2 to the isomorphisms $\mathbb{D}_X : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))^{\text{op}}$, the obtained functor $\text{Sch}^{(\text{op})} \to \text{Cat}_\infty$ encode exactly the exceptional functoriality. $\square$

**Theorem 3.23.** The functor $X \mapsto \mathcal{D}_M^b(X)$, with respect to the $(-)^*$-functoriality, are hypersheaves with respect to $h$-topology.

Proof. By [CD19, Corollary 3.3.38] (or see the proof of [BM21, Proposition 5.11]), it suffices to prove that this functor is an étale sheaf because of localisation and proper base change hold for $\mathcal{D}_M^b$. We will use Corollary 3.13 with a realisation functor (either the $\ell$-adic realisation for Nori motives, or the underlying $\mathbb{Q}$-structure functor for mixed Hodge modules) that we will denote by

$$B: \mathcal{D}_M^b \to \mathcal{D}_c^b.$$

Note that $\mathcal{D}_c^b$ affords étale descent. Now, to apply Corollary 3.13 we need to ensure that the limits ensuring descent exist in $\mathcal{D}_M^b$. This is not obvious. However, for any variety $X$ and $m \in \mathbb{N}$, if we denote by $\mathcal{D}_M^{[-m,m]}(X)$ the full subcategory of $\mathcal{D}_M^b(X)$ of objects having concentrated, for the constructible t-structure, in degrees $[-m, m]$, we have

$$\bigcup_n \mathcal{D}_M^{[-m,m]}(X) = \mathcal{D}_M^b(X).$$

Moreover, the functor $\mathcal{D}_M^b$ is a hyper sheaf if and only if for each $m$, the functor $\mathcal{D}_M^{[-m,m]}$ is a hyper sheaf, and the same holds for $\mathcal{D}_c^b$. Indeed, if $\mathcal{D}_M^b$ is a hyper sheaf, then as being concentrated in degrees $[-m, m]$ can be tested after pulling back by any surjective map of schemes, it turns out that each $\mathcal{D}_M^{[-m,m]}$ is also a hyper sheaf. Conversely, if each $\mathcal{D}_M^{[-m,m]}$ is an hyper sheaf, then because pullback functors $f^*$ are t-exact functors, in turns out that the conditions to test that $\mathcal{D}_M^b$ is a sheaf only take place in $\mathcal{D}_M^{[-m,m]}$.

Thus because $\mathcal{D}_c^{[-m,m]}$ is an étale hyper sheaf, it suffices to check the conditions of Corollary 3.13 for the realisation functor

$$B: \mathcal{D}_M^{[-m,m]} \to \mathcal{D}_c^{[-m,m]}.$$

Let $f_*: Y_* \to X$ be an étale hyper covering. As the realisation is conservative, it suffices to check that:

1. For each complex $M$ in $\mathcal{D}_M^{[-m,m]}(X)$, the limit

$$\lim_{\Delta} \tau^{\leq m} (f_n)_* f_n^* M$$

exists, where $\Delta$ is the simplicial category, where $\tau^{\leq m}$ is the truncation functor for the t-structure.
## 2. The realisation

$$ \mathcal{D}_{\mathcal{M}}^{[-m,m]}(X) \to \mathcal{D}_{c}^{[-m,m]}(X) $$

commutes with such a limit.

In fact as $\mathcal{D}_{\mathcal{M}}^{[-m,m]}(X)$ is a $(2m+1)$-category, the above totalisation in (1) is finite thus exists (because the finite limit $\lim(f_n)_*f_n^*M$ exists in $\mathcal{D}_{\mathcal{M}}^{> -m}$ because it exists in $\mathcal{D}_{\mathcal{M}}^b$, and then we apply the right adjoint $\tau^{\leq m}$), and the realisation clearly commutes with such a finite limit because the realisation

$$ \mathcal{D}_{\mathcal{M}}^{> -m}(X) \to \mathcal{D}_{c}^{> -m}(X) $$

commutes with finite limits (the inclusion $\mathcal{D}^{> -m} \to \mathcal{D}^b$ commutes with limits) and the image by the realisation of the above limit in (1) is the truncation by $\tau^{\leq m}$, which is a right adjoint, of a limit in $\mathcal{D}_{c}^{> -m}(X)$. $\square$

# 4 The realisation functor and applications.

## 4.1 Construction of the realisation.

In this section, we construct realisation functors from the category of étale motives to the derived categories of mixed Hodge modules and perverse Nori motives that commute with the 6 operations. We use Drew and Gallauer result in [DG22] in which they prove that the stable motivic $\mathbb{A}^1$-invariant $\infty$-category $SH$ affords a universal property among systems of $\infty$-categories called coefficient systems: these are functors

$$ \mathrm{Sch}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty) $$

that have the minimal functoriality needed for the 6 operations to exist thanks to Ayoub's thesis (see Appendix A for a precise definition and some results on coefficient systems.) Any coefficient system $C \in \mathrm{CoSys}_k$ receives an unique realisation

$$ \mathcal{SH}_c \to C $$

from the compact objects of the motivic homotopy category (one may need $C$ to take values in idempotent complete $\infty$-categories) that commutes with enough operations to ensure that it commutes with the 6 operations (if $C$ has some good properties, see below).

**Notation 4.1.** For $X$ a separated non reduced finite type $k$-scheme we define

$$ \mathcal{D}^b(\mathrm{MHM}(X)) := \mathcal{D}^b(\mathrm{MHM}(X_{\mathrm{red}})). $$

By localisation for mixed Hodge modules this is well defined. We also denote by $\mathcal{D}_{\mathcal{M}}^b$ the functor $\mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_\infty)$ of Construction 3.15.
**Theorem 4.2.** The functor $\mathcal{D}_{\mathcal{M}}^b$ defines an object of CoSys$_k$.

*Proof.* This is the content of Section 3.2 and Proposition A.4. See Definition A.1 for the precise definition of coefficient systems. □

The above theorem contains the statement that perverse Nori motives and mixed Hodge modules extend to every finite type k-scheme, together with all the operations and h-hyperdescent.

*Remark 4.3.* Using that the mapping spectra in $\mathcal{D}_{\mathcal{M}}^b$, $\mathcal{D}^b(\mathcal{M}_{\text{perv}})$ and $\mathcal{D}^b(\mathcal{M}_{\text{ct}})$ are étale sheaves (for the perverse heart, one has to use affine étale morphisms), one can show that after extension to finite type k-schemes, we still have

$$ \mathcal{D}_{\mathcal{M}}^b(X) \simeq \mathcal{D}^b(\mathcal{M}_{\text{ct}}(X)) \simeq \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) $$

and in the case of perverse Nori motives, the category $\mathcal{M}_{\text{perv}}(X)$ is the universal category as for quasi-projective schemes. The weight structures also extend to that case.

**Theorem 4.4.** There exists an (essentially unique) morphism of coefficients systems $\mathcal{D}\mathcal{M} \to \text{Ind}\mathcal{D}_{\mathcal{M}}^b$ on the category $\text{Sch}_k$. It restricts to a functor

$$ \nu : \mathcal{D}\mathcal{M}_c \to \mathcal{D}_{\mathcal{M}}^b. \qquad (4.4.1) $$

The functor (4.4.1) commutes with the 6 operations, that is all the exchange transformations are isomorphisms.

*Proof.* Thanks to Theorem 4.2 we know that $\mathcal{D}_{\mathcal{M}}^b$ is an object of CoSys$_k$, hence its indization $\text{Ind}\mathcal{D}_{\mathcal{M}}^b$ is an object of CoSys$_k$. Together with Theorem A.2, this gives a morphism of coefficient systems

$$ \mathcal{SH} \to \text{Ind}\mathcal{D}_{\mathcal{M}}^b. $$

As $\text{Ind}\mathcal{D}_{\mathcal{M}}^b$ is $\mathbb{Q}$-linear, this morphism of coefficient systems factors through $\mathcal{SH} \to \mathcal{SH}_{\mathbb{Q}}$ to give a morphism of coefficient systems

$$ \mathcal{SH}_{\mathbb{Q}} \to \text{Ind}\mathcal{D}_{\mathcal{M}}^b. $$

Note that this is in fact a morphism of pullback formalism in the sense of [DG22, Definition 2.2], and as $\text{Ind}\mathcal{D}_{\mathcal{M}}^b$ satisfies non effective étale hyperdescent, this map factors through the étale hyper-sheafification of $\mathcal{SH}_{\mathbb{Q}}$, which is $\mathcal{D}\mathcal{M}$ by [CD19, Theorem 16.2.13], [CD16, Theorem 5.2.2 and Corollary 5.5.7], to give a morphism of coefficient systems

$$ \tilde{\nu} : \mathcal{D}\mathcal{M} \to \text{Ind}\mathcal{D}_{\mathcal{M}}^b. $$

Denote by $\mathcal{D}\mathcal{M}_c(X)$ (resp. by $\mathcal{D}_{\mathcal{M}}^{\text{gm}}(X)$) the thick full subcategory of $\mathcal{D}\mathcal{M}(X)$ (resp. of $\text{Ind}\mathcal{D}_{\mathcal{M}}^b(X)$) spanned by the $f_{\#}\mathbb{Q}_Y(i)$ for $f:Y \to X$ smooth and $i \in \mathbb{Z}$. As $\mathcal{D}_{\mathcal{M}}^b(X) \subset \text{Ind}\mathcal{D}_{\mathcal{M}}^b(X)$ is thick and each $f_{\#}\mathbb{Q}_Y(i)$ is in $\mathcal{D}_{\mathcal{M}}^b(X)$, we have $\mathcal{D}_{\mathcal{M}}^{\text{gm}}(X) \subset \mathcal{D}_{\mathcal{M}}^b(X)$.
Moreover, $\tilde{\nu}$ commutes with all the $f^*$, $f_\sharp$ and is monoidal. Note that the Tate twist $M(i) = M \otimes \mathbb{Q}_X(1)$ is tensorisation by $\mathbb{Q}_X(1) \simeq \text{fib}(p_\sharp \mathbb{Q}_{\mathbb{P}_X^1} \to \mathbb{Q}_X)[-2]$ with $p : \mathbb{P}_X^1 \to X$ and the map $p_\sharp \mathbb{Q}_{\mathbb{P}_X^1} \to \mathbb{Q}_X$ is induced by the inclusion $\infty_X \to \mathbb{P}_X^1$. Therefore, $\tilde{\nu}$ also commutes with Tate twists, and induces a morphism of coefficient systems

$$ \nu^{\text{gm}} : \mathcal{DM}_c \to \mathcal{D}_{\mathcal{M}}^{\text{gm}} \qquad (4.4.2) $$

In particular, at the level of triangulated categories, we obtain a premotivic morphism that verifies all the hypothesis of [CD19, Theorem 4.4.25] : $\mathcal{D}_{\mathcal{M}}^{\text{gm}}$ is $\tau$-generated for the Tate twist, $\mathcal{DM}_c$ is $\tau$-dualisable by absolute purity, $\mathcal{D}_{\mathcal{M}}^{\text{gm}}$ is separated because it is a sub pullback formalism of $\mathcal{D}_{\mathcal{M}}^b$ which satisfy étale descent by Theorem 3.23 (which is equivalent to separateness by [CD19, Corollary 3.3.34] in the $\mathbb{Q}$-linear case) and all Tate twists are invertible in $\mathcal{DM}_c$ by construction.

Therefore, the functor $\nu^{\text{gm}}$ commutes with the 6 operations. One has to be a little careful here: a priori, [CD19, Theorem 4.4.25] implies only that $\nu^{\text{gm}}$ commutes with all the operations when $\mathcal{D}_{\mathcal{M}}^{\text{gm}}$ is endowed with the operations they construct in their book, and not necessarily with the constructions of Saito, Ivorra, Morel as in Corollary 3.22. We know that for $f : Y \to X$ a morphism, $f^*$ is the same for both constructions, hence by adjunction this is also the case for $f_*$.

For $j : U \to X$ an open immersion the $j_\sharp$ are the same, but $j_\sharp = j_!$ hence by Nagata compactification (and the fact that for a proper $p$, $p_* = p_!$) and composition both $f_!$ are isomorphic for any $f : Y \to X$ separated. By adjunction we also have an isomorphism between the $f^!$. This defines a isomorphism of étale sheaves of $\infty$-categories, hence we also have compatibility of operations for non separated morphisms (when they are defined for $\mathcal{DM}$). The tensor product is the same, hence also the internal $\mathcal{Hem}$ by adjunction.

$$ \nu : \mathcal{DM}_c \to \mathcal{D}_{\mathcal{M}}^b \qquad (4.4.3) $$

commute with the 6 operations at the triangulated categories level. But all exchange morphisms exist at the level of $\infty$-categories, and the functor from an $\infty$-category to its homotopy category is conservative, hence $\nu$ commutes with all the operations. $\square$

*Remark 4.5.* As noted by Frédéric Déglise, for the construction of the above realisation one does not actually need to prove directly étale descent as in Theorem 3.23. Indeed as $\text{Ind}\mathcal{D}_{\mathcal{M}}^b$ is a cocomplete coefficient system, as above there is a realisation

$$ \mathcal{SH} \to \text{Ind}\mathcal{D}_{\mathcal{M}}^b $$

which factors through $\mathcal{SH}_{\mathbb{Q}}$ because $\text{Ind}\mathcal{D}_{\mathcal{M}}^b$ is a system of $\mathcal{D}(\mathbb{Q})$-modules. But now by [CD19, Section 16.2] there is a canonical decomposition

$$ \mathcal{SH}_{\mathbb{Q}} \simeq \mathcal{SH}_{\mathbb{Q}}^{+} \times \mathcal{SH}_{\mathbb{Q}}^{-} $$

with the $+$-part being isomorphic to $\mathcal{DM}(-, \mathbb{Q})$ by [CD19, Theorem 16.2.18]. Therefore to obtain the factorisation of

$$ \mathcal{SH}_{\mathbb{Q}} \to \text{Ind}\mathcal{D}_{\mathcal{M}}^b $$
through $DM(-, \mathbb{Q})$ it suffices to show that the image of $\mathbb{Q}^-$ (which is the image of $\mathbb{Q}$ under the projection $SH_{\mathbb{Q}} \to SH_{\mathbb{Q}}^-$) in Ind$\mathcal{D}_M^b$ is zero. But now as the Betti (or $\ell$-adic) realisation is conservative on compact objects, this can be checked in $\mathcal{D}_c^b$, hence étale locally where it is true because $\mathbb{Q}^-$ vanishes whenever $-1$ is a sum of squares in the residual fields of the base scheme.

**Corollary 4.6.** *The realisation functor*

$$SH_c \to \mathcal{D}_M^b$$

commutes with the 6 operations.

*Proof.* Indeed the functor $SH_c \to DM_c$ commutes with the operations: it factors through $SH_c \to SH_{\mathbb{Q},c}$, which commutes with the operations because all of the 6 functors preserve colimits (either because they are left adjoint, or because they are right adjoint of compact preserving left adjoint), so that they commutes with $- \otimes \mathbb{Q}$ which can be written as a colimit. Moreover as seen in Remark 4.5 the functor $SH_{\mathbb{Q}} \to DM$ is only the projection $SH_{\mathbb{Q}} \to SH_{\mathbb{Q}}^+$ on the plus part, which amount to kill an idempotent. As this is a finite colimit (one takes the cofiber of the multiplication by the idempotent), this commutes with the 6 operations. Finally, the composition $SH_c \to DM_c \to \mathcal{D}_M^b$ commutes with the 6 operations. □

**Corollary 4.7.** *There exists a Hodge realisation of $R_H : \mathcal{D}^b(M_{\text{perv}}) \to \mathcal{D}^b(\text{MHM})$ compatible with the six operations. It factors the Hodge realisation that we have constructed.*

*Proof.* By Theorem 4.4, there exists a Hodge realisation from $DM_c(-, \mathbb{Q})$ to the derived category of mixed Hodge modules. Therefore, taking the underlying $\mathbb{Q}$-structure and then perverse $\mathbb{P}H^0$, we obtain a factorisation of the perverse $\mathbb{P}H^0$ of the Betti realisation through $\text{MHM}(X)$. The universal property of perverse Nori motives then gives a faithful exact functor $M_{\text{perv}}(X) \to \text{MHM}(X)$, which induces a functor $\mathcal{D}^b(M_{\text{perv}}(X)) \to \mathcal{D}^b(\text{MHM}(X))$. Now all the constructions of [IM22] are the constructions of the operations for mixed Hodge modules, hence this functor is compatible with the operations. The fact that this functor factors the Hodge realisation we constructed above follows from the universal property of $DM_c$. □

Recall that by [IM22, Corollary 6.18], [Bon11, Proposition 2.3.1] and [Hé11, Theorem 3.3] the categories $\mathcal{D}^b(M_{\text{perv}}(X))$, $\mathcal{D}^b(\text{MHM}(X))$ and $DM_c(X)$ admit weight structures (for the last one, note that Beilinson motives and étale motives are equivalent by [CD19, Theorem 16.1.2].)

**Proposition 4.8.** *Let $X$ be a finite type $k$-scheme. Then the functors*

$$R_H : \mathcal{D}^b(M_{\text{perv}}(X)) \to \mathcal{D}^b(\text{MHM}(X)),$$

$$\nu : DM_c(X) \to \mathcal{D}^b(M_{\text{perv}}(X))$$

and

$$\text{Hdg}^* : DM_c(X) \to \mathcal{D}^b(\text{MHM}(X))$$

are weight-exact.
*Proof.* First note that by definition of the weights on $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$, the $\ell$-adic realisation $R_\ell : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}_c^b(X, \mathbb{Q}_\ell)$ is weight exact and reflects weights. As all weight filtrations and structures considered are bounded, it suffices to show that our functors map zero weight objects to zero weight objects. Let us first consider functors with source $\mathcal{D}\mathcal{M}_c(X)$. For those, it suffices to check that the image of the relative motive of a smooth proper $X$-scheme is pure of weight 0. For the $\ell$-adic realisation this is [Del80] and [Mor19]. Therefore, the functor $\nu : \mathcal{D}\mathcal{M}_c(X) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ is also weight exact. The fact that $\text{Hdg}^*$ is exact is the fact that the mixed Hodge modules associated to a smooth proper $X$-scheme is pure of weight 0, see [Sai06, Theorem 6.7 and Proposition 6.6].

Now is only left $R_H : \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}^b(\text{MHM}(X))$. By [IM22, Corollary 6.27], any pure complex of perverse Nori motives is the direct sum of shifts of its $\text{P}^\text{H}^i$, hence to check that $R_H$ is weight exact it suffices to show that it sends pure objects of $\mathcal{M}_{\text{perv}}(X)$ to pure objects of $\text{MHM}(X)$. By construction of $R_H$, we have a commutative diagram:

$$ \begin{tikzcd} \mathcal{D}\mathcal{M}_c(X) \arrow[r, "H_u"] \arrow[d, "\text{Hdg}^*"'] & \mathcal{M}_{\text{perv}}(X) \arrow[r, "R_H"] \arrow[d, "\text{MHM}(X)"'] & \text{MHM}(X) . \arrow[dl, "\text{Hdg}^*"'] \arrow[dr, "\text{MHM}(X)"'] \end{tikzcd} \quad (4.8.1) $$

With $H_u$ the universal functor defining $\mathcal{M}_{\text{perv}}(X)$. But by [IM22, Corollary 6.27], $H_u$ sends pure objects to pure objects, and any motive $M \in \mathcal{M}_{\text{perv}}(X)$ is a quotient of some $H_u(N)$ for $N \in \mathcal{D}\mathcal{M}_c(X)$. Assume that $M$ is pure of some weight $w$. Then $M$ is a direct factor of a quotient of the weight filtration of $\text{H}^0(N)$. As $\text{Hdg}^*$ is weight exact, it follows that $R_H(M)$ is a direct factor of a weight $w$ mixed Hodge module, hence $R_H$ is weight exact. $\square$

*Remark 4.9.* Our construction of the realisation functor $\mathcal{D}\mathcal{M}_c(X)$ is very similar to that of Ivorra in [Ivo16] (which works for smooth varieties). Indeed, given a smooth variety $X$, in both definitions one first gives the image of the smooth $X$-schemes in $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$. To be more precise, Ivorra only gives the image of smooth affine $X$-schemes and then extend to smooth $X$-schemes by colimits but as the functor $(f : Y \to X) \mapsto f_!f^!\mathbb{Q}_X$ is a Nisnevich cosheaf, its value on smooth $X$-schemes is indeed determined by its value on smooth affine $X$-schemes. Once one has the value of the functor on smooth $X$-schemes, the rest of the construction is just applying the universal property of $\mathcal{D}\mathcal{M}_c(X)$: one has to check that the functor on smooth $X$-schemes is an $\mathbb{A}^1$-invariant étale sheaf and $\mathbb{P}^1$-stable. Therefore, it is easy to see that if both definitions on smooth affine schemes (ours with the 6-operations and Ivorra's using cellular complexes and Beilinson basic lemma) coincide, then our functor $\mathcal{D}\mathcal{M}_c(X) \to \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ and Ivorra's are equivalent. In particular, this would prove that Ivorra's construction is compatible with the Betti realisation constructed by Ayoub on Voevodsky motives, hence that the category of perverse Nori motives as in [IM22] is equivalent as the diagram category of relative pairs studied in [Ivo17] and [IM22, Section 2.10].

This can be reduced to check a simple fact, that the author did not manage to do: in [Ivo16, Proposition 4.15] Ivorra proves that for a given smooth affine map $f : Y \to X$
with $H_u$ the universal homological functor of the definition of perverse Nori motives and $F_X$ an exact equivalence of categories, compatible with the realisation. We identify via $F_X$ the categories $\mathcal{M}_{\text{perv}}(X)$ and $\mathcal{M}(X)$. By the universal property of the bounded derived category Corollary 3.5 we have a commutative diagram

$$ \begin{tikzcd} \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \arrow[r, "\gamma_X"] \arrow[d, "H^0"] & \mathcal{D}\mathcal{M}_c(X) \arrow[r, "\nu_X"] \arrow[d, "H^0"] & \mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \arrow[d, "H^0"'] \\ \mathcal{M}_{\text{perv}}(X) \arrow[r, "\text{Id}"] & \mathcal{M}_{\text{perv}}(X) \arrow[r, "H^0\nu_X"] & \mathcal{M}_{\text{perv}}(X) \end{tikzcd} \quad (4.13.2) $$

compatible with the realisation. Applying Lemma 4.10 to the derived category of perseverse motives $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$ we obtain that $H^0\nu = \text{Id}$. The functor $\nu_X \circ \gamma_X$ is determined by its restriction to the heart $\mathcal{M}_{\text{perv}}(X)$, hence it is the identity. $\square$

**Lemma 4.14.** The functor $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X)) \to \mathcal{D}\mathcal{M}_c(X)$ induces a fully faithful symmetric monoidal embedding $\mathcal{M}_{\text{ct}}(X) \to \mathcal{C}\mathcal{M}(X)$, where $\mathcal{C}\mathcal{M}(X)$ is the heart of the constructible t-structure on $\mathcal{D}\mathcal{M}_c(X)$.

Moreover, the induced functor $\mathcal{D}^b(\mathcal{M}_{\text{ct}}(X)) \to \mathcal{D}\mathcal{M}_c(X)$ coincides with $\gamma_X$ and provides a lift of the family of the $(\gamma_X)$ to a natural transformation

$$ \gamma: \mathcal{D}^b(\mathcal{M}_{\text{perv}}(-)) \Rightarrow \mathcal{D}\mathcal{M}_c(-) $$

in $\text{Fun}(\text{Sch}_k^{\text{op}}, \text{CAlg}(\text{Cat}_\infty))$, where we use the $(-)^*$-functoriality.

*Proof.* Because $\mathcal{D}\mathcal{M}_c(X) \to \mathcal{D}^b(\mathcal{M}_{\text{ct}}(X))$ is t-exact for the constructible t-structure (this can be checked after realisation), we have faithful exact functors $\mathcal{M}_{\text{ct}}(X) \to \mathcal{C}\mathcal{M}(X)$ and $\mathcal{C}\mathcal{M}(X) \to \mathcal{M}_{\text{ct}}(X)$. Whose composition is the identity by Lemma 4.13. Thus the functor $\mathcal{M}_{\text{ct}}(X) \to \mathcal{C}\mathcal{M}(X)$ is fully faithful.

Now by Proposition 3.7 and its proof (see Remark 3.8), we have a natural transformation

$$ r: \mathcal{K}^b(\mathcal{M}_{\text{ct}}(-)) \to \mathcal{D}\mathcal{M}_c(-) $$

of functors with values in $\text{CAlg}(\text{Cat}_\infty)$. Because the composition $\mathcal{K}^b(\mathcal{M}_{\text{ct}}(X)) \to \mathcal{D}\mathcal{M}_c(X) \to \mathcal{D}^b(\mathcal{M}_{\text{ct}}(X))$ is the canonical localisation functor (here we use the universal property of $\mathcal{K}^b(-)$), the functor r sends quasi-isomorphisms to isomorphisms hence the same proof as in Proposition 3.7 provides a natural transformation

$$ \delta: \mathcal{D}^b(\mathcal{M}_{\text{ct}}(-)) \Rightarrow \mathcal{D}\mathcal{M}_c $$

with values in symmetric monoidal stable $\infty$-categories. Because $\nu \circ \delta = \text{Id}_{\mathcal{D}^b(\mathcal{M}_{\text{ct}}(X))}$ and the restriction to $\mathcal{M}_{\text{perv}}(X)$ of $\nu$ is also the identity, we see that the restriction of $\delta$ to $\mathcal{M}_{\text{perv}}(X)$, landing in $\mathcal{M}_{\text{perv}}(X)$, is also the identity, thus by the universal property of $\mathcal{D}^b(\mathcal{M}_{\text{perv}}(X))$, for each finite type $k$-scheme $X$ the functors $\gamma_X$ and $\delta_X$ are canonically identified. $\square$
Proof. (of Theorem 4.12)

We can now show that $\gamma_X \circ \nu_X \simeq \mathrm{Id}_{\mathcal{D}\mathcal{M}_c(X)}$. For this, we first show that $\gamma$ commutes with $f_\sharp$ for $f$ a smooth morphism of finite type $k$-schemes. Because $\gamma$ commutes with $f^*$, there exists a natural transformation $f_\sharp \circ \gamma \to \gamma \circ f_\sharp$. But when applying the conservative functor $\nu$, commutation of $\nu$ with $f_\sharp$ together with Lemma 4.13 show that this natural transformation becomes invertible so that it was before applying $\nu$.

Thus, $\gamma$ is a morphism of coefficient systems, and therefore $\gamma \circ \nu$ is a endomorphism of coefficients systems $\mathcal{D}\mathcal{M}_c \to \mathcal{D}\mathcal{M}_c$. Now let $X$ be a finite type $k$-scheme and let $G$ be the right adjoint of $F := \gamma_X \circ \nu_X$ on the indization $\mathcal{D}\mathcal{M}(X)$. Showing that the unit $\mathrm{Id} \to G \circ F$ is an equivalence can be tested on objects of the form $f_\sharp\mathbb{Q}_Y(j)$ for $f: Y \to X$ a smooth $X$-scheme and $j \in \mathbb{Z}$ an integer. But then we can check that the map $f_\sharp\mathbb{Q}_Y(j) \to G \circ F(f_\sharp\mathbb{Q}_Y(j))$ is an isomorphism by applying the functors $\mathrm{map}_{\mathcal{D}\mathcal{M}(X)}(g_\sharp\mathbb{Q}(i), -)$ with $g$ smooth and $i \in \mathbb{Z}$. Those obtained maps are isomorphisms because $F$ commutes with the operations $f_\sharp$, $g_\sharp$ and twists. Thus $F$ is fully faithful. It is essentially surjective because it clearly reaches the generators $f_\sharp\mathbb{Q}_Y(j)$. This finishes the proof of the theorem. Note that one could also have used the universality of $\mathcal{D}\mathcal{M}$. $\square$

## 4.3 Nori motives as modules in étale motives.

**Notation 4.15.** To be coherent with the notations of [Ayo22], we will denote by Nor* the realisation functor

$$ \mathcal{D}\mathcal{M} \to \mathcal{D}\mathcal{N} $$

where we have denoted by $\mathcal{D}\mathcal{N}$ the indization of the bounded derived category of perverse motives. We will denote by $\mathcal{D}\mathcal{N}_c$ its full subcategory of compact objects.

By construction, the functor Nor* preserves colimits, hence it has a right adjoint $\mathrm{Nor}_*: \mathcal{D}\mathcal{N} \to \mathcal{D}\mathcal{M}$. Denote by

$$ \mathcal{N}_k = \mathrm{Nor}_*\mathrm{Nor}^*\mathbb{Q}_{\mathrm{Spec}\,k} \in \mathcal{D}\mathcal{M}(\mathrm{Spec}\,k, \mathbb{Q}). \qquad (4.15.1) $$

This is a $\mathbb{E}_\infty$-ring object in $\mathcal{D}\mathcal{M}(k)$. For $X$ a finite type $k$-scheme, denote by $\mathcal{N}_{|X} \in \mathcal{D}\mathcal{M}(X)$ the restriction of $\mathcal{N}_k$ to $X$. As in Ayoub's [Ayo22, Construction 1.91], Nor* factors canonically through the functor

$$ \mathrm{Mod}_{\mathcal{N}}(\mathcal{D}\mathcal{M}) : X \mapsto \mathrm{Mod}_{\mathcal{N}_{|X}}(\mathcal{D}\mathcal{M}(X, \mathbb{Q})) \qquad (4.15.2) $$

as

$$ \mathrm{Nor}^* : \mathcal{D}\mathcal{M} \xrightarrow{\mathcal{N} \otimes -} \mathrm{Mod}_{\mathcal{N}}(\mathcal{D}\mathcal{M}) \xrightarrow{\widetilde{\mathrm{Nor}}^*} \mathcal{D}\mathcal{N}. \qquad (4.15.3) $$

**Lemma 4.16.** Let $k$ be a field of characteristic zero and let $\pi_X: X \to \mathrm{Spec}\,k$ be a finite type $k$-scheme. The functor

$$ \widetilde{\mathrm{Nor}}^* : \mathrm{Mod}_{\pi_X^*\mathcal{N}_k}(\mathcal{D}\mathcal{M}(X)) \to \mathcal{D}\mathcal{N}(X) $$

is fully faithful. Moreover the natural map $\pi_X^*\mathrm{Nor}_*\mathbb{Q}_{\mathrm{Spec}\,k} \to \mathrm{Nor}_*\mathbb{Q}_X$ is an equivalence.
Proof. The proof adds nothing new to [Ayo22, Remark 1.97]. Denote by $\widetilde{\mathrm{Nor}}_*$ the right adjoint of $\widetilde{\mathrm{Nor}}^*$. Assume first that $X = \mathrm{Spec} k$. We have to show that the unit map

$$ \mathrm{Id} \to \widetilde{\mathrm{Nor}}_* \widetilde{\mathrm{Nor}}^* $$

is an equivalence. By [Iwa18, Lemma 2.6], the $\infty$-category $\mathrm{Mor}_{\mathcal{N}_k}(\mathcal{DM}(k))$ is compactly generated by the $f_{\sharp} \mathbb{Q}_Y(i) \otimes \mathcal{N}_k$ for $f: Y \to \mathrm{Spec} k$ smooth and $i \in \mathbb{Z}$. As $\widetilde{\mathrm{Nor}}^*$ of those compact generators are compact objects of $\mathcal{DN}(k)$, the right adjoint $\widetilde{\mathrm{Nor}}_*$ preserves colimits. Thus it suffices to check that for any compact object $C \otimes \mathcal{N}_k$ of the above form, the map

$$ C \otimes \mathcal{N}_k \to \widetilde{\mathrm{Nor}}_* \widetilde{\mathrm{Nor}}^* (C \otimes \mathcal{N}_k) \simeq \widetilde{\mathrm{Nor}}_* \mathrm{Nor}^* C $$

is an equivalence. As the forgetful functor $\mathrm{Mor}_{\mathcal{N}_k}(\mathcal{DM}(k)) \to \mathcal{DM}(k)$ is conservative it is even sufficient to prove that

$$ C \otimes \mathcal{N}_k \to \mathrm{Nor}_* \mathrm{Nor}^* C $$

is an equivalence in $\mathcal{DM}(k)$. This is true because over a field, compact objects in $\mathcal{DM}(k)$ are dualisable and for dualisable objects this map is an equivalence by [Ayo14b, Lemma 2.8].

Now for a general $X$ we use that the functor $\widetilde{\mathrm{Nor}}^*$ commutes with the 6 operations. For compact $M, N \in \mathrm{Mod}_{\pi_X^* \mathcal{N}_k}(\mathcal{DM}(X))$ we have:

$$
\begin{align*}
\mathrm{Map}_{\mathrm{Mod}_{\pi_X^* \mathcal{N}_k}(\mathcal{DM}(X))}(M, N) &\simeq \mathrm{Map}_{\mathrm{Mod}_{\pi_X^* \mathcal{N}_k}(\mathcal{DM}(X))}(\mathbb{Q}_X, \mathrm{Hom}(M, N)) \\
&\simeq \mathrm{Map}_{\mathrm{Mod}_{\mathcal{N}_k}(\mathcal{DM}(k))}(\mathbb{Q}_k, (\pi_X)_* \mathrm{Hom}(M, N)) \\
&\simeq \mathrm{Map}_{\mathcal{DN}(k)}(\mathbb{Q}_k, \widetilde{\mathrm{Nor}}^*(\pi_X)_* \mathrm{Hom}(M, N)) \\
&\simeq \mathrm{Map}_{\mathcal{DN}(k)}(\mathbb{Q}_k, (\pi_X)_* \mathrm{Hom}(\widetilde{\mathrm{Nor}}^*M, \widetilde{\mathrm{Nor}}^*N)) \\
&\simeq \mathrm{Map}_{\mathcal{DN}(X)}(\widetilde{\mathrm{Nor}}^*M, \widetilde{\mathrm{Nor}}^*N)
\end{align*}
$$

which finishes the proof of fully faithfulness. Now that we know that the unit

$$ \mathrm{Id} \to \widetilde{\mathrm{Nor}}_* \widetilde{\mathrm{Nor}}^* $$

is an equivalence in $\mathrm{Mod}_{\pi_X^* \mathcal{N}_k}(\mathcal{DM}(X))$, by plugging in $\pi_X^* \mathcal{N}_k$ we obtain

$$ \pi_X^* \mathcal{N}_k \xrightarrow{\sim} \widetilde{\mathrm{Nor}}_* \widetilde{\mathrm{Nor}}^* (\pi_X^* \mathcal{N}_k) $$

and we have

$$ \widetilde{\mathrm{Nor}}^*(\pi_X^* \mathcal{N}_k) \simeq \pi_X^* \widetilde{\mathrm{Nor}}^*(\mathcal{N}_k) \simeq \mathbb{Q}_X. $$

Therefore the canonical map

$$ \pi_X^* \mathcal{N}_k \to \widetilde{\mathrm{Nor}}_* \mathbb{Q}_X $$
is an equivalence, thus it stays an equivalence after applying the forgetful functor, leaving us with the equivalence

$$ \pi_X^* \mathcal{N}_k \xrightarrow{\sim} \mathrm{Nor}_* \mathbb{Q}_X $$

in $\mathcal{DM}(X)$.

□

From the previous lemma there is no more ambiguity for $\mathcal{N}_X = \pi_X^* \mathcal{N}_k \simeq \mathrm{Nor}_* \mathbb{Q}_X \in \mathcal{DM}(X)$.

**Proposition 4.17.** Let $k = \mathrm{colim}_i k_i$ be a filtered colimit of fields of characteristic zero. Then the natural functors

$$ \mathrm{colim}_i \mathcal{DN}_c(\mathrm{Spec} \ k_i) \to \mathcal{DN}_c(\mathrm{Speck}) $$

and

$$ \mathrm{colim}_i \mathcal{DN}(\mathrm{Spec} \ k_i) \to \mathcal{DN}(\mathrm{Speck}) $$

are equivalences in $\mathrm{Cat}_\infty$ and $\mathrm{Pr}^L$ respectively.

The same is true for $\eta = \lim_i U_i$ with $\eta \in X$ the generic point of an integral finite type k-scheme and the $U_i$ are the nonempty open subsets of $X$:

$$ \mathrm{colim}_i \mathcal{DN}_{(c)}(U_i) \to \mathcal{DN}_{(c)}(\eta). $$

*Proof.* As the functor Ind, from small idempotent complete stable $\infty$-categories to compactly generated presentable stable categories is an equivalence and that the inclusion functor $\mathrm{Pr}_\omega^L \to \mathrm{Pr}^L$ preserves colimits, the second statement follows from the first. By [IM22, Proposition 6.8] (or [IM22, Corollary 6.10] for the generic point case), the diagram $\mathcal{M} : I^\Delta \to \mathrm{AbCAt}$ from $I$ to the (2, 1)-category of small abelian categories with exact functors sending $i \in I$ to $\mathcal{M}_{\mathrm{perv}}(\mathrm{Spec} \ k_i)$ and the cone point $v$ to $\mathcal{M}_{\mathrm{perv}}(\mathrm{Spec} \ k)$ is a colimit diagram. By [BCKW24, Theorem 7.4.9] the functor sending an Abelian category to the $\infty$-category of bounded complexes up to homotopy is a left adjoint, hence commutes with colimits, so that we obtain a colimit diagram

$$ \mathcal{K}^b \circ \mathcal{M} : I^\Delta \to \mathrm{Cat}_\infty^{\mathrm{st}}. $$

Now as $\mathcal{D}^b(\mathcal{A})$ is a localisation of $\mathcal{K}^b(\mathcal{A})$ for any abelian category $\mathcal{A}$, and as the transitions of $\mathcal{K}^b \circ \mathcal{M}$ preserve quasi-isomorphisms, $\mathcal{D}^b \circ \mathcal{M}$ is a colimit diagram. □

**Corollary 4.18.** Let $f : \mathrm{Spec} \ k \to \mathrm{Spec} \ \mathbb{Q}$ be a map of fields. Then the natural map $f^* \mathcal{N}_{\mathbb{Q}} \to \mathcal{N}_k$ is an equivalence. Consequently, for any scheme $X$ of finite type over $k$ if we denote by $p_X : X \to \mathrm{Spec} \ \mathbb{Q}$ the unique map (which has to factor through $f$), then $\mathcal{N}_X \simeq p_X^* \mathcal{N}_{\mathbb{Q}}$.
**Proof.** The second claims follows from the first. Write $k = \text{colim}_i k_i$ with $k_i/\mathbb{Q}$ of finite type. Then the same proof as [AGV22, Lemma 3.5.7], using continuity of both $\mathcal{D}\mathcal{M}$ and $\mathcal{D}\mathcal{N}$ for fields (Proposition 4.17) and that these categories a compactly generated gives that the natural map

$$ \text{colim}_i f_i^* \mathcal{N}_{\mathbb{Q}} \to \mathcal{N}_k $$

is an equivalence so that this reduces to the case where $k/\mathbb{Q}$ is of finite type. Then $k$ is the generic point of an integral $\mathbb{Q}$-scheme, and again the same proof as [AGV22, Lemma 3.5.7], using continuity for the generic point of perverse Nori motives, reduces to show that $\pi_U^* \mathcal{N}_{\mathbb{Q}} \to \mathcal{N}_U$ is an equivalence for $U$ of finite type over $\mathbb{Q}$, which is Lemma 4.16. $\square$

Therefore now there is even less ambiguity for $\mathcal{N}_X = (X \to \text{Spec } \mathbb{Q})^* \mathcal{N}_{\mathbb{Q}}$.

**Proposition 4.19.** Let $X = \lim_i X_i$ be a limit of $\mathbb{Q}$-schemes with affine transitions. The natural morphism

$$ \text{colim}_i \text{Mod}_{\mathcal{N}}(\mathcal{D}\mathcal{M}(X_i)) \to \text{Mod}_{\mathcal{N}}(\mathcal{D}\mathcal{M}(X)) \quad (4.19.1) $$

is an equivalence.

**Proof.** By [EHIK21, Lemma 5.1 (ii)], the functor

$$ \text{colim}_i \mathcal{D}\mathcal{M}(X_i, \mathbb{Q}) \to \mathcal{D}\mathcal{M}(X, \mathbb{Q}) \quad (4.19.2) $$

is an equivalence. Therefore, the assignment $i \mapsto (\mathcal{D}\mathcal{M}(X_i, \mathbb{Q}), \mathcal{N}_{|X_i})$ is a colimit diagram in the category $\text{Pr}^{\text{Alg}}$ of [Lur22, Notation 4.8.5.10]. By [Lur22, Theorem 4.8.5.11], the functor $(\mathcal{C}, A)$ that sends a presentable symmetric monoidal category $\mathcal{C}$ together with an algebra object $A \in \mathcal{C}$ to the category $\text{Mod}_A(\mathcal{C})$ has a right adjoint, hence preserves colimits. This is why the assignment $i \mapsto \text{Mod}_{\mathcal{N}_{X_i}}(\mathcal{D}\mathcal{M}(X_i, \mathbb{Q}))$ is a colimit diagram. We found a precise statement giving the symmetric monoidal structure in the paper by Ayoub, Gallauer and Vezzani. The precise statement is [AGV22, Lemma 3.5.6]. $\square$

**Proposition 4.20.** The functor

$$ \widetilde{\text{Nor}}^* : \text{Mod}_{\mathcal{N}}(\mathcal{D}\mathcal{M}(X)) \to \mathcal{D}\mathcal{N}(X) \quad (4.20.1) $$

is an equivalence for every finite type $k$-scheme $X$.

**Proof.** The functor is fully faithful by Lemma 4.16. We first prove that it is essential surjective for $X = \text{Spec } K$ the spectrum of a field, not necessarily finite over $k$. As $\mathcal{D}\mathcal{N}(\text{Spec } K)$ is compactly generated it suffices to show that $\widetilde{\text{Nor}}^*$ reaches all compact objects $M \in \mathcal{D}^b(\mathcal{M}_{\text{perv}}(K))$. As the functor is already fully faithful, by dévissage it suffices to check that any object of the heart $M \in \mathcal{M}_{\text{perv}}(K)$ is in the image of $\widetilde{\text{Nor}}^*$. Any such object has a weight filtration whose graded pieces are direct factors of
$H^{n+2j}(\pi_X)_*\mathbb{Q}_X(i)$ with $\pi_X: X \to \text{Spec } K$ a smooth projective morphism by Arapura's [HMS17, Theorem 10.2.5]. Note that here we used that if $K$ is a finite type extension of $k$, then the category $\mathcal{M}_{\text{perv}}(\text{Spec } K)$ obtained by colimit as motives over the generic point of a variety over $k$ coincides with Nori motives over $K$. This holds with a beautiful proof in [Ter24, Proposition 1.15] or can be proven by using that the $\ell$-adic realisation is absolute, i.e. does not depend on the base field. Therefore by dévissage again it suffices to check that any such $H^{n+2j}(\pi_X)_*\mathbb{Q}_X(i)$ is in the image. But as $(\pi_X)_*\mathbb{Q}_X(i)$ is pure of weight $-2i$, it is the direct sum of its $H^n[-n]$, thus is suffices to show that each $(\pi_X)_*\mathbb{Q}_X(i)$ is in the image, which is obvious by compatibility with the 6 operations.

Now for a finite type $k$-scheme, the proposition is proven by Noetherian induction. Assume now that $X$ is such that the statement is true for any proper closed subset of $X$. We take an object $M \in \mathcal{DN}(X)$ that we may assume to be compact. Let $\eta$ be a generic point of $X$. By the case of a field, there exists $N \in \text{Mod}_{\mathcal{N}}(\mathcal{DM}(\eta))$, compact, such that $\widetilde{\text{Nor}}^*(N) = M_\eta$. By Proposition 4.19 there exists an open subset $U$ of $X$ and an object $N' \in \text{Mod}_{\mathcal{N}}(\mathcal{DM}(U))$, compact, such that $(N')_\eta = N$. The functor $\text{colim}_{\eta \in U}\mathcal{DN}(U) \to \mathcal{DN}(\eta)$ is fully faithful by proposition 4.17. Thus the isomorphism $\widetilde{\text{Nor}}^*(N) \to M_\eta$ lifts to a smaller open subset $V$ of $U$: we have $\widetilde{\text{Nor}}^*((N')_{|V}) \simeq M_{|V}$. Denote by $Z$ the reduced closed complement of $V$ in $X$. By induction hypothesis, there exists a $N'' \in \text{Mod}_{\mathcal{N}}(\mathcal{DM}(Z))$ such that $\widetilde{\text{Nor}}^*(N'') = M_{|Z}$. The cofiber sequence $\widetilde{\text{Nor}}^*(j_!(N')_{|V}) \to M \to \widetilde{\text{Nor}}^*(i_*N'')$ with $j: U \to X$ and $i: Z \to X$ the immersion then ensures that $M$ is the image of $\widetilde{\text{Nor}}^*$, which finishes the proof. $\square$

*Remark 4.21.* The same arguments would prove that the $\infty$-category

$$ \mathcal{DH}^{\text{geo}}(X) \subset \text{Ind}\mathcal{D}^b(\text{MHM}(X)) $$

of geometric origin objects in the indization of the derived category of mixed Hodge modules is also the category of modules over some algebra $\mathcal{H}_X$ in $\mathcal{DM}(X)$. Note that the lack of continuity of Mixed Hodge modules would prevent things like $p_X^*\mathcal{H}_{\text{Spec } \mathbb{Q}} \simeq \mathcal{H}_X$ of happening so one would have to be careful of which algebra is used and an additional argument over the generic point will be needed, similarly to the case of the Betti realisation. Similar ideas are developed in [Dre18] and the fact that one would obtain $(X \to \text{Spec } k)^*\mathcal{H}_{\text{Spec } k} \simeq \mathcal{H}_X$ identifies Drew's category as the full subcategory $\mathcal{DH}^{\text{geo}}(X)$ of geometric origin mixed Hodge modules. His considerations with enriched categories of motives also have an application here: his results, together with the same proof as above, would show that his category **DH** of motivic enriched Hodge modules is the smallest full subcategory of $\text{Ind}\mathcal{D}^b(\text{MHM}(-))$ which is stable under truncation and filtered colimits, and that contains the $f_*g^*H$ for $f: Y \to X$ proper, $g: Y \to \text{Spec } \mathbb{C}$ the structural morphism and $H \in \mathcal{D}(\text{IndMHS}^p)$. See [Tub24] for a detailed discussion.

**Corollary 4.22.** The $\infty$-functor $X \mapsto \mathcal{D}^b(M_{\text{perv}}(X))$ has an unique extension $\mathcal{DN}_c(X)$ to quasi-compact and quasi-separated schemes of characteristic zero such that for all
limit $X = \lim_i X_i$ of such schemes with affines transitions the natural functor

$$ \mathrm{colim}_i \mathcal{D}\mathcal{N}_c(X_i) \to \mathcal{D}\mathcal{N}_c(X) $$

is an equivalence of $\infty$-categories.

**Proof.** Let $\mathcal{C}$ be category of schemes that are of finite type over a field of characteristic zero. Let $\mathrm{Sch}_{\mathbb{Q}}$ be the category of quasi-compact and quasi-separated (qcqs) $\mathbb{Q}$-schemes, and $\iota: \mathcal{C} \to \mathrm{Sch}_{\mathbb{Q}}$ the inclusion. Nori motives, together with the pullback functoriality, define a functor $\mathcal{C}^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_{\infty})$ which sends limits of schemes with affine transitions to colimits of $\infty$-categories by Proposition 4.19 together with the fact that the colimit is taken in $\mathrm{Pr}^L$ hence restricts to compact objects. We can denote by $\mathcal{D}\mathcal{N}_c$ its left Kan extension along $\iota$. $\square$

**Corollary 4.23.** Let $X$ be a finite type $k$-scheme. The $\infty$-category $\mathcal{D}\mathcal{N}(X)$ is compactly generated by the $f_{\sharp}\mathbb{Q}_Y(i)$ for $f: Y \to X$ smooth and $i \in \mathbb{Z}$.

**Proof.** Indeed, this the case for $\mathcal{D}\mathcal{M}(X)$, thus for $\mathrm{Mod}_{\mathcal{N}_X}(\mathcal{D}\mathcal{M}(X)) \simeq \mathcal{D}\mathcal{N}(X)$ by [Iwa18, Lemma 2.6]. $\square$

We finish with the proof of the following natural corollary, whose proof was surprisingly harder than expected for a result that the author took for granted for a long time.

**Corollary 4.24.** Let $X$ be a quasi-projective $k$-scheme. The two functors

$$ \mathcal{D}\mathcal{M}_c(X) \to \mathcal{M}_{\mathrm{perv}}(X) $$

given by $\mathrm{pH}^0 \circ \mathrm{Nor}^*$ and the universal functor $\mathrm{H}_{\mathrm{univ}}$ are canonically equivalent.

**Proof.** First, note that the functor $\mathrm{pH}^0 \circ \mathrm{Nor}^*$ gives a factorisation of the $\mathrm{pH}^0$ of the Betti realisation of étale motives

$$ \mathcal{D}\mathcal{M}_c(X) \to \mathrm{Perv}(X). $$

By the universal property of perverse Nori motives, this induces a canonical functor

$$ F: \mathcal{M}_{\mathrm{perv}}(X) \to \mathcal{M}_{\mathrm{perv}}(X) $$

such that $R_B \circ F \simeq R_B$ and $\mathrm{pH}^0 \circ \mathrm{Nor}^* \simeq F \circ \mathrm{H}_{\mathrm{univ}}$. Now the compatibility of Nor* with the operations ensures that for any perverse t-exact operation $h$ the functor $F$ commutes with the operation $h$. Hence $F$ commutes with $i_*$ for $i$ a closed immersion and for $f^*[d]$ for $f$ a smooth map of relative dimension $d$. We will still denote by $F$ the functor induced by it on $\mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X))$. As $F$ commutes with $i_*$, there exists an exchange morphism $i_* \circ F \to F \circ i_*$. Applying the Betti (or $\ell$-adic in the case the field $k$ is too big) realisation where this exchange morphism is an equivalence, we see that $F$ commutes with $i_*$. The same argument works to show that $F$ commutes with $f_*$ for
smooth $f$. For a smooth morphism $f$, there is a natural transformation $f_{\sharp} \circ F \to F \circ f_{\sharp}$ which is an equivalence after realisation, hence $F$ also commutes with the $f_{\sharp}$ functors. It also commutes with external tensor product because it is perverse t-exact, hence with the internal tensor product which is obtained from the external one by pulling back along the diagonal.

The functor $F$ induces a compact and colimit preserving functor

$$F : \mathrm{Ind}\mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X)) \to \mathrm{Ind}\mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X))$$

which commutes with all pullbacks, pushforwards and left adjoint to pushforward by smooth maps. Denote by $G$ its right adjoint. We claim that the natural transformation

$$\mathrm{Id} \to GF$$

is an equivalence. Indeed, by Corollary 4.23 we have that $\mathrm{Ind}\mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X))$ is generated by colimits by the $f_{\sharp}\mathbb{Q}_Y(i)$ for $f:Y \to X$ a smooth map and $i \in \mathbb{Z}$. This means that it suffices to check that for $M \in \mathrm{Ind}\mathcal{D}^b(\mathcal{M}_{\mathrm{perv}}(X))$, the map

$$\mathrm{Map}(f_{\sharp}\mathbb{Q}_Y(i), M) \to \mathrm{Map}(f_{\sharp}\mathbb{Q}_Y(i), GF(M))$$

is an equivalence. As $G$ and $F$ commute with filtered colimits and $f_{\sharp}\mathbb{Q}_Y(i)$ is compact, we can assume that $M$ is as well compact. Moreover, compact objects are constructed by extensions, finite limits, finite colimits and direct factors from the same $f_{\sharp}\mathbb{Q}_Y(i)$, thus by dévissage we can assume that $M = g_{\sharp}\mathbb{Q}_Z(j)$ for some smooth map $g:Z \to X$ and $j \in \mathbb{Z}$. Using that $G$ is a right adjoint and that $F$ commutes with the operations, the above map is equivalent to the map

$$\mathrm{Map}(f_{\sharp}\mathbb{Q}_Y(i), g_{\sharp}\mathbb{Q}_Z(j)) \to \mathrm{Map}(f_{\sharp}F(\mathbb{Q}_Y)(i), g_{\sharp}F(\mathbb{Q}_Z)(j))$$

and as $F(\mathbb{Q}) = \mathbb{Q}$, this finishes the proof.

Now that $F$ is fully faithful, we note that it is an equivalence whenever $X = \mathrm{Spec} K$ is the spectrum of a field by [HMS17, Corollary 10.1.7]. Thus an easy dévissage using the localisation sequence and continuity proves that $F$ is essentially surjective, finishing the proof that $F$ is an isomorphism. Now, as $F$ commutes with the operations, we have by universality of $\mathcal{DM}$ that $F \circ {}^{\mathrm{P}}\mathrm{H}^0 \circ \mathrm{Nor}^* \simeq {}^{\mathrm{P}}\mathrm{H}^0 \circ \mathrm{Nor}^*$, thus $F \circ F \circ \mathrm{H}_{\mathrm{univ}} \simeq {}^{\mathrm{P}}\mathrm{H}^0 \circ \mathrm{Nor}^*$ and the universal property of $\mathcal{M}_{\mathrm{perv}}(X)$ tells us that $F \circ F \simeq F$, so that $F \simeq \mathrm{Id}$. $\square$

# A Coefficient systems.

Recall the following definition of Drew and Gallauer in [DG22] :

**Definition A.1.** A functor $C : \mathrm{Sch}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_{\infty}^{\mathrm{st}})$ taking values in symmetric monoidal stable $\infty$-categories and exact symmetric monoidal functors is called a *coefficient system* if it satisfies the following properties.
1. (Pushforwards) For every $f: Y \to X$ in $\mathrm{Sch}_k$, the pullback functor $f^*$ admits a right adjoint $f_* : C(Y) \to C(X)$.

2. (Internal homs) For every $X \in \mathrm{Sch}_k$, the symmetric monoidal structure on $C(X)$ is closed.

3. For each smooth morphism $p: Y \to X \in \mathrm{Sch}_k$, the functor $p^*: C(X) \to C(Y)$ admits a left adjoint $p_\sharp$, and:

(a) (Smooth base change) For each cartesian square

$$ \begin{array}{ccc} Y' & \xrightarrow{p'} & X' \\ \bigdownarrow & & \bigdownarrow \\ Y & \xrightarrow{p} & X \end{array} $$

in $\mathrm{Sch}_k$, the exchange transformation $p'_\sharp(f')^* \to f^*p_\sharp$ is an equivalence.

(b) (Smooth projection formula) The exchange transformation

$$ p_{\sharp}(p^{*}(-) \otimes -) \to - \otimes p_{\sharp}(-) $$

is an equivalence of functors $C(X) \times C(Y) \to C(Y)$.

4. (Localisation) $C(\emptyset) \simeq 0$ and for each closed immersion $Z \to X$ in $\mathrm{Sch}_k$ with complementary open immersion $j: U \to X$, the square

$$ \begin{array}{ccc} C(Z) & \xrightarrow{i_*} & C(X) \\ \bigdownarrow & & \bigdownarrow \\ 0 & \longrightarrow & C(U) \end{array} $$

is cartesian in $\mathrm{Cat}_\infty^{\mathrm{st}}$.

5. (A¹-homotopy invariance) For each $X \in \mathrm{Sch}_k$, if $\pi_{A^1}: A^1_X \to X$ denotes the canonical projection then the functor $\pi_{A^1}^*: C(X) \to C(A^1_X)$ is fully faithful.

6. (T-stability) The composite $\pi_{A^1, \sharp} s_*: C(X) \to C(X)$ is an equivalence.

A morphism of coefficient systems is a natural transformation $\phi: C \to C'$ such that for each smooth morphism $p: Y \to X$ in $\mathrm{Sch}_k$, the exchange transformation

$$ p_{\sharp} \phi_Y \to \phi_X p_{\sharp} $$

is an equivalence. This defines a sub $\infty$-category $\mathrm{CoSy}_k \subset \mathrm{Fun}(\mathrm{Sch}_k^{\mathrm{op}}, \mathrm{CAlg}(\mathrm{Cat}_\infty^{\mathrm{st}}))$. We say that a coefficient system $C$ is *cocomplete* if it takes values in symmetric monoidal cocomplete $\infty$-categories and colimit preserving functors. We denote by $\mathrm{CoSy}_k^c$ their $\infty$-category.
Recall the main theorem of [DG22]:

**Theorem A.2 (Drew-Gallauer).** The stable $\mathbb{A}^1$-homotopy $\infty$-category defines an object $SH \in \mathrm{CoSy}_k^c$. It is the initial object.

**Definition A.3.** We let QPCoSy$_k$ the $\infty$-category defined in the same terms as **Definition A.1** but with functors

$$ \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_{\infty}^{\mathrm{st}}) $$

with source quasi-projective varieties instead of finite type k-schemes.

**Proposition A.4.** The restriction functor

$$ \mathrm{CoSy}_k \to \mathrm{QPCoSy}_k $$

is an equivalence of $\infty$-categories with quasi-inverse the right Kan extension.

*Proof.* By [Gal22, Corollary 2.19] or [DG22, Proposition 7.13], any coefficient system (on all finite type k-schemes or varieties) affords Nisnevich descent, thus Zariski descent. Restriction and right Kan extension give an equivalence of categories between Zariski sheaves

$$ \mathrm{Shv}_{\mathrm{Zar}}(\mathrm{Var}_k, \mathrm{CAlg}(\mathrm{Cat}_{\infty}^{\mathrm{st}})) \simeq \mathrm{Shv}_{\mathrm{Zar}}(\mathrm{Sch}_k, \mathrm{CAlg}(\mathrm{Cat}_{\infty}^{\mathrm{st}})) $$

because any finite type k-scheme admits a Zariski covering by affine schemes. Therefore it suffices to check that if $C \in \mathrm{QPCoSy}_k$, its right Kan extension to $\mathrm{Sch}^{\mathrm{op}}$ is an object of $\mathrm{CoSy}_k$, and that a morphism in $\mathrm{QPCoSy}_k$ gives a morphism in $\mathrm{CoSy}_k$ after Kan extension.

We denote by $\mathcal{D}$ the right Kan extension of the functor

$$ C : \mathrm{Var}_k^{\mathrm{op}} \to \mathrm{CAlg}(\mathrm{Cat}_{\infty}) $$

along the inclusion $\mathrm{Var}_k^{\mathrm{op}} \to \mathrm{Sch}_k^{\mathrm{op}}$. Because $C$ is a Zariski sheaf, for $X \in \mathrm{Sch}_k$ we may choose a Zariski covering $p : U \to X$ with $U$ affine and as $p$ is quasi-projective, all $U_n := U^{\times n+1}$ are quasi-projective over $k$ and we have

$$ \mathcal{D}(X) \xrightarrow{\sim} \lim_{\Delta} C(U_n) $$

in $\mathrm{CAlg}(\mathrm{Cat}_{\infty})$. But then if $f : Y \to X$ is a smooth morphism, base change induces a map of simplicial schemes $f_{\bullet} : V_{\bullet} \to U_{\bullet}$ with $V_n = U_n \times_X Y$ and we also have

$$ \mathcal{D}(Y) \xrightarrow{\sim} \lim_{\Delta} C(V_n) $$

in $\mathrm{CAlg}(\mathrm{Cat}_{\infty})$. In fact, the functor

$$ f^* : \mathcal{D}(X) \to \mathcal{D}(Y) $$

is the limit of the functors

$$ f_n^* : C(U_n) \to C(V_n) $$
Then as above, the commutative square

$$
\begin{tikzcd}
\mathcal{D}(X) \arrow[r, "f^*"'] & \mathcal{D}(Y) \\
g^* \arrow[u, "\downarrow"'] & (g')^* \arrow[u, "\downarrow"'] \\
\mathcal{D}(X') \arrow[r, "(f')^*"'] & \mathcal{D}(Y')
\end{tikzcd}
$$

can be seen as a map $f^* \to (f')^*$ in $\mathrm{Fun}(\Delta^1, \mathrm{Cat}_\infty)$, which itself is a limit of maps $b'_n \to b_n$ in $\mathrm{Fun}(\Delta^1, \mathrm{Cat}_\infty)$. As this last simplicial diagram of maps takes values in $\mathrm{Fun}(\Delta^1, \mathrm{Cat}_\infty)$ because of the smooth base change holds for maps in $\mathrm{Var}_k$, the limit is left adjointable, giving the smooth base for the pair $(f,g)$.

The existence of pushforwards and internal $\mathcal{Hem}$ is proven in a similar way to the existence of the $f_\sharp$ (using right adjointable squares instead of left adjointable). The only thing to check in order to obtain those functors as functors to a limit of $\infty$-categories is that over quasi-projective schemes these functors commute with pullbacks by Zariski covering so that they define a compatible system as in (A.4.1). This is the case because these pullbacks $\pi^*$ are right adjoints to $\pi_\sharp$ and the smooth base change and the projection formula ensures the commutativity of $\pi_\sharp$ with the left adjoints of internal $\mathcal{Hem}$ and pushforwards. Localisation follows from the fact that the category $\mathrm{Sq}^{\mathrm{cart}}(\mathrm{Cat}_\infty)$ of cartesian squares in $\mathrm{Cat}_\infty$ is closed under limits in the category of commutative squares of $\infty$-categories. The properties of $\mathrm{A}^1$-invariance and T-stability can be checked Zariski locally thanks to the smooth projection formula and the smooth base change, hence they hold. The same is true for extensions of morphisms of coefficient systems. $\square$

Remark A.5. The above proposition on extension of coefficient system from quasi-projective to finite type objects works verbatim if one replaces $\mathrm{Spec} k$ with any noetherian finite dimensional scheme $\mathcal{B}$.

Finally recall the following theorem:

**Theorem A.6** (Ayoub-Cisinski-Déglise-Röndings-Voevodsky). *Any coefficient system affords a six functors formalism.*

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Swann Tubach swann.tubach@ens-lyon.fr

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