On the Nori and Hodge realisations of Voevodsky motives

Abstract

We show that the derived categories of perverse Nori motives investigate and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations. As a result, we 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.

1. 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 $\mathcal{MM}(k)$ , the target of the universal cohomology theory

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

on k-varieties in the sense that any other reasonable cohomology theory on k-varieties would factor uniquely through $M^*$ . This is beyond the reach of the current technology. However, 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 $\mathrm{DM}_\mathrm{gm}(k,\mathbb{Q})$ [VSF00], a candidate for the bounded derived category of mixed motives. The second is the abelian category of Nori motives $\mathcal{M}_\mathrm{Nori}(k)$ [Fak00], a candidate for $\mathcal{MM}(k)$ . These categories have realisation functors factoring the known cohomology theories of varieties. Indeed, if k is a subfield of $\mathbb{C}$ , Huber [Hub00], Levine [Lev05] and Nori have constructed a Hodge realisation functor

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

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 and Gallauer [CGAdS17]. When k is arbitrary they also constructed a $\ell$ -adic realisation functor with values in the derived category of $\ell$ -adic Galois representations.

There exist relative versions of the categories of Voevodsky and Nori. The triangulated category of constructible rational étale motives $\mathrm{DM}^{\mathrm{\acute{e}t}}_c(-)$ 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 ${\mathcal{M}_{\mathrm{perv}}}(-)$ constructed by Ivorra and S. Morel [IM22] gives a category of motivic sheaves generalising the abelian category of Nori. Both settings have a six-functor formalism, the tensor product on perverse Nori motives being constructed by Terenzi [Ter24]. Note that another approach based on constructible sheaves and using Nori’s ideas has been considered by Arapura in [Ara13, Ara23] but his construction does not include all six operations. If X is a quasi-projective smooth k-variety and k is a subfield of $\mathbb{C}$ , Ivorra [Ivo16] constructed a realisation functor

$$\mathrm{DM}^\mathrm{\acute{e}t}_c(X)\to \mathrm{D}^b(\mathrm{MHM}(X))$$

to Saito’s derived category of mixed Hodge modules [Sai90]. 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 six operations that exist on $\mathrm{DM}^\mathrm{\acute{e}t}_c(-)$ and $\mathrm{D}^b(\mathrm{MHM}(-))$ . The reason for this difficulty is that the functor is built by constructing an explicit complex $K^\bullet\in\mathrm{Ch}^b(\mathrm{MHM}(X))$ that computes the relative homology sheaf $f_!f^!\mathbb{Q}_X\in\mathrm{D}^b(\mathrm{MHM}(X))$ of a smooth affine X-scheme $f:Y\to X$ , and that we do not know how to lift the six operations explicitly on complexes: a priori they are only defined over the derived category.

On the other hand, we know since Robalo’s thesis [Rob15] that the $\infty$ -category $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ (which is a stable $\infty$ -category lifting $\mathrm{DM}^\mathrm{\acute{e}t}_c(X)$ ) has an universal property that enables one to construct realisation functors to any stable $\infty$ -category that has reasonable properties. Indeed, he shows that $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)=\mathrm{Ind}\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ is the target of the universal symmetric monoidal functor from $\mathrm{Sm}_X$ 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 any contravariant functor $\mathcal{C}$ from finite-type k schemes to symmetric monoidal $\mathbb{Q}$ -linear presentable $\infty$ -categories that satisfies non-effective étale descent and $\mathbb{A}^1$ -invariance, together with $\mathbb{P}^1$ -stability and smooth base change, receives a natural transformation from $\mathcal{DM}^\mathrm{\acute{e}t}(-)$ compatible with smooth base change functoriality. Moreover, Cisinski and Déglise (see also [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 $\mathcal{C}$ has the six operations, and that any natural transformation $\mathcal{DM}^{\mathrm{\acute{e}t}}\to\mathcal{C}$ 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 $\mathrm{Hdg}^*:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b(\mathrm{MHM}(X))$ compatible with the six operations, it suffices to lift the six 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 $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to \mathcal{DH}_c(X)$ commuting with all the operations. If $X = \mathrm{Spec} k$ is the spectrum of a field, then $\mathcal{DH}_c(\mathrm{Spec} k)\hookrightarrow \mathcal{D}^b(\mathrm{MHS}^p_k)$ 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 $\mathcal{D}^b(\mathrm{MHM}(X))$ 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 $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ (see Proposition 6.16 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.

Theorem 1.1 (Theorem 6.1, Theorem 6.5 and Proposition 6.9). The six operations on the categories $\mathrm{D}^b(\mathrm{MHM}(-))$ and $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))$ constructed by Saito, Ivorra, Morel and Terenzi admit $\infty$ -categorical lifts and are defined over any finite-type k-scheme. For every finite-type k-scheme X there exists an $\infty$ -functor

\[\mathrm{Nor}^*: \mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X)),\]

and if k is a subfield of $\mathbb{C}$ then there exist $\infty$ -functors

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

and

\[\mathrm{Hdg}^*: \mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b(\mathrm{MHM}(X)).\]

The three functors commute with the six operations and we have $R_H\circ\mathrm{Nor}^*\simeq \mathrm{Hdg}^*$ . Moreover, $\mathrm{Nor}^*$ is compatible with the Betti and $\ell$ -adic realisation functors, $R_H$ is t-exact and all functors are compatible with weights.

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, it is very easy to construct $\infty$ -categorical lifts of derived functors. Most of the six operations are not t-exact (even on one side) for the perverse t-structure. However, pullback functors are t-exact 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 $\mathrm{D}^b_c(X(\mathbb{C}),\mathbb{Q})$ on the $\mathbb{C}$ -points of an algebraic variety X over a subfield of $\mathbb{C}$ is the derived category of its constructible heart. We show that one can adapt his argument to mixed Hodge modules and perverse Nori motives, and obtain the following result.

Theorem 1.2 (Corollary 4.19). Let X be a quasi-projective k-variety. Denote by $\mathrm{MHM}_c(X)$ (respectively, by ${\mathcal{M}_{\mathrm{ct}}}(X)$ ) the heart of the constructible t-structure on $\mathrm{D}^b(\mathrm{MHM}(X))$ (respectively, on $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ ). The canonical $\infty$ -functors

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

if k is a subfield of $\mathbb{C}$ , and

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

for an arbitrary field k, are equivalences of $\infty$ -categories.

This theorem enables us to make the handy change of variables $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\simeq\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ . As pullback functors $f^*$ are easily seen to be $\infty$ -functors this implies they are derived functors of their restrictions to the constructible hearts, so that the functoriality of the constructible heart, which can be written by hand, immediately gives the functoriality of the derived $\infty$ -category! This consideration was the starting idea of this paper.

Let $\mathcal{DN}(X)=\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ be the indization of the bounded derived $\infty$ -category of ${\mathcal{M}_{\mathrm{perv}}}(X)$ . This is a compactly generated stable presentably symmetric monoidal $\infty$ -category whose category of compact objects is $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ . The functor $\mathrm{Nor}^*:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ extends to a functor

\[\mathrm{Nor}^*:\mathcal{DM}^{\mathrm{\acute{e}t}}(X)\to\mathcal{DN}(X)\]

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

Proposition 1.3 (Proposition 6.16). For each scheme X of finite type over some field k of characteristic zero, denote by $p_X:X\to\mathrm{Spec}\mathbb{Q}$ the unique morphism. Then $p_X^*\mathscr{N}_\mathbb{Q}\simeq\mathscr{N}_X$ and the natural functor

\[\mathrm{Mod}_{\mathscr{N}_X}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X))\to\mathcal{DN}(X)\]

is an equivalence of categories that commutes with pullback functors.

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

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

Corollary 1.4 (Corollary 6.18). The $\infty$ -functor $X\mapsto\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ has a unique extension $\mathcal{DN}_c$ to quasi-compact and quasi-separated schemes of characteristic zero such that for every projective system $(X_i)_i$ of such schemes with affine transitions morphisms the natural functor

\[ \mathrm{colim}_i\mathcal{DN}_c(X_i)\to\mathcal{DN}_c(\lim_i X_i)\]

is an equivalence of $\infty$ -categories.

Now that a well-behaved comparison functor $\mathrm{Nor}^*:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{DN}_c(X)$ is available, one wants to see to what extent we can compare both categories. We have a result in that direction, conditional on the existence of a motivic t-structure on $\mathrm{DM}_\mathrm{gm}(K)$ for every field K of characteristic zero, a deep conjecture that we know implies all standard conjectures in characteristic zero [Bei12]. Moreover, by the work of Bondarko in [Bon15, Theorem 3.1.4], the existence of motivic t-structure for all fields of characteristic zero implies that for all finite-type k-schemes, we have a perverse and a constructible t-structure on $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ .

Theorem 1.5 (Theorem 6.22). Assume that a motivic t-structure exists for all fields of characteristic zero. Let X be a finite-type k scheme for such a field k. Then the heart of the perverse t-structure of $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ is canonically equivalent to the category ${\mathcal{M}_{\mathrm{perv}}}(X)$ of perverse Nori motives. Moreover, the functor $\mathrm{Nor}^* : \mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is an equivalence of stable $\infty$ -categories. This implies that $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ is the derived category of both its perverse and its constructible heart.

In particular, over a field we would have $\mathrm{DM}_\mathrm{gm}(k,\mathbb{Q})\simeq\mathrm{D}^b(\mathcal{MM}(k))$ , showing that the ‘even more optimistic’ expectation of [And04, 21.1.8] is in fact no more optimistic than expecting the motivic t-structure to exist.

2. Organisation of the paper

After some recollections on motivic constructions in § 3, in § 4 we review Nori’s proof in [Nor02] that the category of cohomologically constructible complexes of sheaves on the $\mathbb{C}$ -points of a quasi-projective variety X is equivalent to the derived category of the abelian 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 new idea here was to replace local systems with dualisable objects and cohomology of the global sections with $\mathrm{Hom}_{\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))}(\mathbb{Q}_X,-[q])$ . The main point of the proof is to show effaceability of cohomology of the constant object on an affine n-space.

We divide § 5 into two parts. In the first part 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 six operations on mixed Hodge modules and on perverse Nori motives can be lifted to $\infty$ -categorical functors of stable $\infty$ -categories.

In § 6, we use what has been proven on $\infty$ -categorical enhancements to construct the realisation functors. We then prove that Nori motives are modules over an algebra in $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ . The arguments more or less follow Ayoub’s proof of the similar statement for the Betti realisation in [Ayo22]. Finally, we prove the result related to the t-structure conjecture. The argument is the repeated use of the universal properties of $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ and ${\mathcal{M}_{\mathrm{perv}}}(X)$ that forces the Nori realisation functor to be an equivalence when $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ has a perverse t-structure.

2.1 Forthcoming and future work

In [Tub24], using the $\infty$ -categorical enhancement of mixed Hodge modules, we will extend them to Artin stacks, together with the operations, t-structures and weights. In joint work with Raphaël Ruimy [RT24] we construct an integral version of the derived category of perverse Nori motives. This gives an Abelian category of motivic sheaves with integral coefficients over any finite-dimensional scheme of characteristic zero.

2.2 Notations and conventions

Throughout the paper, k is a field of characteristic zero. When k is a subfield of $\mathbb{C}$ , if X is a finite-type k-scheme, we denote by $\mathrm{D}_c^b(X,\mathbb{Q})$ 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 $\mathrm{Perv}(X,\mathbb{Q})$ defined in [BBD82]. For a general field k, we denote by $\mathrm{D}^b_c(X,\mathbb{Q}_\ell)$ the category of constructible bounded complexes of $\mathbb{Q}_\ell$ -adic étale sheaves on X as in [HRS23].

We use the language of $\infty$ -categories as developed in [Lur18a] by Lurie, after Joyal’s work on quasi-categories. We fix an universe and call an $\infty$ -category ‘small’ if its mapping spaces and core are in this universe. We denote by $\Delta$ the simplicial category. We denote by $\mathrm{Cat}_\infty$ the $\infty$ -category of small $\infty$ -categories and by $\mathrm{Pr}^L$ the $\infty$ -category $\infty$ -category of presentable $\infty$ -categories. If $\mathcal{C}$ is a symmetric monoidal $\infty$ -category we denote by $\mathrm{CAlg}(\mathcal{C})$ the commutative algebras in $\mathcal{C}$ , and if $A\in\mathcal{C}$ 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 $H\mathbb{Q}$ -modules in spectra $\mathrm{Mod}_\mathbb{Q} := \mathrm{Mod}_{H\mathbb{Q}}(\mathrm{Sp})$ is the unbounded derived category of the abelian category of $\mathbb{Q}$ -vector spaces. We always try to put the index $\mathcal{C}$ when mentioning Hom, as in $\mathrm{Hom}_\mathcal{C}(A,B)$ . When $\mathcal{C}$ is an ordinary category, this is the Hom set, and when $\mathcal{C}$ is an $\infty$ -category this is the $\pi_0$ of the mapping space $\mathrm{Map}_\mathcal{C}(A,B)$ . In a stable $\infty$ -category $\mathcal{C}$ , the mapping spectra will be denoted by $\mathrm{map}_\mathcal{C}(-,-)$ .

3. Recollections

3.1 Categories of étale motives

Ayoub [Ayo10, Ayo14a], Cisinski and Déglise [CD19, CD16] and Voevodsky [Voe02] have constructed triangulated categories of motivic sheaves that are the relative versions of the triangulated category of geometric motives $\mathrm{DM}_{\mathrm{gm}}(k)$ . In this paper, we are interested in the étale versions. We use the stable $\infty$ -categorical construction due to Robalo [Rob15, Section 2.4]. Also, we only consider $\mathbb{Q}$ -linear categories unless otherwise specified.

Let S be a scheme. One starts with the category $\mathrm{Sm}_S$ of smooth S-schemes, and considers the $\infty$ -category $\mathrm{PSh}(\mathrm{Sm}_S,\mathrm{Sp})$ of presheaves of spectra on it. The category of rational effective étale motivic sheaves is the full subcategory of $\mathrm{PSh}(\mathrm{Sm}_S,\mathrm{Sp})$ whose objects are the presheaves $\mathcal{F}$ that are $\mathbb{Q}$ -linear (i.e., whose image lands in $\mathcal{D}(\mathbb{Q})\simeq\mathrm{Mod}_{H\mathbb{Q}}\subset \mathrm{Sp}$ ), $\mathbb{A}^1$ -invariant (i.e., the natural map $\mathcal{F}(X)\to\mathcal{F}(\mathbb{A}^1_X)$ is an equivalence for any smooth S-scheme X), and satisfy étale hyper-descent (i.e., for any étale hypercover $U_\bullet\to X$ of a smooth S-scheme X, the natural map $\mathcal{F}(X)\to \lim_{\Delta}\mathcal{F}(U_n)$ is an equivalence). It is a symmetric monoidal stable $\infty$ -category, the tensor product being inherited from the monoidal structure of $\mathrm{Sm}_S$ given by the fiber product. Inverting the object $\mathbb{Q}(1):=\mathrm{cofib}(S\xrightarrow{1}\mathrm{Gm}_S)$ for the tensor product gives the category that we will denote by $\mathcal{DM}^{\mathrm{\acute{e}t}}(S)$ and call the category of Voevodsky (étale) motives.

In symbols, we have

(3.1) \begin{equation}\mathcal{DM}^{\mathrm{\acute{e}t}}(S):= (\mathrm{L}_{\mathbb{A}^1}\mathrm{Shv}_{\acute{e}t}^\wedge(\mathrm{Sm}_S,\mathcal{D}(\mathbb{Q})))[({\mathbb{G}_m}_S/1_S)^{-1}].\end{equation}

This is a stable presentably symmetric monoidal $\infty$ -category.

We will denote by $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(S)$ the full subcategory of $\mathcal{DM}^\mathrm{\acute{e}t}(S)$ whose objects are compact objects (i.e., the M in $\mathcal{DM}^{\mathrm{\acute{e}t}}(S)$ such that $\mathrm{Hom}_{\mathcal{DM}^{\mathrm{\acute{e}t}}}(M,-)$ commutes with small filtered colimits). If X is a smooth S-scheme and j is an integer, we denote by M(X)(j) the image of X under the Yoneda functor $\mathrm{Sm}_S\to\mathcal{DM}^\mathrm{\acute{e}t}(S)$ , tensored by $\mathbb{Q}(1)^{\otimes j}$ . 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 $\mathrm{M}(X)(-n)$ for X a smooth S-scheme and $n\in\mathbb{N}$ . As $\mathcal{DM}^{\mathrm{\acute{e}t}}(S)$ is compactly generated, we have $\mathrm{Ind}\mathcal{DM}^{\mathrm{\acute{e}t}}_c(S)=\mathcal{DM}^{\mathrm{\acute{e}t}}(S)$ . Moreover, if $S =\mathrm{Spec} k$ is the spectrum of a field, then there is a natural equivalence $\mathrm{ho}(\mathcal{DM}^{\mathrm{\acute{e}t}}_c(\mathrm{Spec} k))\simeq \mathrm{DM}_\mathrm{gm}(k,\mathbb{Q})$ between the homotopy category of étale motives and the classical triangulated category of geometric Voevodsky motives over a field.

Let S be a scheme. The categories $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ for X ranging through S-schemes can be put together to give a functor $\mathrm{Sch}_S^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Pr}^L)$ from the category of finite-type S-schemes to the $\infty$ -category of presentable symmetric monoidal $\infty$ -categories (see [Rob14, Section 9.1]). Moreover, this functor $\mathcal{DM}^{\mathrm{\acute{e}t}}$ is a coefficient system in the sense of Drew and Gallauer (see Definition 7.1), and the $\infty$ -category of constructible objects $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(-)$ is part of a six-functor formalism (see [CD19, Section A.5] for a detailed definition), which includes the fact that all pullbacks $f^*$ have right adjoints $f_*$ , that there exists exceptional functoriality consisting of an adjoint pair $(f_!,f^!)$ , such that $f^!\simeq f^*$ for f étale and $p_*\simeq p_!$ for p proper. If one replaces the étale topology by the Nisnevich topology in the definition of $\mathcal{DM}^\mathrm{\acute{e}t}(S)$ given in (3.1), one obtains $\mathcal{SH}_\mathbb{Q}(S)$ , the rationalisation of the stable motivic homotopy category $\mathcal{SH}(S)$ , which can be obtained by further replacing in (3.1) the $\infty$ -category $\mathcal{D}(\mathbb{Q})$ with the $\infty$ -category Sp of spectra and which is the stabilisation of the $\mathbb{A}^1$ -homotopy category introduced in [MV99]. Except for the computation at $S= \mathrm{Spec}(k)$ , the results stated in this paragraph and the previous one also hold for $\mathcal{SH}(S)$ and $\mathcal{SH}_\mathbb{Q}(S)$ .

When S is of finite type over a subfield of the complex numbers, Ayoub constructed in [Ayo10] the Betti realisation of $\mathcal{DM}^{\mathrm{\acute{e}t}}(S)$ with values in the derived category of sheaves on the analytification $S^{an}$ of S which restricts to $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(S)$ and then lands into the stable category $\mathcal{D}_c^b(S(\mathbb{C}),\mathbb{Q})$ of bounded complexes of sheaves with constructible cohomology. This category is both the derived category of the abelian category 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 on étale motives 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 $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(S)\to\mathcal{D}_c^b(S,\mathbb{Q}_\ell)$ , the latter category being enhanced using condensed coefficients in [HRS23] where it is denoted by $\mathcal{D}_{\mathrm{cons}}(S,\mathbb{Q}_\ell)$ . The $\ell$ -adic realisation functor also has a version on the big category $\mathcal{DM}^{\mathrm{\acute{e}t}}(S)$ by taking indization. All the realisation functors commute with the six operations.

3.2 Perverse Nori motives

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

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

\[\rho_\ell:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to \mathcal{D}^b(X_{\acute{e}t},\mathbb{Q}_\ell)\]

with the perverse cohomology functor

\[{\ ^{\mathrm{p}}\mathrm{H}}^0:\mathcal{D}^b(X_{\acute{e}t},\mathbb{Q}_\ell)\to\mathrm{Perv}(X,\mathbb{Q}_\ell)\]

to obtain a homological functor ${\ ^{\mathrm{p}}\mathrm{H}}^0_\ell\colon \mathcal{DM}_c(X)\to \mathrm{Perv}(X,\mathbb{Q}_\ell).$ Let $\mathcal{R}(\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X))$ be the abelian category of coherent modules over $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ : it is the full subcategory of additive presheaves $\mathcal{F}:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)^\mathrm{op}\to\mathrm{Ab}$ that fit in exact sequences

\[y_M\to y_N\to\mathcal{F}\to 0,\]

where $M\to N$ is a morphism in $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ .

The functor $\mathrm{H}^0_\ell$ factors through $\mathcal{R}(\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X))$ , giving a functor

$$\widetilde{{\ ^{\mathrm{p}}\mathrm{H}}^0_\ell}:\mathcal{R}(\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X))\to \mathrm{Perv}(X,\mathbb{Q}_\ell).$$

Denote by $Z_\ell$ the kernel of $\widetilde{{\ ^{\mathrm{p}}\mathrm{H}}^0_\ell}$ , that is, the objects mapping to zero. The category of perverse Nori motives ${\mathcal{M}_{\mathrm{perv}}}(X)$ is the quotient $\mathcal{R}(\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X))/Z_\ell$ . It comes with a universal functor $\mathrm{H}_\mathrm{univ}\colon \mathcal{DM}^\mathrm{\acute{e}t}_c(X)\to{\mathcal{M}_{\mathrm{perv}}}(X)$ .

By construction, ${\mathcal{M}_{\mathrm{perv}}}(X)$ has the following universal property: any homological functor $H:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{A}$ to an abelian category $\mathcal{A}$ , such that for all $M\in\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ we have $H(M)=0$ whenever ${\ ^{\mathrm{p}}\mathrm{H}}^0_\ell(M)=0$ , factors uniquely through the universal functor

\[\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\xrightarrow{\mathrm{H}_\mathrm{univ}}{\mathcal{M}_{\mathrm{perv}}}(X).\]

A priori this category depends on the chosen $\ell$ , but one can show, using a continuity argument, that this is not the case. Indeed, for finite-type fields (or more generally, for fields embeddable into $\mathbb{C}$ ), the Betti realisation can also be used to define ${\mathcal{M}_{\mathrm{perv}}}(X)$ and the comparison between the $\ell$ -adic realisation functor and the Betti realisation functor (see, for example, the proof of [Ayo24, Proposition 6.10.] that works over any finite-type $\mathbb{C}$ -scheme) gives that the two constructions of perverse Nori motives agree.

In fact, every realisation functor existing on $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ gives us a realisation functor on ${\mathcal{M}_{\mathrm{perv}}}(X)$ : we have the $\ell$ -adic realisation $R_\ell$ and the Betti realisation $R_B$ . Ivorra and Morel have proven that ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec} k)$ coincides with the category of cohomological motives constructed by Nori, and thus also has an Hodge realisation functor.

Ivorra and Morel show in [IM22, Theorem 5.1] that the triangulated derived category of the abelian category of perverse motives $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))$ 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 $f^*,f_*,f_!,f^!$ , together with Verdier duality $\mathbb{D}$ , verifying $f^!\simeq \mathbb{D}\circ f^*\circ\mathbb{D}$ and $f_!\simeq\mathbb{D}\circ f_*\circ\mathbb{D}$ . In his PhD thesis Terenzi [Ter24] constructed the remaining two operations: tensor product and internal $\mathscr{{H om}}\,$ . This proves that $\mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))$ is part of a full six-functor formalism. The six operations commute with the realisation functors $R_\ell$ and $R_B$ making them morphisms of symmetric monoidal stable homotopical 2-functors. The construction of the operations on perverse Nori motives is very similar to Saito’s construction of mixed Hodge modules.

3.3 Mixed Hodge modules

Let X be a separated finite-type $\mathbb{C}$ -scheme. In [Sai90], Saito constructed an abelian category $\mathrm{MHM}(X)$ of mixed Hodge modules. He also constructed the six operations on $\mathrm{D}^b(\mathrm{MHM}(-))$ . There is a faithful exact functor $\mathrm{MHM}(X)\to \mathrm{Perv}(X^\mathrm{an},\mathbb{Q})$ to the abelian category of perverse sheaves on the analytification of X. This induces a functor $\mathrm{D}^b(\mathrm{MHM}(X))\to\mathrm{D}^b(\mathrm{Perv}(X^\mathrm{an},\mathbb{Q}))\to\mathrm{D}^b_c(X^\mathrm{an},\mathbb{Q})$ and gives a natural transformation $\mathrm{D}^b(\mathrm{MHM}(-))\to\mathrm{D}^b_c(-,\mathbb{Q})$ that commutes with the six operations. If $X=\mathrm{Spec} \mathbb{C}$ , then there is a natural equivalence $\mathrm{MHM}(\mathrm{Spec} \mathbb{C})\simeq\mathrm{MHS}^p_\mathbb{Q}$ between polarisable mixed Hodge structures and mixed Hodge modules. One of the most interesting properties of the category 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 the polarisable variations of mixed Hodge structures. We will not go into details about the construction of Saito’s category, and 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.

4. Adapting Nori’s argument to perverse Nori motives

4.1 Hypothesis on the coefficient system

Let k be a field of characteristic zero. For this section we will denote by $\mathrm{Var}_k$ either the category of quasi-projective k-varieties or that of separated reduced k-schemes of finite type, and we will call objects of $\mathrm{Var}_k$ varieties. Choose your favourite $\mathbb{Q}$ -linear triangulated category of coefficients $\mathrm{D}:\mathrm{Var}_k^\mathrm{op}\to \mathrm{Triang}$ with a six-functor formalism (by this we mean a symmetric monoidal stable homotopy 2-functor as in Ayoub [Ayo07]), and a conservative realisation functor $R:\mathrm{D}\to\mathrm{D}^b_c$ compatible with the operations, where $\mathrm{D}^b_c$ is either the derived category of constructible sheaves on the analytification (if k is a subfield of $\mathbb{C}$ ), or the derived category of $\ell$ -adic constructible sheaves. For $X\in\mathrm{Var}_k$ , we will refer to the elements of $\mathrm{D}(X)$ as ‘motives’. For example, D could be the derived category of perverse Nori motives ([IM22, Theorem 5.1] and [Ter24, Main theorem]) or of mixed Hodge modules [Sai90, Theorem 0.1]. Note that in the mixed Hodge modules situation the 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. We assume that for each variety X the category $\mathrm{D}(X)$ has a perverse t-structure for which R is t-exact when $\mathrm{D}^b_c(X)$ is endowed with its perverse t-structure of heart the abelian category of perverse sheaves $\mathrm{Perv}(X)$ .

Recall the following definition.

Definition 4.1. The constructible t-structure on $\mathrm{D}(X)$ is the unique t-structure on $\mathrm{D}(X)$ that makes all pullback functors t-exact. Explicitly, it is the t-structure given by

$$\ ^\mathrm{ct}\mathrm{D}^{\leqslant 0}(X) := \{M \in \mathrm{D}(X)\mid \forall x\in X,\text{ closed, } x^*M\in \mathrm{D}^{p\leqslant 0}(\mathrm{Spec}\kappa(x))\}$$

and

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

That this indeed gives a t-structure can be proven by gluing, using that over a smooth variety, a lisse object (see Definition 4.2) is positive (respectively, negative) for the constructible t-structure if and only it is positive (respectively, negative) for the perverse t-structure when shifted by the dimension.

Denote by ${\mathcal{M}_{\mathrm{ct}}}(X)$ the heart of the constructible t-structure on $\mathrm{D}(X)$ . As R is t-exact and conservative, all known t-exactness results about the six functors are true in D. We will call elements of ${\mathcal{M}_{\mathrm{ct}}}(X)$ constructible motives. Denote by $\mathscr{{H om}}\,(-,-)$ and $\mathbb{Q}_X$ the internal Hom and unit object of $\mathrm{D}(X)$ .

We assume that $\mathrm{D}(X)$ has enough structure so that there exists a natural triangulated functor $\mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{D}(X)$ extending the inclusion of the constructible heart (e.g., if $\mathrm{D}(X)$ is a derived category as shown by Beilinson, Bernstein, Deligne and Gabber in [BBD82, section 3.1] or more generally if $\mathrm{D}(X)$ 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 $\mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec}k))\to\mathrm{D}(\mathrm{Spec} k)$ is an equivalence. This is true for perverse Nori motives and mixed Hodge modules.

From now on in this section by the ‘t-structure’ we mean the constructible t-structure, unless specified otherwise.

Definition 4.2. Let X be a k-variety. A motive L is a lisse object if its realisation R(L) is a local system, that is, a locally constant bounded complex of finite-dimensional $\mathbb{Q}$ -vector spaces. We denote by $\mathrm{D}^{liss}(X)$ the full subcategory of lisse objects in $\mathrm{D}(X)$ .

We will denote by $\mathbb{Q}_X\in{\mathcal{M}_{\mathrm{ct}}}(X)$ the unit object for the tensor structure. Its realisation is the constant sheaf.

Remark 4.3. Note that as $\mathrm{D}(X)$ is closed as a tensor category, given an object $M\in\mathrm{D}(X)$ , we have a natural candidate for the strong dual of M: it is $\mathscr{{Hom}}\,(M,\mathbb{Q}_X)$ . Therefore, an object M is dualisable if and only if the map

(4.1) \begin{equation}\mathscr{H{om}}\,(M,\mathbb{Q}_X)\otimes M \to \mathscr{{H{om}}}\,(M,M), \end{equation}

obtained by the $\otimes$ - $\mathscr{{Hom}}\,$ adjunction from the map $\mathrm{ev}_M\otimes \mathrm{Id}_M : \mathscr{{Hom}}\,(M,\mathbb{Q}_X)\otimes M\otimes M\to \mathbb{Q}\otimes M\simeq M$ , is an isomorphism. We will denote by $M^\vee = \mathscr{{Hom}}\,(M,\mathbb{Q}_X)$ the dual of M when M is dualisable.

Lemma 4.4. Let X be a k-variety. An object $K\in \mathrm{D}(X)$ is lisse if and only if it is dualisable. In particular, for every object $N\in \mathrm{D}(X)$ , there exists a dense open on which N is dualisable. If an object $L\in{\mathcal{M}_{\mathrm{ct}}}(X)$ is lisse, then its dual $L^\vee$ is also an object of ${\mathcal{M}_{\mathrm{ct}}}(X)$ .

Proof. For the first point, dualisable objects in $\mathrm{D}^b_c(X,\mathbb{Q})$ are exactly local systems by Ayoub [Ayo22, Lemma 1.24] in the analytic case, and by [HRS23, Proposition 7.6] together with [MW22, Corollary 7.4.12.] in the $\ell$ -adic case. As the realisation functor is conservative, we can test if the map (4.1) is an isomorphism after applying the realisation functor. If $L\in{\mathcal{M}_{\mathrm{ct}}}(X)$ is dualisable, the functor $\mathscr{{Hom}}\,(L,-) = -\otimes L^\vee$ is t-exact because it is left and right adjoint to the exact functor $-\otimes L$ . Therefore, $L^\vee=\mathscr{H{ om}}\,(L,\mathbb{Q}_X)\in{\mathcal{M}_{\mathrm{ct}}}(X)$ lies in the heart.

The second point follows from the fact that after realisation any constructible motive is locally constant on a stratification, hence, on some dense open subset.

4.2 Cohomology of motives over an affine variety

Definition 4.5. (i) 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.

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

Lemma 4.6. Let X be a variety and M be a constructible motive on $\mathbb{A}^1_X$ . Assume that there is a smooth open subset $U\subset \mathbb{A}^1_X$ on which M is lisse and such that the restriction of $\pi:\mathbb{A}^1_X\to X$ to $Z = \mathbb{A}^1_X\setminus U$ is finite and surjective. Assume that ${M}_{\mid {Z}} = 0$ . Then there exists $N\in {\mathcal{M}_{\mathrm{ct}}}(\mathbb{A}^1_X)$ such that $\pi_*N = 0$ and that M injects into N.

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

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

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

Moreover, Nori [Nor02] proves that for all $p\geqslant 0$ we have $\mathrm{H}^p(\pi_*N)=0$ , which is equivalent to $\pi_*N=0$ .

If 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.

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 4.7 (Cf. [Nor02, Theorem 1]). For every $n\in\mathbb{N}$ , the motive $\mathbb{Q}_{\mathbb{A}^n_k}\in{\mathcal{M}_{\mathrm{ct}}}(\mathbb{A}^n_k)$ 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 $M\in {\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec} k)$ , as $\mathrm{D}(\mathrm{Spec} k)= \mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(\mathrm{Spec} k))$ , the functors $\mathrm{Hom}_{\mathrm{D}(\mathrm{Spec} k)}(M,-[q])$ are effaceable by [BBD82, Proposition 3.1.16].

Suppose that $n\geqslant 1$ and that the result is known for $n-1$ . Let $M\in {\mathcal{M}_{\mathrm{ct}}}(\mathbb{A}^n_k)$ and $q>0$ . Let $f\in k[T_1,\dots,T_n]$ be a polynomial such that M is lisse when restricted to the open subset $U=D(f)\subset \mathbb{A}^n_k$ . Up to a change of coordinates, we may assume that the projection $\pi:\mathbb{A}^n_k\to\mathbb{A}^{n-1}_k$ on the $n-1$ first coordinates gives a finite map $\pi_{|V(f)}:V(f)\to \mathbb{A}^{n-1}_k$ when restricted to $Z=V(f)$ . Denote by j and i the inclusions of U and Z in $\mathbb{A}^n_k$ .

By Lemma 4.6, $j_!{M}_{\mid {U}}$ is a submotive of a constructible motive N’ such that $\pi_*N'=0$ . Taking N to be the pushout of the maps $j_!{M}_{\mid {U}}\to M$ and $j_!{M}_{\mid {U}}\to N'$ we get a morphism of exact sequences

with $\gamma$ also injective.

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

By the induction hypothesis, we can find an injection $g:\pi_*N\to K$ with K a constructible motive such that $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^{n-1}_k)}(\mathbb{Q}_{\mathbb{A}^{n-1}_k},g[q]) = 0$ . Let L be the pushout of the counit $\pi^*\pi_*N \to N$ and the injection $\pi^*g:\pi^*\pi_*N\to \pi^*K$ (note that $\pi^*$ is t-exact). We get a morphism of exact sequences.

(4.2)

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

$$\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},\pi^*\pi_*N[q])\simeq\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^{n-1}_k)}(\mathbb{Q}_{\mathbb{A}^{n-1}_k},\pi_*\pi^*\pi_*N[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},N[q]) $$

is an isomorphism. Moreover, the map

$$\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},\pi^*\pi_*N[q])\xrightarrow{\pi^*g} \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},\pi^*K[q]), $$

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

Therefore, when applying $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^n_k)}(\mathbb{Q}_{\mathbb{A}^n_k},-[q])$ to the left square of (4.2), we see that the map

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

induced by h, factors through the zero map and hence is zero.

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

Corollary 4.8. If X is affine, the object $\mathbb{Q}_X$ is admissible.

Proof. Let $q>0$ . Take $i:X\to \mathbb{A}^n_k$ a closed immersion and $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$ . By Theorem 4.7, there is an injection $i_*M\to K$ with K a constructible motive such that $\mathrm{Hom}_{\mathrm{D}(\mathbb{A}^{n})}({\mathbb{Q}}_{\mathbb{A}^{n}_k},i_*M[q])\to \mathrm{Hom}_{\mathrm{D}(\mathbb{A}^{n})}({\mathbb{Q}}_{\mathbb{A}^{n}_k},K[q])$ is the zero map. We have an inclusion $f:i^*i_*M\simeq M\to N:=i^*K$ , and a commutative diagram

giving that $f\circ-$ is zero, thus finishing the proof.

4.3 Admissibility of constant motives on any variety

Notation 4.9. 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 $M[U]:=j_!{M}_{\mid {U}}$ . Note that by localisation we have $M/M[U]\simeq i_*{M}_{\mid {Z}}$ with $i:Z=X\setminus U\to X$ the closed complement.

Lemma 4.10 (Stability of admissibility). Let

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

be an exact sequence of constructible motives on X.

1. (i) Assume that M” is admissible and at least one of M’, M is admissible. Then all three are admissible.

2. (ii) Assume that M’ and M are admissible. If the functor

   $$\mathrm{coker}\ (\mathrm{Hom}_{\mathrm{D}(X)}(M,-)\to\mathrm{Hom}_{\mathrm{D}(X)}(M',-))$$
   is effaceable, then M” is admissible.

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

Corollary 4.11 [Nor02, Lemma 3.5] Let $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$ .

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

2. (ii) If V,W are open subsets 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 (ii) of Lemma 4.10. For part (i), 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 $f\colon\mathrm{Hom}_{\mathrm{D}(X)}(M,Q)\to\mathrm{Hom}_{\mathrm{D}(X)}(M[U],Q)$ 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,

$$\mathrm{Hom}_{\mathrm{D}(X)}(M,i_*{P}_{\mid {Z}})\oplus \mathrm{Hom}_{\mathrm{D}(U)}({M}_{\mid {U}},{P}_{\mid {U}})\simeq\mathrm{Hom}_{\mathrm{D}(X)}(M,Q)$$

and

$$\mathrm{Hom}_{\mathrm{D}(U)}({M}_{\mid {U}},{P}_{\mid {U}}) \simeq 0\oplus \mathrm{Hom}_{\mathrm{D}(X)}(j_!{M}_{\mid {U}},j_*{P}_{\mid {U}}) \simeq \mathrm{Hom}_{\mathrm{D}(X)}(M[U],Q), $$

thus is f surjective as it is just the projection on the second factor.

For part (ii), first one has the Mayer–Vietoris exact sequence

(4.3) \begin{equation}0\to M[V\cap W]\to M[V]\oplus M[W] \to M[V\cup W]\to 0.\end{equation}

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

(4.4) \begin{equation}\mathrm{Hom}_{\mathrm{D}(X)}(M[V],Q)\xrightarrow{a^*} \mathrm{Hom}_{\mathrm{D}(X)}(M[U],Q)\end{equation}

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

Proposition 4.12 (Avoiding the use of injective objects, after Nori [Nor02, Remark 3.8]). Let $\mathcal{A},\mathcal{B}$ and $\mathcal{C}$ be abelian categories. Let $G:\mathcal{B}\to\mathcal{A}$ be a functor which has an exact left adjoint. Let $H:\mathcal{A}\to \mathcal{C}$ be an effaceable functor. Then HG is also effaceable.

Proof. Let F be a left adjoint of G. Let B be 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 $\varepsilon:FG(B)\to B$ . Let B’ be the pushout of these two maps.

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

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

Corollary 4.13. Let X and Y be varieties. Let $F:\mathrm{D}(X)\to \mathrm{D}(Y)$ be a t-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 4.12, $H\circ G$ is therefore effaceable. We have a natural isomorphism $H\circ G \simeq \mathrm{Hom}_{\mathrm{D}(Y)}(F(M),-[q])$ because G is t-exact, which gives the claim.

Corollary 4.14. Let X be a variety. Let $M\in{\mathcal{M}_{\mathrm{ct}}}(X)$ be an admissible motive.

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

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

Proof. We use Corollary 4.13. For the first point, we have the adjunction $(j_!,j^*)$ where $j_!$ 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 $\mathscr{{H om}}\,(L,-)$ and right adjoint to $\mathscr{H{ om}}\,(L^\vee,-)$ , hence is t-exact. The same can be said about $\mathscr{{H om}}\,(L,-)=-\otimes L^\vee$ .

Proposition 4.15 [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 the open immersion. We prove the theorem by induction on the number n of affines needed to cover U. If U is affine, Corollary 4.8 ensures that $\mathbb{Q}_U$ is admissible, hence by Corollary 4.14, $j_!\mathbb{Q}_U=\mathbb{Q}_X[U]$ is admissible. For the induction step, 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 $\mathbb{Q}_X[V]$ , $\mathbb{Q}_X[V\cap W]$ and $\mathbb{Q}_X[W]$ are admissible. Then Corollary 4.11 ensures that $\mathbb{Q}_X[U]$ is admissible.

Corollary 4.16 [Nor02, Theorem 2]. Let X be a variety over k. Then the constant motive $\mathbb{Q}_X$ is admissible, that is, for every $q>0$ , every constructible motive $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$ can be embedded in a constructible motive $N\in {\mathcal{M}_{\mathrm{ct}}}(X)$ such that the map

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

vanishes.

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

4.4 Lisse motives enter the stage

Proposition 4.17. Let X be a variety. Any lisse motive in ${\mathcal{M}_{\mathrm{ct}}}(X)$ is admissible.

Proof. By Corollary 4.16 the unit object $\mathbb{Q}_X$ on X is admissible. By Corollary 4.14, $L \simeq \mathbb{Q}_X\otimes L$ is admissible.

Theorem 4.18. Let X be a variety. Then any $M\in {\mathcal{M}_{\mathrm{ct}}}(X)$ is admissible, that is, for every $q>0$ , the functor $\mathrm{Hom}_{\mathrm{D}(X)}(M,-[q]):{\mathcal{M}_{\mathrm{ct}}}(X)\to \mathrm{Vect}_\mathbb{Q}$ 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 ${M}_{\mid {V}} = 0$ . We follow the proof of [Nor02, Proposition 3.10].

Let $S^d$ be the union of the irreducible components of maximal dimension of S. We denote by d the dimension of S. Let $U_\mathrm{liss}$ 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 the intersection $U_1\cap S^d$ is contained in $U_\mathrm{liss}$ and is non-empty. Let $U_2'$ be the open subset of X obtained by removing from $U_1$ the irreducible components of S that are not of maximal dimension. We have that $U_2'\cap S$ is non-empty and contained in $U_\mathrm{liss}$ . We choose a non-empty affine open subset $U_2$ of $U_2'$ such that $U_2\cap S$ is non-empty and a Noether normalisation $g_2\colon U_2\cap S\to \mathbb{A}^d_k$ . It extends to a map $f_2\colon U_2\to \mathbb{A}^d_k$ . Indeed, because $U_2\cap S$ is closed in $U_2$ and the three schemes $U_2$ , $U_2\cap S$ and $\mathbb{A}^d_k$ are affine, we may choose in $\mathscr{O}_{U_2}(U_2)$ lifts of the images of the coordinates of $\mathbb{A}^d_k$ in $\mathscr{O}_{U_2\cap S}(U_2\cap S)$ . Let $W\subset \mathbb{A}^d_k$ be a non-empty open subset over which $g_2$ is smooth. As $g_2$ is of relative dimension 0 in fact $g_2$ is étale over W. We set $U := f_2^{-1}(W)$ .

Let $Z = U\cap S$ and $g\colon Z\to W$ be the restriction of $g_2$ to Z. By construction we have $Z = g_2^{-1}(W)$ , thus the map $g\colon Z\to W$ is finite and étale. We also let $f:U\to W$ be the restriction of $f_2$ to U. The situation may be summarised in a commutative diagram

where we have denoted by i the inclusion of Z inside U.

Note that because the restriction of M to Z is lisse and g is finite étale, the object $g_*M_{\mid Z}$ is still lisse, so that $L:=f^*g_*M_{\mid Z}$ is lisse. The restriction $i^*L$ to Z of L is $g^*g_*M_{\mid Z}$ which has $M_{\mid Z}$ as a direct factor because g is étale.

Because the support of $M_{\mid U}$ is contained in Z we have that $i_*M_{\mid Z}= i_*i^*M_{\mid U}\simeq M_{\mid U}$ . By the above, this shows that $M_{\mid U}$ is a direct factor of $i_*L_{\mid Z}$ . Let j be the inclusion of the open complement of Z in U. By localisation we have that $i_*L_{\mid Z} \simeq L/j_!j^*L$ . By Proposition 4.3 the objects L and $j^*L$ are admissible, thus, as by Corollary 4.14, the object $j_!j^*L$ is then admissible, we deduce by Corollary 4.11 that the object $i_*L_{\mid Z}$ is admissible. Thus, the restriction $M_{\mid U}$ of M to U is admissible as a direct factor of an admissible object.

If $\iota\colon U\to X$ is the open immersion then $\iota_!M_{\mid U}$ is admissible by Corollary 4.14. We have an exact sequence

$$0\to \iota_!M_{\mid U}\to M\to M/\iota_!M_{\mid U}\to 0,$$

in which the support of $M/\iota_!M_{\mid U}$ is strictly smaller than the support of M. By Noetherian induction $M/\iota_!M_{\mid U}$ is admissible, thus by Lemma 4.10 we finally obtain that M is admissible.

Corollary 4.19. Let X be a variety. The natural functor $\mathrm{D}^b({\mathcal{M}_{\mathrm{ct}}}(X))\to\mathrm{D}(X)$ is an equivalence.

Proof. We apply [BBD82, Proposition 3.1.16]. The conditions are verified because of the previous Theorem 4.18.

5. Higher categorical enhancements

5.1 Categorical preliminaries

Recall the following fundamental definition due to Lurie.

Definition 5.1. An $\infty$ -category $\mathcal{C}$ is stable if it has all finite limits and colimits, if initial objects coincide with final objects, 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 $\mathrm{ho}(\mathcal{C})$ of a stable $\infty$ -category is always a triangulated category [Lur17, Theorem 1.1.2.14], and exact functors induce triangulated functors. By definition, a t-structure on a stable $\infty$ -category is a t-structure on its homotopy category.

Definition 5.2. 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 $\mathrm{Ch}^b(\mathcal{A})$ of bounded complexes of objects of $\mathcal{A}$ . It is a stable $\infty$ -category with a t-structure, and its homotopy category is the usual derived category $\mathrm{D}^b(\mathcal{A})$ .

If $F\colon \mathcal{A}\to \mathcal{C}$ is a functor between an abelian category and a stable $\infty$ -category, we will say that F is exact if it preserves finite coproducts and it sends short exact sequences to exact triangles.

We will use the following result from Bunke, Cisinski, Kasprowski and Winges.

Theorem 5.3 [BCKW24, Corollary 7.4.12]. Let $\mathcal{C}$ be a stable $\infty$ -category and let $\mathcal{A}$ be an abelian category. Then restriction to the heart gives an equivalence of $\infty$ -categories

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

between $\infty$ -categories of exact functors.

There is an easy generalisation with multiple variables.

Corollary 5.4. Let $\mathcal{C}$ be a stable $\infty$ -category and let $\mathcal{A}_1,\dots,\mathcal{A}_n$ be abelian categories. Denote by $\mathrm{Fun}^{\mathrm{nex}}(\prod_i\mathcal{D}^b(\mathcal{A}_i),\mathcal{C})$ the $\infty$ -category of n-multi-exact functors $\prod_i \mathcal{D}^b(\mathcal{A}_i)\to\mathcal{C}$ , that is, functors that are exact in each variable. Denote also similarly by $\mathrm{Fun}^{\mathrm{nex}}(\prod_i\mathcal{A}_i,\mathcal{C})$ the $\infty$ -category of n-multi-exact functors. Then the restriction to the hearts functor

(5.1) \begin{equation}\mathrm{Fun}^{\mathrm{nex}}\Big(\prod_i\mathcal{D}^b(\mathcal{A}_i),\mathcal{C}\Big)\to\mathrm{Fun}^{\mathrm{nex}}\Big(\prod_i\mathcal{A}_i,\mathcal{C}\Big)\end{equation}

is an equivalence of $\infty$ -categories.

Proof. We give the proof for $n=2$ ; the general case is similar and can be reduced to the case $n=2$ by induction. We have canonical equivalence of $\infty$ -categories $\mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_1)\times\mathcal{D}^b(\mathcal{A}_2),\mathcal{C})\simeq \mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_1),\mathrm{Fun}(\mathcal{D}^b(\mathcal{A}_2),\mathcal{C}))$ . This equivalences induces

$$\mathrm{Fun}^{\mathrm{2ex}}(\mathcal{D}^b(\mathcal{A}_1)\times\mathcal{D}^b(\mathcal{A}_2),\mathcal{C})\simeq\mathrm{Fun}^\mathrm{ex}(\mathcal{D}(\mathcal{A}_1),\mathrm{Fun}^\mathrm{ex}(\mathcal{D}^b(\mathcal{A}_2),\mathcal{C})).$$

By Theorem 5.3 the latter category is exactly $\mathrm{Fun}^{\mathrm{ex}}(\mathcal{A}_1,\mathrm{Fun}^{\mathrm{ex}}(\mathcal{A}_2,\mathcal{C}))$ , which again is equivalent to $ \mathrm{Fun}^{\mathrm{2ex}}(\mathcal{A}_1\times\mathcal{A}_2,\mathcal{C})$ .

We will also need a result on the monoidal structure of the bounded derived category. We denote by $\mathrm{SymMono}_1$ the 2-category of symmetric monoidal 1-categories and symmetric monoidal functors.

Proposition 5.5. Let $\mathcal{B}$ be a symmetric monoidal abelian category such that the tensor product is exact in each variable. Then there is a canonical symmetric monoidal structure on the stable $\infty$ -category $\mathcal{D}^b(\mathcal{B})$ such that the inclusion functor $\mathcal{B}\to \mathcal{D}^b(\mathcal{B})$ is symmetric monoidal. Moreover, for any category I and any functor $\mathcal{A}: I\to \mathrm{SymMono}_1$ such that for each $i\in I$ the category $\mathcal{A}(i)$ is a symmetric monoidal abelian category with exact tensor product as above, and such that the transition functors are exact, there is a lift of the functor $\mathcal{D}^b\circ \mathcal{A}\colon I\to \mathrm{Cat}_\infty$ to a functor $I\to \mathrm{CAlg}(\mathrm{Cat}_\infty).$

Proof. The first case follows from the second with $I=*$ . Let $\mathcal{A}:I\to \mathrm{SymMono}_1$ be a diagram of symmetric monoidal abelian categories and symmetric monoidal exact functors, and such that for each $i\in I$ the tensor product on $\mathcal{A}(i)$ is exact in both variables. We can compose this functor with $\mathrm{Ch}^b(-)$ to obtain a diagram of symmetric monoidal additive categories (the monoidal structure on $\mathrm{Ch}^b(\mathcal{A}(i))$ is induced by that of $\mathcal{A}(i)$ , using the sign trick). By [ML98, Chapter XI, Section 1, Theorem 1] we know that any symmetric monoidal 1-category has all higher coherences. Written differently as in Lurie [Lur17, Corollary 5.1.1.7], the forgetful functor induces an equivalence of 2-categories

$$\mathrm{CAlg}(\mathrm{Cat}_1)\to \mathrm{SymMono}_1.$$

Thus, we can see $\mathrm{Ch}^b\circ\mathcal{A}$ as a functor

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

Now by [Lur17, Theorem 2.4.3.18 and Proposition 2.4.2.5] the data of this functor is equivalent to the data of a $I^\amalg$ -monoid, which is classified by a cocartesian fibration $p\colon \mathfrak{C}\to I^\amalg$ (we denote by $I^\amalg$ the $\infty$ -operad constructed in [Lur17, Construction 2.4.3.1]). By [Hin16, Proposition 2.1.4] we can fiberwise invert quasi-isomorphisms to obtain a cocartesian fibration $q\colon \mathfrak{D}\to I^\amalg$ which classifies (using unstraightening ¹ and [Lur17, Theorem 2.4.3.18 and Proposition 2.4.2.5] again) a functor $I\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$ 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 $\mathcal{A}(i)$ .

We also have a version of the universal property of $\mathcal{D}^b(\mathcal{A})$ (see Theorem 5.3) that works in families.

Proposition 5.6. Let $\mathcal{C}: I\to \mathrm{CAlg}(\mathrm{Cat}_\infty)$ 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 $\mathcal{D}^b(\mathcal{C}(i)^\heartsuit)\to \mathcal{C}(i)$ assemble to give a natural transformation $\mathcal{D}^b(\mathcal{C}^\heartsuit)\Rightarrow \mathcal{C}$ of functors $I\to \mathrm{CAlg}(\mathrm{Cat}_\infty)$ .

Proof. For $\mathcal{B}$ an abelian 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 $\mathrm{Ch}^b(\mathcal{B})$ 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

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

from the $\infty$ -category of additive $\infty$ -categories and coproduct-preserving functors to the $\infty$ -category $\mathrm{Cat}_\infty^\mathrm{ex}$ of small stable $\infty$ -categories and exact functors. Moreover, this functor is left adjoint to the forgetful functor

$$\mathrm{ff}\colon \mathrm{Cat}_\infty^\mathrm{ex}\to\mathrm{Cat}_\infty^\mathrm{add}.$$

By Cisinski [Cis19, Theorem 6.1.22] this induces an adjunction

(5.2)

where the category $I^\amalg$ is as introduced in the proof of the above proposition. Our assumptions on $\mathcal{C}$ imply that if we denote by $\mathcal{A}(i)$ the heart of the t-structure on $\mathcal{C}(i)$ , we have a natural transformation $\mathcal{A}\Rightarrow\mathcal{C}$ of functors $I\to\mathrm{CAlg}(\mathrm{Cat}_\infty^\mathrm{add}$ ). Now, as in the proof of Proposition 5.5, we have from Lurie [Lur17, Theorem 2.4.3.18 and Proposition 2.4.2.5] that the data of this natural transformation is equivalent to the data of a natural transformation $\mathfrak{A}\Rightarrow \mathfrak{C}$ of $I^\amalg$ -monoids $I^\amalg\to \mathrm{Cat}_\infty^\mathrm{add}$ (see [Lur17, Definition 2.4.2.1] for a definition). By (5.2), this gives a natural transformation

$$\mathrm{can}\colon\mathcal{K}^b\circ \mathfrak{A} \Rightarrow \mathfrak{C}$$

of functors $I^\amalg\to \mathrm{Cat}_\infty^\mathrm{ex}$ . We claim that $\mathcal{K}^b\circ\mathfrak{A}$ is still a $I^\amalg$ -monoid. Indeed, it suffices to prove that $\mathcal{K}^b$ preserves finite products, but both in $\mathrm{Cat}_\infty^\mathrm{ex}$ and in $\mathrm{Cat}_\infty^\mathrm{add}$ finite product and coproduct agree (this can be proven exactly as in [BCKW24, Lemma 2.1.38]), so that this follows from $\mathcal{K}^b$ being a left adjoint which thus preserves colimits.

Unravelling the definitions, we have proven that the natural transformation $\mathcal{K}^b(\mathcal{A})\Rightarrow \mathcal{C}$ is well defined and is symmetric monoidal. We will now show that this factors through $\mathcal{D}^b(\mathcal{A})$ , and that everything stays symmetric monoidal. For this, we will see our map of $I^\amalg$ -monoid $\mathcal{K}^b\circ\mathfrak{A}\Rightarrow \mathfrak{C}$ through straightening, thus as a commutative triangle

where $\kappa$ and $\gamma$ are the cocartesian fibrations classifying the functors $\mathcal{K}^b\circ\mathfrak{A}$ and $\mathfrak{C}$ . We consider a marking W on $\int_{I^\amalg}\mathcal{K}^b\circ\mathfrak{A}=:\mathscr{K}$ to be the set of arrows that are products of quasi-isomorphisms. That is, an arrow f in $\mathscr{K}$ is marked if there exist an integer n and an object $\underline{i}=(i_1,\dots,i_n)\in I^\amalg_{\langle n\rangle}$ such that f is a map in the fiber $\prod_{j=1}^n\mathcal{K}^b(\mathcal{A}(i_j))$ above $\underline{i}$ and is of the form $f_1\times\cdots\times f_n$ with each $f_j$ being a quasi-isomorphism. Then the cocartesian fibration $\kappa$ , together with the data of W and the marking of isomorphisms in $I^\amalg$ , is a marked cocartesian fibration in the sense of Hinich [Hin16, Definition 2.1.1]. Because each $\mathcal{A}(i)$ is the heart of a t-structure on $\mathcal{C}(i)$ , all maps in W are sent to isomorphisms in $\int_{I^\amalg}\mathfrak{C}=:\mathscr{C}$ . Indeed, over a multi-index $(i_1,\dots,i_n)$ in $I^\amalg$ , the functor $\mathfrak{r}$ becomes the functor

\[\prod_{j=1}^n \mathcal{K}^b(\mathcal{A}(i_j))\to \prod_{j=1}^n\mathcal{C}(i_j),\]

which sends marked maps to isomorphisms by [BCKW24, Proposition 7.4.11]. Thus, the map

$$\mathscr{K}\to\mathscr{C}$$

factors through

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

By [Hin16, Proposition 2.1.4], the map $\mathscr{K}[W^{-1}]\to I^\amalg$ is a cocartesian fibration, and moreover, the fiber above an $\underline{i}=(i_1,\dots,i_n)$ 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 $I^\amalg$ is a 1-category, to check that the diagram

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

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

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

Remark 5.7. The proof of Proposition 5.6 also gives 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 5.8 [NRS20, Theorem 3.3.1]. 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 5.9. Let $\mathcal{C}$ and $\mathcal{D}$ be stable $\infty$ -categories. Let $G:\mathcal{D}\to\mathcal{C}$ be an exact functor. Then G admits a left (respectively, 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.

Theorem 5.10 [Cis19, Theorem 7.6.10]. 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.

Recall that by [Lur18a, Tag 01F1], we have the following proposition.

Proposition 5.11. 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 [Lur18a, Tag 01F1], the restriction

(5.3) \begin{equation}\iota^* : \mathrm{Fun}(\mathcal{C},\mathcal{D})\to\mathrm{Fun}(A,\mathcal{D}) \end{equation}

is an isofibration of simplicial sets (cf. [Cis19, 3.3.15]). The data of $f_a$ defines an isomorphism $\gamma$ in $\mathrm{Fun}(A,\mathcal{D})$ between $F_{\mid A}$ and the map of simplicial sets $A\to\mathcal{D}$ defined by $a\mapsto d_a$ . By the lifting property of isofibrations, there exist 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 precisely the statement of the proposition.

To prove descent statements 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 5.12. Let $\chi : (\mathcal{C}^\mathrm{op})^{\lhd}\to \mathrm{Cat}_\infty$ be a functor. For any $f:c\to c'$ in $(\mathcal{C}^\mathrm{op})^{\lhd}$ , denote $f^*=\chi(f)$ , and assume that any such $f^*$ has a right adjoint $f_*$ . For each $c\in\mathcal{C}$ , denote by $f_c:c\to v$ the unique map from c to the end point v (we have that $(\mathcal{C}^\mathrm{op})^\lhd \simeq (\mathcal{C}^\rhd)^\mathrm{op}$ ). Let $\overline{\pi}:\overline{\mathcal{D}}\to (\mathcal{C}^\mathrm{op})^{\lhd}$ be the cocartesian fibration that classifies $\chi$ , and let $\pi:\mathcal{D}\to \mathcal{C}^\mathrm{op}$ be its pullback by the inclusion $\mathcal{C}^\mathrm{op}\to(\mathcal{C}^\mathrm{op})^{\lhd}$ .

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

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

2. (ii) For any cocartesian section $X:\mathcal{C}^\mathrm{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 [Lur18a, Construction 02T0], there is a diffraction functor

(5.4) \begin{equation}\mathrm{Df}:\overline{\mathcal{D}}_v\to \mathrm{Fun}_{/\mathcal{C}^\mathrm{op}}^\mathrm{Cocart}(\mathcal{C}^\mathrm{op},\mathcal{D}) \end{equation}

that fits in a commutative triangle.

(5.5)

By the diffraction criterion [Lur18a, Theorem 02T8], $\chi$ is a limit diagram if and only if the restriction morphism in (5.5) is an equivalence. In that case, by [Lur18a, Remark 02TF], the functor Df is an equivalence.

Assume that for any cocartesian section $X:\mathcal{C}^\mathrm{op}\to \mathcal{D}$ of $\pi$ , the functor $c\mapsto (f_c)_*X(c)$ admits a limit in $\overline{D}_v$ . By the preservation of limits of any right adjoint in $(\mathcal{C}^\mathrm{op})^\lhd$ and [Lur18a, Corollary 0311 and Corollary 02KY], this is equivalent to saying that for any cocartesian section $X:\mathcal{C}^\mathrm{op}\to\mathcal{D}$ of $\pi$ , the lifting problem

(5.6)

admits a solution $\overline{X}$ which is a $\overline{\pi}$ -limit [Lur18a, Definition 02KG]. By [Lur18a, Example 02ZA], in that case $\overline{X}$ is the right Kan extension functor along the inclusion $i:\mathcal{C}^\mathrm{op}\to(\mathcal{C}^\mathrm{op})^\lhd$ . This easily implies that the restriction functor $\mathrm{res}:\mathrm{Fun}_{/(\mathcal{C}^\mathrm{op})^\lhd}^\mathrm{Cocart}((\mathcal{C}^\mathrm{op})^\lhd,\overline{\mathcal{D}})\to \mathrm{Fun}_{/\mathcal{C}^\mathrm{op}}^\mathrm{Cocart}(\mathcal{C}^\mathrm{op},\mathcal{D})$ is fully faithful as the left adjoint of the right Kan extension along $i : \mathcal{C}^\mathrm{op}\to(\mathcal{C}^\mathrm{op})^\lhd$ .

Therefore, assuming point (ii) of the proposition, $\chi$ is a limit diagram if and only if the restriction functor is conservative by [Lur18a, Corollary 03UZ]. But as the family of evaluations $\mathrm{ev}_c : \mathrm{Fun}(\mathcal{C}^\mathrm{op},\mathcal{D})\to\mathcal{D}$ is conservative, this is equivalent to the family of functors $\mathrm{ev}_c\circ\mathrm{res}=\mathrm{ev}_c\circ\mathrm{Df}\circ\mathrm{ev}_v = f_c^*$ being conservative, which is point (i) of the proposition.

By (5.5), if $\chi$ is a limit diagram, Df is an equivalence which implies that (i) holds. To finish the proof of the proposition, we therefore only have to show that if $\chi$ is a limit diagram, then (ii) holds. Let $\overline{X}$ be a cocartesian section of $\overline{\pi}$ . Assume first that $\overline{X}$ is a $\overline{\pi}$ -limit diagram. By [Lur18a, Example 03ZA] this is equivalent to supposing that $\overline{X}$ is the $\overline{\pi}$ -right Kan extension of its restriction X to $\mathcal{C}^\mathrm{op}$ . Then, by [Lur18a, Corollary 0307], the limit of $c\mapsto (f_c)_*X(c)$ exists and the fact that $\overline{X}$ is cocartesian is equivalent to the fact that for every $c\in\mathcal{C}$ , the canonical map

(5.7) \begin{equation}f_c^*(\lim_{c'} (f_{c'})_*X(c))\to X(c) \end{equation}

is an equivalence.

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

Corollary 5.13 (See also [Lur17, Corollary 5.2.2.37]). Let $\chi,\chi':(\mathcal{C}^\mathrm{op})^{\lhd}\to\mathrm{Cat}_\infty$ be two functors as in Proposition 5.12, and assume one has a natural transformation $B:\chi\to\chi'$ which commutes with the right adjoints. Suppose also that the limits of point (ii) in Proposition 5.12 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 $\chi$ .

5.2 Enhancement of the six-functor formalism

In this subsection, we will work with perverse Nori motives and mixed Hodge modules interchangeably. Therefore, we denote by $\mathrm{Var}_k$ 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 $\mathrm{Var}_k$ ‘varieties’ or ‘k-varieties’. For $X\in\mathrm{Var}_k$ , we will denote by ${\mathcal{M}_{\mathrm{perv}}}(X)$ 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 $R: \mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathrm{D}^b_c(X)$ 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 $\mathrm{D}^b_c(X)$ is the derived category of bounded cohomologically constructible sheaves on $X^\mathrm{an}$ in the two first cases, and the derived category of constructible $\ell$ -adic étale sheaves on X. The triangulated category $\mathrm{D}^b_c(X)$ is endowed with a perverse and a constructible t-structures and the functor R is t-exact (for both t-structures) and conservative.

We begin with a recollection on the $\infty$ -categorical enhancement of derived categories of constructible sheaves.

Proposition 5.14. There exists a functor

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

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 and étale sheaves, for $X\in\mathrm{Var}_k$ , the category $\mathrm{D}^b_c(X)$ is embedded in a bigger category $\mathcal{D}(X)$ which is of the form $\mathcal{D}(\mathrm{Shv}(\mathfrak{X},\mathscr{O}))$ with $\mathfrak{X}$ a topos and $\mathscr{O}$ a sheaf of rings on $\mathfrak{X}$ and is naturally a symmetric monoidal stable $\infty$ -category. Indeed, in the case of analytic sheaves one can take $\mathfrak{X}$ to be the categories of analytic opens of the complex points of X, and $\mathscr{O}$ is the constant sheaf $\mathbb{Q}_X$ . In the case of $\ell$ -adic sheaves, following [HRS23], one can take $\mathfrak{X}$ to be the proétale topos of X, and $\mathscr{O}$ to be the sheaf of rings 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 [Lur18b, Remark 2.1.0.5]. As the constructions $X\mapsto \mathfrak{X}$ and $\mathfrak{X}\mapsto \mathcal{D}(\mathrm{Shv}(\mathfrak{X},\mathscr{O}))$ 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 $f^*\colon \mathcal{D}(Y)\to\mathcal{D}(X)$ 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.

Construction 5.15. We have a symmetric monoidal homotopy 2-functor

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

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 5.5 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 in Corollary 4.19 lifts to an equivalence of $\infty$ -categories thanks to Theorem 5.10. Proposition 5.11 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 5.16. 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 in.

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 Theorem 5.3.

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, $p^*(M)=M\boxtimes \mathbb{Q}_Y$ , 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 5.9 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.

Corollary 5.17. Let $f:Y\to X$ be a map in $\mathrm{Var}_k^\mathrm{op}$ . Then the $\infty$ -functor

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

admits a right adjoint $f_*$ . Moreover, if f is smooth, then $f^*$ admits a left adjoint $f_\sharp$ .

Proof. These adjoints exist on the homotopy categories, thus they have $\infty$ -categorical lifts by Corollary 5.9.

Proposition 5.18. For each $X\in\mathrm{Var}_k$ , the monoidal structure obtained in Construction 5.15 induces on the homotopy categories the same monoidal structure as that constructed by 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 $\boxtimes :{\mathcal{M}_{\mathrm{perv}}}(X)\times{\mathcal{M}_{\mathrm{perv}}}(X)\to{\mathcal{M}_{\mathrm{perv}}}(X \times X)$ that is exact in both variables and hence induces by Corollary 5.4 an $\infty$ -functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X\times X))$ 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 structural morphisms of the monoidal structure are obtained similarly. Denote by $\otimes^T : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ Saito’s or Terenzi’s tensor product and denote by $\otimes^\infty : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ the one we just constructed. We want to check that $\mathrm{ho}(\otimes^\infty)=\mathrm{ho}(\otimes^T)$ . For this there are several isomorphisms to check (see [ML98, Section XI.1]):

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

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

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

4. (iv) Modulo point (i) 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 ${\mathcal{M}_{\mathrm{ct}}}(X)\times{\mathcal{M}_{\mathrm{ct}}}(X)$ . By Corollary 5.4 this gives (i). We explain how to obtain (iii) and the other checks will be left to the reader as they are very similar. Denote by

$$t_1^\infty : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$$

the functor sending M,N,P to $M\otimes^\infty (N\otimes^\infty P)$ and by

$$t_2^\infty : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\times\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$$

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 images by the restriction functor of the morphisms $c^\infty$ and $c^T$ are the same, hence, up to the equivalence of (i), we obtain (iii).

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

Corollary 5.19. Let X be a variety. The tensor structure on $ \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is closed.

Proof. Again this is true on the homotopy categories and we can apply Corollary 5.9.

Proposition 5.20. Let X be a variety. The realisation functor

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

admits a canonical lift to a map in $\mathrm{CAlg}(\mathrm{Cat}_\infty)$ .

Moreover, it induces a natural transformation

$$\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))\Rightarrow \mathcal{D}^b_c(-)$$

on functors

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

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

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

thanks to Nori’s theorem. This enables us to consider the functor $\Delta^1\times\mathrm{Var}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_1)$ that sends (0,f) to $f^*:{\mathcal{M}_{\mathrm{ct}}}(Y)\to {\mathcal{M}_{\mathrm{ct}}}(X)$ , the edge (1,f) to $f^*:\mathrm{Cons}(Y)\to\mathrm{Cons}(X)$ and $(0\to 1,X)$ to the realisation functor. Therefore, by Proposition 5.5 we obtain the result.

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

$$\mathrm{real}_X:\mathcal{D}^b(\mathrm{Cons}(X,\mathbb{Q}_\ell))\to \mathcal{D}^b_c(X,\mathbb{Q}_\ell)$$

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

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

Hence, we have to check that the $\mathrm{real}_X$ assemble to give a symmetric monoidal natural transformation

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

This is Proposition 5.6.

Corollary 5.21. The realisation functor

$$\mathcal{R} \colon \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))\to\mathcal{D}^b_c(-)$$

commutes with all the operations constructed above. This includes pullbacks $f^*$ , pushforwards $f_*$ , left adjoints $f_\sharp$ when f is a smooth map, the tensor product $-\otimes-$ and the internal homomorphism $\mathscr{{H om}}\,(-,-)$ .

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.

Verdier duality also lifts.

Proposition 5.22. Let X be a k-variety. Then the Verdier duality functor $\mathbb{D}_X$ has a $\infty$ -categorical lift $\mathbb{D}_X\colon \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))^\mathrm{op}$ which commutes with the Betti realisation. Moreover, there is a canonical isomorphism $\mathbb{D}_X\circ\mathbb{D}_X\simeq \mathrm{Id}_{\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))}$ .

Proof. Indeed, the functor $\mathbb{D}_X$ , both in Ivorra and Morel and in Saito, is constructed as the trivial derived functor of the exact functor $\mathbb{D}_X\colon {\mathcal{M}_{\mathrm{perv}}}(X)\to{\mathcal{M}_{\mathrm{perv}}}(X)^\mathrm{op}$ , thus it is an $\infty$ -functor. Thus, the functor also commutes with the Betti realisation as an $\infty$ -functor (as in fact we apply the functor $\mathcal{D}^b(-)$ to the commutative square involving Verdier duality of ${\mathcal{M}_{\mathrm{perv}}}(X)$ , $\mathrm{Perv}(X)$ and R). The fact that $\mathbb{D}_X\circ\mathbb{D}_X\simeq \mathrm{Id}_{\mathcal{D}^b_\mathcal{M}(X)}$ holds can be checked on the homotopy category as well.

For free, we also obtain the exceptional functoriality.

Corollary 5.23. The exceptional functors $f^!$ and $f_!$ constructed by Ivorra, Morel and Saito have $\infty$ -categorical lifts that assemble to give functors

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

Moreover, Verdier duality is a natural isomorphism 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 $\mathbb{D}\circ f_* \simeq f_!\circ\mathbb{D}$ tells us that if we apply Proposition 5.11 to the isomorphisms $\mathbb{D}_X : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))^\mathrm{op}$ , the functor $\mathrm{Sch}^{(\mathrm{op})}\to\mathrm{Cat}_\infty$ obtained encodes exactly the exceptional functoriality.

Theorem 5.24. The functor $X\mapsto \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ , with respect to the $(-)^*$ -functoriality, is a hypersheaf with respect to h-topology.

Proof. Denote ${\mathcal{D}^b_\mathcal{M}}(X):=\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ . 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 hypersheaf because of localisation and proper base change hold for ${\mathcal{D}^b_\mathcal{M}}$ . We will use Corollary 5.13 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

$$B\colon {\mathcal{D}^b_\mathcal{M}} \to\mathrm{D}^b_c.$$

Note that $\mathrm{D}^b_c$ has étale hyperdescent. Now, to apply Corollary 5.13 we need to make sure that the limits ensuring descent exist in ${\mathcal{D}^b_\mathcal{M}}$ . This is not obvious. However, for any variety X and $m\in\mathbb{N}$ , if we denote by $\mathcal{D}^{[-m,m]}_\mathcal{M}(X)$ the full subcategory of ${\mathcal{D}^b_\mathcal{M}}(X)$ of objects concentrated, for the constructible t-structure, in degrees $[-m,m]$ , we have

\[\bigcup_m \mathcal{D}^{[-m,m]}_\mathcal{M}(X) = {\mathcal{D}^b_\mathcal{M}}(X).\]

Moreover, the functor ${\mathcal{D}^b_\mathcal{M}}$ is a hypersheaf if and only if for each m, the functor $\mathcal{D}^{[-m,m]}_\mathcal{M}$ is a hypersheaf, and the same holds for $\mathrm{D}^b_c$ . Indeed, if ${\mathcal{D}^b_\mathcal{M}}$ is a hypersheaf, 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 $\mathcal{D}^{[-m,m]}_\mathcal{M}$ is also a hypersheaf. Conversely, if each $\mathcal{D}^{[-m,m]}_\mathcal{M}$ is a hypersheaf, then because pullback functors $f^*$ are t-exact functors, in turns out that the conditions to test that ${\mathcal{D}^b_\mathcal{M}}$ is a hypersheaf only occur in $\mathcal{D}^{[-m,m]}_\mathcal{M}$ : if $U_\bullet\to X$ is a hypercovering, full faithfulness of the sheaf condition is computed in $\mathcal{D}^{[-m,m]}_\mathcal{M}(X)$ for some m, and then essential surjectivity holds because, given an element $K=(K_n)_n$ of the limit, there exists a fixed m such that K has its $K_0$ , hence also all its factors $K_n$ in $\mathcal{D}^{[-m,m]}_\mathcal{M}(U_n)$ .

Thus, because $\mathrm{D}^{[-m,m]}_c$ is an étale hypersheaf, it suffices to check the conditions of Corollary 5.13 for the realisation functor

$$B\colon \mathcal{D}^{[-m,m]}_\mathcal{M}\to\mathrm{D}^{[-m,m]}_c.$$

Let $f_\bullet\colon Y_\bullet\to X$ be an étale hypercovering. As the realisation functor is conservative, it suffices to check the following assertions.

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

   \[\lim_\Delta \tau^{\leqslant m}(f_n)_*f_n^*M\]
   exists, where $\Delta$ is the simplicial category and $\tau^{\leqslant m}$ is the truncation functor for the t-structure.

2. (ii) The realisation functor

   \[\mathcal{D}_\mathcal{M}^{[-m,m]}(X)\to\mathrm{D}_c^{[-m,m]}(X)\]
   commutes with such a limit.

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

\[\mathcal{D}^{\geqslant -m}_\mathcal{M}(X)\to\mathrm{D}_c^{\geqslant -m}(X)\]

commutes with finite limits (the inclusion $\mathrm{D}^{\geqslant -m}\to\mathrm{D}^b$ commutes with limits) and the image by the realisation functor of the above limit in (i) is the truncation by $\tau^{\leqslant m}$ , which is a right adjoint, of a limit in $\mathrm{D}^{\geqslant -m}_c(X)$ .

6. The realisation functor and applications

6.1 Construction of the realisation functor

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 six operations. Denote by $\mathrm{Sch}_k$ the category of finite-type k-schemes. We use Drew and Gallauer’s result in [DG22] in which they prove that the stable motivic $\mathbb{A}^1$ -invariant $\infty$ -category $\mathcal{SH}$ has 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 six operations to exist thanks to Ayoub’s thesis (see Appendix A for a precise definition and some results on coefficient systems). Any coefficient system $C\in\mathrm{CoSys}_k$ with values in idempotent complete $\infty$ -categories receives a unique realisation functor

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

from the compact objects of the motivic homotopy category that commutes with enough operations to ensure that it commutes with the six operations (if C has some good properties, see below).

Theorem 6.1. Denote by $\mathcal{D}_\mathcal{M}^b$ the functor $\mathrm{Var}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{\mathrm{Cat}_\infty})$ of Construction 5.15. The functor $\mathcal{D}_\mathcal{M}^b$ defines an object of $\mathrm{CoSys}_k$ .

Proof. This is the content of Section 5.2 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 (the only non-trivial statement here is the h-hyperdescent, but this follows from the fact that the Kan extension of a hypersheaf on a basis of a site to the full site is still a hypersheaf; see, for example, [Aok23, Theorem A.6]).

Remark 6.2. Using that the mapping spectra in $\mathcal{D}^b_\mathcal{M}$ , $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}})$ and $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}})$ are étale sheaves (for the perverse heart, one has to use affine étale morphisms) and the fact that a limit of a diagram of stable $\infty$ -categories with t-structures and t-exact transition functors is endowed with a t-structure (see [RS20, Lemma 3.2.18]), one can show that after extension to finite-type k-schemes, we still have

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

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

Notation 6.3. For X a finite-type k-scheme, we will denote by $\mathcal{D}^b(\mathrm{MHM}(X))$ (respectively, by $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ ) the value at X of the functor ${\mathcal{D}^b_\mathcal{M}}$ in the mixed Hodge modules case (respectively, in the Nori motives case).

Before going into the proof of the main theorem, we recall a construction originally due to Fabien Morel, and we will use [CD19, Section 16.2] as a reference. Let X be a finite-type k-scheme. There is a canonical decomposition

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

of the rational part of the motivic homotopy category $\mathcal{SH}(X)$ over X. It comes from the decomposition in $\mathcal{SH}_\mathbb{Q}(X)$ of the unit $\mathbb{Q}_X$ as a direct sum

\[\mathbb{Q}_X\simeq \mathbb{Q}_X^+\oplus \mathbb{Q}_X^-.\]

Moreover, the object $\mathbb{Q}_X^+$ has a commutative algebra structure (because it is idempotent) in $\mathcal{SH}(X)_\mathbb{Q}$ such that the projection map $\mathbb{Q}_X\to\mathbb{Q}_X^+$ is a map of algebras. By [CD19, Theorem 16.2.13], there is a canonical identification of $\mathcal{SH}_\mathbb{Q}^+(X)\simeq \mathrm{Mod}_{\mathbb{Q}^+}(\mathcal{SH}_\mathbb{Q}(X))$ with étale motives $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ . In particular, a symmetric monoidal colimit-preserving $\infty$ -functor

\[F\colon \mathcal{SH}_\mathbb{Q}(X)\to \mathcal{D}\]

to a stable presentably symmetric monoidal $\infty$ -category $\mathcal{D}$ factors through $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ if and only if $F(\mathbb{Q}^-_X)=0$ . By [CD19, Corollary 16.2.14] the object $\mathbb{Q}^-_X\in\mathcal{SH}_\mathbb{Q}(X)$ vanishes whenever $-1$ is a sum of squares in the residual fields of X. We will need the following lemma.

Lemma 6.4. The maps $\mathcal{SH}_\mathbb{Q}(X)\to\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ and $\mathcal{SH}(X)\to\mathcal{SH}_\mathbb{Q}(X)$ are part of morphisms of coefficient systems $\mathcal{SH}_\mathbb{Q}\to \mathcal{DM}^{\mathrm{\acute{e}t}}$ and $\mathcal{SH}\to\mathcal{SH}_\mathbb{Q}$ .

Proof. This is a particular case of a theorem by Drew [Dre18, Theorem 8.10] (although in the second case of rationalisation a simpler proof could have been given; see [Ayo14a, Lemma A.11]), where for example in the first case we take (in his notation) $\mathcal{V} = \mathcal{SH}_\mathbb{Q}(\mathrm{Spec}(k))$ , $\mathcal{M}^*=\mathcal{SH}_\mathbb{Q}$ and $A = \mathbb{Q}^+_{\mathrm{Spec}(k)}$ .

Theorem 6.5. There exists an (essentially unique) morphism of coefficients systems $\mathcal{DM}^{\mathrm{\acute{e}t}}\to \mathrm{Ind} {\mathcal{D}^b_\mathcal{M}}$ on the category $\mathrm{Sch}_k$ . It restricts to a functor

(6.1) \begin{equation}\nu:\mathcal{DM}^{\mathrm{\acute{e}t}}_c\to {\mathcal{D}^b_\mathcal{M}}.\end{equation}

The functor (6.1) commutes with the six operations, that is, all the exchange transformations are isomorphisms.

In the case of mixed Hodge modules we will denote the realisation functor by $\mathrm{Hdg}^*$ , and in the case of Nori motives we will denote it by $\mathrm{Nor}^*$ .

Proof. Thanks to Theorem 6.1 we know that $\mathcal{D}^b_\mathcal{M}$ is an object of $\mathrm{CoSys}_k$ , hence its indization $\mathrm{Ind}\mathcal{D}^b_\mathcal{M}$ is an object of $\mathrm{CoSys}_k^\mathrm{c}$ . Together with Theorem A.2, this gives a morphism of coefficient systems

$$\rho_{\mathcal{SH}}\colon\mathcal{SH}\to \mathrm{Ind}\mathcal{D}^b_\mathcal{M}.$$

Note that the functor $\mathrm{Ind}\mathcal{D}^b_\mathcal{M}$ naturally takes values in $\mathrm{CAlg}(\mathrm{Pr}^L)_{\backslash \mathrm{Ind}\mathcal{D}^b_\mathcal{M}(\mathrm{Spec}(k))}$ , the coslice $\infty$ -category, because $\mathrm{Spec}(k)$ is the initial object of $\mathrm{Sch}_k^\mathrm{op}$ . Using the functor $\rho_{\mathcal{SH}}$ over $\mathrm{Spec}(k)$ , we see that $\mathrm{Ind}\mathcal{D}^b_\mathcal{M}$ takes values in $\mathrm{CAlg}(\mathrm{Pr}^L)_{\backslash \mathcal{SH}(\mathrm{Spec}(k))}$ , the $\infty$ -category of $\mathcal{SH}(\mathrm{Spec}(k))$ -algebras in $\mathrm{Pr}^L$ , where we endowed $\mathrm{Pr}^L$ with the Lurie tensor product constructed in [Lur17, Section 4.8.1]. We claim that $\rho_{\mathcal{SH},k}\colon\mathcal{SH}(\mathrm{Spec}(k))\to\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(\mathrm{Spec}(k))$ factors through $\mathcal{DM}^\mathrm{\acute{e}t}(\mathrm{Spec}(k))$ .

First, note that because $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(\mathrm{Spec}(k))$ is $\mathbb{Q}$ -linear, the functor $\rho_{\mathcal{SH},k}$ factors through $\mathcal{SH}_\mathbb{Q}(\mathrm{Spec}(k))$ to give a symmetric monoidal functor

\[\rho_{\mathcal{SH}_\mathbb{Q},k}\colon \mathcal{SH}_\mathbb{Q}(\mathrm{Spec}(k))\to\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(\mathrm{Spec}(k)).\]

Moreover, by the universal property of the localisation, if K is a finite extension of k in which we added a square root of $-1$ , the square

commutes, where $g\colon \mathrm{Spec}(K)\to\mathrm{Spec}(k)$ is the structural morphism. Now because $\mathbb{Q}^-_{\mathrm{Spec}(k)}$ is a direct factor of $\mathbb{Q}_{\mathrm{Spec}(k)}$ , its image under $\rho_{\mathcal{SH}_\mathbb{Q},k}$ lands in ${\mathcal{D}^b_\mathcal{M}}(\mathrm{Spec}(k))$ . By étale descent of ${\mathcal{D}^b_\mathcal{M}}$ proven in Theorem 5.24, the functor $g^*$ is conservative on ${\mathcal{D}^b_\mathcal{M}}(\mathrm{Spec}(k))$ , and as $\mathbb{Q}^-_{\mathrm{Spec}(K)}=0$ , we see that the image of $\mathbb{Q}^-_{\mathrm{Spec}(k)}$ in $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(\mathrm{Spec}(k))$ vanishes. By the previous discussion Lemma 6.4, this proves that the functor $\rho_{\mathcal{SH}_\mathbb{Q},k}$ factors through $\mathcal{DM}^\mathrm{\acute{e}t}(\mathrm{Spec}(k))$ to give a functor that we will denote by $\widetilde{\nu_k}$ . Through $\widetilde{\nu_k}$ we see that our functor $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}$ takes values in $\mathcal{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec}(k))$ -algebras in $\mathrm{Pr}^L$ . By [Lur17, Remark 7.3.2.13] applied to the left adjoint functor

\[-\otimes_{\mathcal{SH}(\mathrm{Spec}(k))}\mathcal{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec}(k))\colon \mathrm{Mod}_{\mathcal{SH}(\mathrm{Spec}(k))}(\mathrm{Pr}^L)\to\mathrm{Mod}_{\mathcal{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec}(k))}(\mathrm{Pr}^L),\]

the forgetful functor

\[\mathrm{ff}\colon\mathrm{CAlg}(\mathrm{Pr}^L)_{\backslash \mathcal{DM}^\mathrm{\acute{e}t}(\mathrm{Spec}(k))}\to \mathrm{CAlg}(\mathrm{Pr}^L)_{\backslash \mathcal{SH}(\mathrm{Spec}(k))}\]

is right adjoint to the functor

\[-\otimes_{\mathcal{SH}(\mathrm{Spec}(k))}\mathcal{DM}^\mathrm{\acute{e}t}(\mathrm{Spec}(k)).\]

Applying this adjunction termwise, we obtain by [Cis19, Theorem 6.1.22] an adjunction between functors on $\mathrm{Sch}^\mathrm{op}_k$ . In particular, the natural transformation $\rho_{\mathcal{SH}}$ may be seen as a map $\mathcal{SH}\to \mathrm{ff}(\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}})$ in the $\infty$ -category

$$ \mathrm{Fun}(\mathrm{Sch}_k^\mathrm{op},\mathrm{CAlg}(\mathrm{Pr}^L)_{\backslash \mathcal{SH}(\mathrm{Spec}(k))}).$$

By adjunction, we obtain a natural transformation

\[\mathcal{SH}\otimes_{\mathcal{SH}(\mathrm{Spec}(k))}\mathcal{DM}^\mathrm{\acute{e}t}(\mathrm{Spec}(k))\to \mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}\]

of functors on $\mathrm{Sch}_k^\mathrm{op}$ and with values in $\mathrm{CAlg}(\mathrm{Pr}^L)_{\backslash \mathcal{DM}^\mathrm{\acute{e}t}(\mathrm{Spec}(k))}$ . The left-hand sides send a scheme X to the $\infty$ -category

\[\mathcal{SH}(X)\otimes_{\mathcal{SH}(\mathrm{Spec}(k))}\mathcal{DM}^{\mathrm{\acute{e}t}}{(\mathrm{Spec}(k))}.\]

By using that

$$\mathcal{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec}(k))\simeq \mathrm{Mod}_{\mathbb{Q}^+_k}(\mathcal{SH}(\mathrm{Spec}(k)))$$

and [Lur17, Theorem 4.8.4.6], we see that

$$\mathcal{SH}(X)\otimes_{\mathcal{SH}(\mathrm{Spec}(k))}\mathcal{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec}(k))\simeq \mathrm{Mod}_{\mathbb{Q}^+_X}(\mathcal{SH}(X))\simeq\mathcal{DM}^\mathrm{\acute{e}t}(X).$$

This proves that $\rho_\mathcal{SH}$ factors through the functor $\mathcal{DM}^\mathrm{\acute{e}t}$ , giving us a natural transformation

\[\widetilde{\nu}\colon\mathcal{DM}^\mathrm{\acute{e}t}\to\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}\]

of functors on $\mathrm{Sch}^\mathrm{op}_k$ with values in $\mathrm{CAlg}(\mathrm{Pr}^L)$ . The factorisation is given by the unit of the adjunction between functor categories described above.

We claim that we obtained a morphism of coefficient systems. Let $f\colon X\to Y$ be a smooth map. Because $\mathcal{DM}^\mathrm{\acute{e}t}(X)$ is generated under colimits by objects of the form $a_\mathrm{\acute{e}t}^\mathbb{Q}(M)$ with $M\in\mathcal{SH}(X)$ and $a_\mathrm{\acute{e}t}^\mathbb{Q}$ the morphism $\mathcal{SH}\to\mathcal{DM}^\mathrm{\acute{e}t}$ , it suffices to prove that the natural exchange morphism

\[f_\sharp(\widetilde{\nu}(a_\mathrm{\acute{e}t}(M)))\to \widetilde{\nu}(f_\sharp^{\mathcal{DM}^\mathrm{\acute{e}t}}(a_\mathrm{\acute{e}t}(M)))\]

is an equivalence. This follows from the following sequence of equivalences:

\begin{eqnarray*}f_\sharp(\widetilde{\nu}(a_\mathrm{\acute{e}t}(M))) & \simeq & f_\sharp (\rho(M))\\ & \overset{(1)}{\simeq} & \rho(f_\sharp^{\mathcal{SH}}(M))\\ & \simeq & \widetilde{\nu}(a_\mathrm{\acute{e}t}^\mathbb{Q}(f_\sharp^\mathcal{SH}(M)))\\ &\overset{(2)}{\simeq} & \widetilde{\nu}(f_\sharp^{\mathcal{DM}^\mathrm{\acute{e}t}}(a_\mathrm{\acute{e}t}^\mathbb{Q}(M))) ,\end{eqnarray*}

where we used in (1) the fact that $\rho_{\mathcal{SH}}$ is a morphism of coefficient systems and in (2) the fact that $a_\mathrm{\acute{e}t}^\mathbb{Q}$ is a morphism of coefficient systems by Lemma 6.4.

Denote by $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ (respectively, by $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}(X)$ ) the thick full subcategory of $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ (respectively, of $\mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(X)$ ) spanned by the $f_\sharp \mathbb{Q}_Y(i)$ for $f:Y\to X$ smooth and $i\in \mathbb{Z}$ . As ${\mathcal{D}^b_\mathcal{M}}(X)\subset \mathrm{Ind}{\mathcal{D}^b_\mathcal{M}}(X)$ is thick and each $f_\sharp \mathbb{Q}_Y(i)$ is in ${\mathcal{D}^b_\mathcal{M}}(X)$ , we have $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}(X)\subset {\mathcal{D}^b_\mathcal{M}}(X)$ . Because $\widetilde{\nu}$ is a morphism of coefficient systems, it induces a morphism of coefficient systems

(6.2) \begin{equation}\nu^{\mathrm{gm}}:\mathcal{DM}^{\mathrm{\acute{e}t}}_c\to \mathcal{D}^{\mathrm{gm}}_\mathcal{M}. \end{equation}

In particular, at the level of triangulated categories, we obtain a premotivic morphism that satisfies all the hypotheses of [CD19, Theorem 4.4.25]: $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}$ is $\tau$ -generated for $\tau$ the Tate twist, $\mathcal{DM}^{\mathrm{\acute{e}t}}_c$ is $\tau$ -dualisable by absolute purity, $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}$ is separated because it is a sub-pullback formalism of ${\mathcal{D}^b_\mathcal{M}}$ which satisfies étale descent by Theorem 5.24 (which is equivalent to separateness in the $\mathbb{Q}$ -linear case by [CD19, Corollairy 3.3.34]) and all Tate twists are invertible in $\mathcal{DM}^{\mathrm{\acute{e}t}}_c$ by construction.

Therefore, the functor $\nu^{\mathrm{gm}}$ commutes with the six operations. One has to be a little careful here: a priori, [CD19, Theorem 4.4.25] implies only that $\nu^{\mathrm{gm}}$ commutes with all the operations when $\mathcal{D}^{\mathrm{gm}}_\mathcal{M}$ is endowed with the operations they construct in their book, and not necessarily with the constructions of Saito, Ivorra and Morel as in Corollary 5.23. 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, we have $p_*\simeq p_!$ ) 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 $\mathcal{DM}^{\mathrm{\acute{e}t}}$ ). The tensor product is the same, hence also the internal $\mathscr{H} \mathscr {om}\,$ by adjunction.

We have obtained that the functor

(6.3) \begin{equation}\nu : \mathcal{DM}^{\mathrm{\acute{e}t}}_c\to {\mathcal{D}^b_\mathcal{M}} \end{equation}

commutes with the six 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.

Corollary 6.6. The realisation functor

$$\mathcal{SH}_c\to {\mathcal{D}^b_\mathcal{M}}$$

commutes with the six operations.

Proof. Indeed, it suffices to check that the functor $\mathcal{SH}_c\to\mathcal{DM}^{\mathrm{\acute{e}t}}_c$ commutes with the operations. This is the content of the next lemma.

This lemma is probably well known to experts, but the author could not find a reference.

Lemma 6.7. The morphisms $\rho_\mathbb{Q}\colon\mathcal{SH}_c\to\mathcal{SH}_{\mathbb{Q},c}$ and $a_{\mathrm{\acute{e}t}}\colon\mathcal{SH}_{\mathbb{Q},c}\to\mathcal{DM}^\mathrm{\acute{e}t}_c$ commute with the six operations.

Proof. The case of $\rho_\mathbb{Q}$ follows from [Ayo14a, Théorème A.15], thus we focus on $a_{\mathrm{\acute{e}t}}$ . We know that $a_{\mathrm{\acute{e}t}}$ is a morphism of coefficient systems generated by objects of the form $g_\sharp\mathbf{1}_Y(n)$ for $g\colon Y\to X$ a smooth map and $n\in\mathbb{Z}$ , thus to prove that it commutes with the six operations it suffices to check that it commutes with pushforwards (see the proof of [CD19, Theorem 4.4.25]). Let $f\colon Y\to X$ be a morphism of schemes. As the functor $a_{\mathrm{\acute{e}t}}$ commutes with $f^*$ its right adjoint $\mathrm{ff}^{\mathrm{\acute{e}t}}$ commutes with $f_*$ , and because $\mathrm{ff}^{\mathrm{\acute{e}t}}$ is conservative it suffices to check that the natural transformation

\[\mathrm{ff}^{\mathrm{\acute{e}t}}\circ a_{\mathrm{\acute{e}t}}\circ f_*\to f_*\circ\mathrm{ff}^{\mathrm{\acute{e}t}}\circ a_{\mathrm{\acute{e}t}}\]

is an equivalence. Moreover, by [Lur17, Corollary 4.2.4.8] there is a canonical identification of functors

\[\mathrm{ff}^{\mathrm{\acute{e}t}}\circ a_{\mathrm{\acute{e}t}} \simeq -\otimes {\mathbb{Q}^+}.\]

Thus, we have to prove that the natural map

(6.4) \begin{equation}\ f_*(-)\otimes \mathbb{Q}^+ \to f_*(-\otimes \mathbb{Q}^+)\end{equation}

is an equivalence. This is the case because as $\mathbb{Q}^+$ is a direct factor of $\mathbb{Q}$ , it is a dualisable object of $\mathcal{SH}_\mathbb{Q}$ so that we can apply [Ayo14b, Lemma 2.8], using that $\mathbb{Q}^+_Y\simeq f^*\mathbb{Q}^+_X$ .

Corollary 6.8. There exists a Hodge realisation functor $R_H:\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}})\to\mathcal{D}^b(\mathrm{MHM})$ compatible with the six operations. It factors the Hodge realisation that we have constructed in Theorem 6.5.

Proof. By Theorem 6.5, there exists a Hodge realisation from $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(-,\mathbb{Q})$ to the derived category of mixed Hodge modules. Therefore, taking the underlying $\mathbb{Q}$ -structure and then perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$ , we obtain a factorisation of the perverse ${\ ^{\mathrm{p}}\mathrm{H}}^0$ of the Betti realisation through $\mathrm{MHM}(X)$ . The universal property of perverse Nori motives then gives a faithful exact functor ${\mathcal{M}_{\mathrm{perv}}}(X)\to \mathrm{MHM}(X)$ , which induces a functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to \mathcal{D}^b(\mathrm{MHM}(X))$ . 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 $\mathcal{DM}^{\mathrm{\acute{e}t}}_c$ and the fact that $R_H$ can be promoted to a natural transformation of $\infty$ -functors on $\mathrm{Sch}_k^\mathrm{op}$ , using that it is also the derived functor of its restriction to the constructible heart.

Recall that by [IM22, Corollary 6.18], [Bon11, Proposition 2.3.1] and [Héb11, Theorem 3.3] the categories $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ , $\mathcal{D}^b(\mathrm{MHM}(X))$ and $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ admit weight structures (for the last one, note that Beilinson motives and étale motives are equivalent by [CD19, Theorem 16.1.2]).

Proposition 6.9. Let X be a finite-type k-scheme. Then the functors

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

\[\mathrm{Nor}^*:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\]

and

\[\mathrm{Hdg}^*:\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b(\mathrm{MHM}(X))\]

are compatible with weights.

Proof. First note that by definition of the weights on $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ , the $\ell$ -adic realisation $R_\ell :\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b_c(X,\mathbb{Q}_\ell)$ is compatible with weights and reflects weights (i.e., if a complex $K\in\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ satisfies that $R_\ell(K)$ is of weights less than or equal to w, then K is of weights less than or equal to w). As all weight filtrations and structures considered are bounded, it suffices to show that our functors map weight-zero objects to weight-zero objects. Let us first consider functors with source $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ . For these, is suffices to check that the image of the relative motive of a smooth proper X-scheme is pure of weight zero. For the $\ell$ -adic realisation this is [Del80] and [Mor19]. Therefore, the functor $\mathrm{Nor}^* : \mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is also weight exact. The fact that $\mathrm{Hdg}^*$ is exact is the fact that the mixed Hodge module is associated to a smooth proper X-scheme is pure of weight zero; see [Sai06, Theorem 6.7 and Proposition 6.6].

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

(6.5)

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

Remark 6.10. Our construction of the realisation functor $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ 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 $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ . To be more precise, Ivorra only gives the image of smooth affine X-schemes and then extends 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 a matter of applying the universal property of $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ : 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 six operations and Ivorra’s using cellular complexes and Beilinson’s basic lemma) coincide, then our functor $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ 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 to the diagram category of relative pairs studied in [Ivo17] and [IM22, Section 2.10].

This can be reduced to checking a simple fact, which 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 X a smooth variety, there is a specific cellular stratification $Y^\bullet$ of Y such that the cellular complex induced on Y is an $\mathrm{H}^0f_!$ -acyclic resolution of $f^!\mathbb{Q}_X$ , hence that there is an isomorphism in $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ between Ivorra’s functor at Y and the homology of Y, that is, $\mathrm{ra}^{\mathcal{M}_{\mathrm{perv}}}_X(Y,Y_\bullet)\simeq f_!f^!\mathbb{Q}_X$ . This isomorphism induces a canonical and functorial isomorphism on each ${\ ^{\mathrm{p}}\mathrm{H}}^i$ . The question is whether one can construct such an isomorphism in a functorial way, as this would give the desired isomorphism of functors. This does not seem easy, and it is unlikely that one can arrange those specific stratifications so that they are compatible with a given morphism of smooth affine X-schemes.

6.2 Nori motives as modules in étale motives

Notation 6.11. Recall that to be coherent with the notation of [Ayo22], we will denote by $\mathrm{Nor}^*$ the realisation functor

$$\mathcal{DM}^{\mathrm{\acute{e}t}}\to\mathcal{DN,}$$

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

By construction, the functor $\mathrm{Nor}^*$ preserves colimits, hence it has a right adjoint $\mathrm{Nor}_*:\mathcal{DN}\to \mathcal{DM}^{\mathrm{\acute{e}t}}$ . Denote

(6.6) \begin{equation}\mathscr{N}_k = \mathrm{Nor}_*\mathrm{Nor}^*\mathbb{Q}_{\mathrm{Spec} k}\in\mathcal{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec} k,\mathbb{Q}).\end{equation}

This is an $\mathbb{E}_\infty$ -ring object in $\mathcal{DM}^{\mathrm{\acute{e}t}}(k)$ . For X a finite-type k-scheme, denote by $\mathscr{N}_{\mid X}\in\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ the restriction of $\mathscr{N}_k$ to X. As in [Ayo22, Construction 1.91], $\mathrm{Nor}^*$ factors canonically through the functor

(6.7) \begin{equation}\mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}) : X\mapsto \mathrm{Mod}_{\mathscr{N}_{\mid X}}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X,\mathbb{Q}))\end{equation}

as

(6.8) \begin{equation} \mathrm{Nor}^* : \mathcal{DM}^{\mathrm{\acute{e}t}}\xrightarrow{\mathscr{N}\otimes -} \mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}) \xrightarrow{\widetilde{\mathrm{Nor}^*}} \mathcal{DN}.\end{equation}

Lemma 6.12. 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

\[\widetilde{\mathrm{Nor}^*}:\mathrm{Mod}_{\pi_X^*\mathscr{N}_k}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X))\to \mathcal{DN}(X)\]

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 $\mathrm{Mor}_{\mathscr{N}_k}(\mathcal{DM}^{\mathrm{\acute{e}t}}(k))$ is compactly generated by $f_\sharp\mathbb{Q}_Y(i)\otimes\mathscr{N}_k$ 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 $C\otimes\mathscr{N}_k$ of the above form, the map

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

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

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

is an equivalence in $\mathcal{DM}^{\mathrm{\acute{e}t}}(k)$ . This is true because over a field, compact objects in $\mathcal{DM}^{\mathrm{\acute{e}t}}(k)$ 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 six operations (this follows from the fact that $\mathscr{N}_X$ is a filtered colimit of dualisable objects, thus the projection formulae of the form $f_*(M)\otimes \mathscr{N} \simeq f_*(M\otimes \mathscr{N})$ are true by [Ayo14b, Lemma 2.8], and this implies commutation with all the operations as in the proof of [CD19, Theorem 4.4.2.5]). For compact $M,N\in \mathrm{Mod}_{\pi_X^*\mathscr{N}_k}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X))$ we have

\begin{eqnarray*}\mathrm{map}_{\mathrm{Mod}_{\pi_X^*\mathscr{N}_k}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X))}(M,N) & \simeq & \mathrm{map}_{\mathrm{Mod}_{\pi_X^*\mathscr{N}_k}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X))}(\mathbb{Q}_X,\mathscr{{Hom}}\,(M,N)) \\ & \simeq & \mathrm{map}_{\mathrm{Mod}_{\mathscr{N}_k}(\mathcal{DM}^{\mathrm{\acute{e}t}}(k))}(\mathbb{Q}_k,(\pi_X)_*\mathscr{{Hom}}\,(M,N)) \\ & \simeq & \mathrm{map}_{\mathcal{DN}(k)}(\mathbb{Q}_k,\widetilde{\mathrm{Nor}^*}(\pi_X)_*\mathscr{{Hom}}\,(M,N)) \\ & \simeq & \mathrm{map}_{\mathcal{DN}(k)}(\mathbb{Q}_k,(\pi_X)_*\mathscr{{Hom}}\,(\widetilde{\mathrm{Nor}^*}M,\widetilde{\mathrm{Nor}^*}N)) \\ & \simeq &\mathrm{map}_{\mathcal{DN}(X)}(\widetilde{\mathrm{Nor}^*}M,\widetilde{\mathrm{Nor}^*}N), \end{eqnarray*}

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

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

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

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

and we have

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

Therefore, the canonical map

$$\pi_X^*\mathscr{N}_k\to \widetilde{\mathrm{Nor}_*}\mathbb{Q}_X$$

is an equivalence, thus it stays an equivalence after applying the forgetful functor, leaving us with the equivalence

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

in $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ .

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

Proposition 6.13. Let $k=\mathrm{colim}_i k_i$ be a filtered colimit of fields of characteristic zero. Then the natural functors

\[ \mathrm{colim}_i\mathcal{DN}_c(\mathrm{Spec} k_i)\to \mathcal{DN}_c(\mathrm{Spec} k)\]

and

\[\mathrm{colim}_i\mathcal{DN}(\mathrm{Spec} k_i)\to \mathcal{DN}(\mathrm{Spec} k)\]

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 non-empty 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 the inclusion functor $\mathrm{Pr}^L_\omega\to\mathrm{Pr}^L$ 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 $\mathcal{M}:I^{\rhd}\to \mathrm{AbCAt}$ 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

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

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 6.14. Let $f:\mathrm{Spec} k\to\mathrm{Spec} \mathbb{Q}$ be a map of fields. Then the natural map $f^*\mathscr{N}_\mathbb{Q}\to\mathscr{N}_k$ 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 $\mathscr{N}_X\simeq p_X^*\mathscr{N}_\mathbb{Q}$ .

Proof. The second claim follows from the first. Write $k=\mathrm{colim}_i k_i$ with $k_i/\mathbb{Q}$ of finite type. Then the same proof as in [AGV22, Lemma 3.5.7], using continuity of both $\mathcal{DM}^{\mathrm{\acute{e}t}}$ and $\mathcal{DN}$ for fields (Proposition 6.13) and that these categories are compactly generated, gives that the natural map

$$\mathrm{colim}_i f_i^*\mathscr{N}_\mathbb{Q} \to \mathscr{N}_k$$

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 in [AGV22, Lemma 3.5.7], using continuity for the generic point of perverse Nori motives, reduces to showing that $\pi_U^*\mathscr{N}_\mathbb{Q} \to \mathscr{N}_U$ is an equivalence for U of finite type over $\mathbb{Q}$ , which is Lemma 6.12.

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

Proposition 6.15. Let $X=\lim_i X_i$ be a limit of $\mathbb{Q}$ -schemes with affine transitions. The natural morphism

(6.9) \begin{equation} \mathrm{colim}_i \mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X_i))\to \mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X)) \end{equation}

is an equivalence where the colimit is computed in $\mathrm{Pr}^{L}$ .

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

(6.10) \begin{equation}\mathrm{colim}_i \mathcal{DM}^{\mathrm{\acute{e}t}}(X_i,\mathbb{Q}) \to \mathcal{DM}^{\mathrm{\acute{e}t}}(X,\mathbb{Q}) \end{equation}

is an equivalence. Therefore, the assignment $i\mapsto (\mathcal{DM}^{\mathrm{\acute{e}t}}(X_i,\mathbb{Q}),\mathscr{N}_{\mid X_i})$ is a colimit diagram in the category $\mathrm{Pr}^{\mathrm{Alg}}$ of [Lur17, Notation 4.8.5.10]. By [Lur17, 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 $\mathrm{Mod}_A(\mathcal{C})$ has a right adjoint, hence preserves colimits. This is why the assignment $i\mapsto \mathrm{Mod}_{\mathscr{N}_{X_i}}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X_i,\mathbb{Q}))$ 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].

Proposition 6.16. The functor

(6.11) \begin{equation}\widetilde{\mathrm{Nor}^*} :\mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X)) \to\mathcal{DN}(X)\end{equation}

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

Proof. The functor is fully faithful by Lemma 6.12. We first prove that it is essential surjective for $X=\mathrm{Spec} K$ the spectrum of a field, not necessarily finite over k. As $\mathcal{DN}(\mathrm{Spec} K)$ is compactly generated it suffices to show that $\widetilde{\mathrm{Nor}^*}$ reaches all compact objects $M\in\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(K))$ . As the functor is already fully faithful, by dévissage it suffices to check that any object of the heart $M\in{\mathcal{M}_{\mathrm{perv}}}(K)$ is in the image of $\widetilde{\mathrm{Nor}^*}$ . Any such object has a weight filtration whose graded pieces are direct factors of $\mathrm{H}^{n+2j}(\pi_X)_*\mathbb{Q}_X(i)$ with $\pi_X:X\to\mathrm{Spec} K$ a smooth projective morphism by Arapura [HMS17, Theorem 10.2.5]. Note that here we used that if K is a finite-type extension of k, then the category ${\mathcal{M}_{\mathrm{perv}}}(\mathrm{Spec} K)$ 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, that is, does not depend on the base field. Therefore, by dévissage again it suffices to check that any such $\mathrm{H}^{n+2j}(\pi_X)_*\mathbb{Q}_X(i)$ is in the image. But as $(\pi_X)_*\mathbb{Q}_X(i)$ is pure of weight $-2i$ , it is the direct sum of its $\mathrm{H}^n[-n]$ , thus it suffices to show that each $(\pi_X)_*\mathbb{Q}_X(i)$ is in the image, which is obvious by compatibility with the six operations.

Now for a finite-type k-scheme, the proposition is proven by Noetherian induction. Assume 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 $N\in\mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}(\eta))$ , compact, such that $\widetilde{\mathrm{Nor}^*}(N)=M_\eta$ . By Proposition 6.15 there exist an open subset U of X and an object $N'\in\mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}(U))$ , compact, such that $(N')_\eta = N$ . The functor $\mathrm{colim}_{\eta\in U}\mathcal{DN}(U)\to\mathcal{DN}(\eta)$ is fully faithful by Proposition 6.13. Thus, the isomorphism $\widetilde{\mathrm{Nor}^*}(N)\to M_\eta$ lifts to a smaller open subset V of U : we have $\widetilde{\mathrm{Nor}^*}((N')_{\mid V})\simeq M_{\mid V}$ . Denote by Z the reduced closed complement of V in X. By induction hypothesis, there exists a $N''\in\mathrm{Mod}_\mathscr{N}(\mathcal{DM}^{\mathrm{\acute{e}t}}(Z))$ such that $\widetilde{\mathrm{Nor}^*}(N'')=M_{\mid Z}$ . The cofiber sequence $\widetilde{\mathrm{Nor}^*}(j_!(N')_{\mid V})\to M \to \widetilde{\mathrm{Nor}^*}(i_*N'')$ with $j:U\to X$ and $i:Z\to X$ the immersion then ensures that M is the image of $\widetilde{\mathrm{Nor}^*}$ , which finishes the proof.

Remark 6.17. The same arguments would prove that the $\infty$ -category

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

of geometric origin objects in the indization of the derived category of mixed Hodge modules is also the category of modules over some algebra $\mathscr{H}_X$ in $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ . Note that the lack of continuity of mixed Hodge modules would prevent things like $p_X^*\mathscr{H}_{\mathrm{Spec} \mathbb{Q}}\simeq \mathscr{H}_{X}$ 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 $(X\to \mathrm{Spec} k)^*\mathscr{H}_{\mathrm{Spec} k}\simeq \mathscr{H}_X$ identifies Drew’s category as the full subcategory $\mathcal{DH}^\mathrm{geo}(X)$ 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 $\mathbf{DH}$ of motivic enriched Hodge modules is the smallest full subcategory of $\mathrm{Ind}\mathcal{D}^b(\mathrm{MHM}(-))$ that is stable under truncation and filtered colimts, and that contains the $f_*g^*H$ for $f:Y\to X$ proper, $g:Y\to\mathrm{Spec} \mathbb{C}$ the structural morphism, and $H\in\mathcal{D}(\mathrm{Ind}\mathrm{MHS}^p)$ . See [Tub24] for a detailed discussion.

Corollary 6.18. The $\infty$ -functor $X\mapsto\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ has a unique extension $\mathcal{DN}_c(X)$ to quasi-compact and quasi-separated schemes of characteristic zero such that for all limits $X=\lim_i X_i$ of such schemes with affine transitions the natural functor

\[ \mathrm{colim}_i\mathcal{DN}_c(X_i)\to\mathcal{DN}_c(X)\]

is an equivalence of $\infty$ -categories, where we compute the colimit in $\mathrm{Pr}^{L}$ .

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 $\mathbb{Q}$ -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 6.15 together with the fact that the colimit is taken in $\mathrm{Pr}^L$ and hence restricts to compact objects. We can denote by $\mathcal{DN}_c$ its left Kan extension along $\iota$ .

Corollary 6.19. Let X be a finite-type k-scheme. The $\infty$ -category $\mathcal{DN}(X)$ 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 $\mathcal{DM}^{\mathrm{\acute{e}t}}(X)$ , thus for $\mathrm{Mod}_{\mathscr{N}_X}(\mathcal{DM}^{\mathrm{\acute{e}t}}(X))\simeq \mathcal{DN}(X)$ by [Iwa18, Lemma 2.6].

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 6.20. Let X be a quasi-projective k-scheme. The two functors

$$\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to {\mathcal{M}_{\mathrm{perv}}}(X),$$

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

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

$$\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to \mathrm{Perv}(X).$$

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 ${\ ^{\mathrm{p}}\mathrm{H}}^0\circ\mathrm{Nor}^*\simeq F\circ \mathrm{H}_\mathrm{univ}$ . Now the compatibility of $\mathrm{Nor}^*$ 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 with $f^*[d]$ for f a smooth map of relative dimension d. We will still denote by F the functor induced by F 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 realisation (or the $\ell$ -adic realisation if the field k is too big) where this exchange morphism is an equivalence, we see that F commutes with $i^*$ . The same argument works to show that F commutes with $f_\sharp$ for smooth f: because F commutes with $f^*$ 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 is it 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, tensor products and $f_\sharp$ for f-1pt, a smooth map. Denote by G its right adjoint. We claim that the natural transformation

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

is an equivalence. Indeed, by Corollary 6.19 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, it is automatically essentially surjective because we know it reaches the generators $f_\sharp\mathbb{Q}(i)$ of $\mathrm{Ind}\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ . Now, as F commutes with the operations, we have by universality of $\mathcal{DM}^{\mathrm{\acute{e}t}}$ that $F\circ \mathrm{Nor}^*\simeq \mathrm{Nor}^*$ . Indeed, because we proved that F commutes with the tensor product and because it is t-exact for the constructible t-structure we can see F as the derived functor of a symmetric monoidal functor on the constructible heart, apply Proposition 5.5 and deduce that F is a morphism of coefficient systems. This proves that the compositions of the morphisms $\mathrm{Nor}^*$ and $F\circ\mathrm{Nor}^*$ with the morphism $\mathcal{SH}\to\mathcal{DM}^\mathrm{\acute{e}t}$ agree by universality of $\mathcal{SH}$ , thus also their factorisation through the localisation $\mathcal{DM}^\mathrm{\acute{e}t}$ of $\mathcal{SH}$ . In particular over X, $F\circ {\ ^{\mathrm{p}}\mathrm{H}}^0\circ \mathrm{Nor}^* \simeq {\ ^{\mathrm{p}}\mathrm{H}}^0\circ\mathrm{Nor}^*$ , hence $F\circ F\circ \mathrm{H}_\mathrm{univ} \simeq {\ ^{\mathrm{p}}\mathrm{H}}^0\circ \mathrm{Nor}^*$ , and the universal property of ${\mathcal{M}_{\mathrm{perv}}}(X)$ tells us that $F\circ F\simeq F$ , so that $F\simeq \mathrm{Id}$ .

6.3 Relations with the t-structure conjecture

Let k be a field of characteristic zero. Recall the following conjecture ([And04, Section 21.1.7] and [Bei12]).

Conjecture 6.21. There exists a non-degenerate t-structure on $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(k)$ compatible with the tensor product and such that the $\ell$ -adic realisation functor is t-exact.

In [Bei12] Beilinson proves that if the characteristic of k is zero the conjecture implies that the realisation functors of $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(k)$ are conservative. In [Bon15, Theorem 3.1.4], Bondarko proves that if such a t-structure exists for all fields of characteristic zero, then there exists a perverse t-structure on $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ for each finite-type k-scheme X. The joint conservativity of the family of $x^*$ for $x\in X$ and the conservativity of the realisation functor over fields implies that in that case, the $\ell$ -adic realisation over X is conservative and t-exact, when its target is endowed with the perverse t-structure.

Theorem 6.22. Assume Conjecture 6.21 for all fields of characteristic zero. Let X be a finite-type k-scheme. Then the heart of the perverse t-structure of $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ is canonically equivalent to the category ${\mathcal{M}_{\mathrm{perv}}}(X)$ of perverse Nori motives. Moreover, the functor $\mathrm{Nor}^* : \mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ is an equivalence of stable $\infty$ -categories. This implies that $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ is the derived category of the abelian category of both its perverse and constructible heart.

From now on, we assume that Conjecture 6.21 is true. We begin with two lemmas, in which we denote by $\mathrm{Var}_k$ the category of quasi-projective k-varieties.

Lemma 6.23. Let X be quasi-projective scheme. Denote by $\mathcal{M}(X)$ the heart of the perverse t-structure on $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ . There is a faithful exact functor $F_X:{\mathcal{M}_{\mathrm{perv}}}(X)\to \mathcal{M}(X)$ that commutes with the $\ell$ -adic realisation. Denote by

$$\gamma_X : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$$

the canonical functor induced by the universal property of the bounded derived category applied to the functor

$${\mathcal{M}_{\mathrm{perv}}}(X)\xrightarrow{F_X}\mathcal{M}(X)\to\mathcal{DM}_c^\mathrm{\acute{e}t}(X).$$

The composition $\mathrm{Nor}^*_X\circ \gamma_X$ is equivalent to the identity functor $\mathrm{Id}_{\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))}$ . Moreover, the $\mathrm{ho}(\gamma_X)$ fit in a morphism $\gamma$ of functors $\mathrm{Var}^\mathrm{op}_k\to\mathrm{SymMono}_1$ with values in symmetric monoidal categories, using the $(-)^*$ -functoriality.

Proof. By assumption the restriction to the heart of the $\ell$ -adic realisation functor

$$\rho_\ell\colon\mathcal{M}(X)\to\mathrm{Perv}(X,\mathbb{Q}_\ell)$$

is faithful exact and the composition $\rho_\ell \circ \, \rm ^P{H^0} $ with the perverse $$\rm ^P{H^0}$$ on $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ is isomorphic to the perverse $\mathrm{H}^0$ of the $\ell$ -adic realisation. By the universal property of perverse Nori motives [IM22, Proposition 6.10] this provides a faithful exact functor $F_X\colon{\mathcal{M}_{\mathrm{perv}}}(X)\to\mathcal{M}(X)$ such that the composition $\rho_\ell\circ F_X$ is isomorphic to the $\ell$ -adic realisation $R_\ell$ of perverse Nori motives and the composition $F_X\circ\mathrm{H}_\mathrm{univ}$ with the universal functor $\mathrm{H}_\mathrm{univ}\colon\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to{\mathcal{M}_{\mathrm{perv}}}(X)$ is the perverse $$\rm ^P{H^0}$$ on $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ : we have a commutative (up to natural isomorphism) diagram.

Now consider the endofunctor

\[{\mathcal{M}_{\mathrm{perv}}}(X)\xrightarrow{F_X}\mathcal{M}(X)\xrightarrow{\mathrm{Nor}^*} {\mathcal{M}_{\mathrm{perv}}}(X)\]

of ${\mathcal{M}_{\mathrm{perv}}}(X)$ . Because $\mathrm{Nor}^*$ is perverse t-exact (this can be checked after $\ell$ -adic realisation), if we compose $F_X$ with the universal $\mathrm{H}_\mathrm{univ}$ we obtain

\[\mathrm{Nor}^*\circ F_X\circ \mathrm{H}_\mathrm{univ}\simeq \mathrm{Nor}^*\circ {\ ^{\mathrm{p}}\mathrm{H}}^0 \simeq {\ ^{\mathrm{p}}\mathrm{H}}^0\circ\mathrm{Nor}^*\]

as functors $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to{\mathcal{M}_{\mathrm{perv}}}(X)$ . But by Corollary 6.20 the functor ${\ ^{\mathrm{p}}\mathrm{H}}^0\circ\mathrm{Nor}^*$ can canonically be identified with $\mathrm{H}_\mathrm{univ}$ . Thus, we have a commutative diagram.

By the universal property of ${\mathcal{M}_{\mathrm{perv}}}(X)$ , the only vertical faithful exact functor fitting in the above diagram and making the two outer triangles commute is (up to natural isomorphism) the identity functor: this implies that there exists a natural equivalence $\mathrm{Nor}^*\circ F_X\simeq\mathrm{Id}_{{\mathcal{M}_{\mathrm{perv}}}(X)}$ .

Now by the universal property of the bounded derived category we have a functor $\mathcal{D}^b(\mathcal{M}(X))\to\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ , thus by composition with the (trivially) derived functor of $F_X$ there is a functor

\[\gamma_X\colon\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X).\]

By definition, the restriction to ${\mathcal{M}_{\mathrm{perv}}}(X)$ of $\gamma_X$ is precisely the functor $F_X$ composed with the inclusion $\mathcal{M}(X)\to\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ . In particular, the composition $\mathrm{Nor}^*_X\circ\gamma_X$ induces the functor $\mathrm{Nor}^*_X\circ F_X$ between the hearts. By the universal property of $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ and the above paragraph, this implies that there exists a natural equivalence $\mathrm{Nor}^*_X\circ\gamma_X\simeq \mathrm{Id}_{\mathcal{D}({\mathcal{M}_{\mathrm{perv}}}(X))}$ .

Using [IM22, Section 2.2] and the t-exactness of the $\ell$ -adic realisation of étale motives we see that there exist natural isomorphisms $i_*\circ \gamma\simeq \gamma\circ i_*$ for i a closed immersion, $f^*[d]\circ\gamma\simeq \gamma\circ f^*[d]$ for f a smooth morphism of relative dimension d, and $\gamma(-)\boxtimes\gamma(-)\simeq \gamma(-\boxtimes -)$ , with $\boxtimes$ the external tensor product. Indeed, for the pullback by a smooth map $f\colon Y\to X$ we have a commutative diagram

obtained by the universal property of ${\mathcal{M}_{\mathrm{perv}}}(X)$ , which induces a commutative square

that gives the desired natural isomorphism by applying the universal property of the bounded derived category. The cases of $i_*$ and $\boxtimes$ are similar. By adjunction there is a natural transformation $i^*\circ \gamma\Rightarrow \gamma\circ i^*$ , which is an isomorphism because $\mathrm{Nor}^*$ is conservative, commutes with $i^*$ and is a retraction of $\gamma$ . Using the fact that any morphism of quasi-projective varieties can be written as a composition of a closed immersion and a smooth map, together with the formula for the tensor product using the external tensor product and pullbacks, we see that $\mathrm{ho}(\gamma)$ commutes with pullbacks and tensor products, giving the last claim of the lemma.

By the lemma above we know that we have a natural transformation

$$\gamma\colon \mathrm{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))\to \mathrm{ho}(\mathcal{DM}^\mathrm{\acute{e}t}_c(-))$$

of contravariant functors on quasi-projective with values in symmetric monoidal 1-categories. We need an $\infty$ -categorical enhancement of this functor. This is the next lemma. Note that as for perverse Nori motives, the existence of a perverse t-structure on étale motives $\mathcal{DM}^\mathrm{\acute{e}t}_c(X)$ implies that there is also a constructible t-structure, whose heart we denote by $\mathcal{CM}(X)$ . The functors $\gamma_X$ and $\mathrm{Nor}^*_X$ are t-exact for the constructible t-structure, as this can be checked after $\ell$ -adic realisation.

Lemma 6.24. Let X be a quasi-projective scheme. The functor $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ induces a symmetric monoidal functor ${\mathcal{M}_{\mathrm{ct}}}(X)\to \mathcal{CM}(X)$ . This gives a natural transformation

\[\delta\colon \mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))\Rightarrow \mathcal{DM}^{\mathrm{\acute{e}t}}_c(-)\]

in $\mathrm{Fun}(\mathrm{Var}_k^\mathrm{op},\mathrm{CAlg}(\mathrm{Cat}_\infty))$ , where we use the $(-)^*$ -functoriality. Moreover, $\delta$ is a morphism of coefficient systems and satisfies $\mathrm{Nor}^*\circ\delta\simeq\mathrm{Id}_{\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))}$ . In fact, we have $\mathrm{ho}(\delta)\simeq \gamma$ .

Proof. Because $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\xrightarrow{\gamma_X}\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ and $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\xrightarrow{\mathrm{Nor}^*}\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ are t-exact for the constructible t-structure, we have faithful exact functors ${\mathcal{M}_{\mathrm{ct}}}(X)\to\mathcal{CM}(X)$ and $\mathcal{CM}(X)\to {\mathcal{M}_{\mathrm{ct}}}(X)$ whose composition is the identity of ${\mathcal{M}_{\mathrm{ct}}}(X)$ by Lemma 6.23.

Now by Proposition 5.6 there is a natural transformation

\[\mathcal{D}^b(\mathcal{CM}(-))\to\mathcal{DM}_c^\mathrm{\acute{e}t}\]

of functors $\mathrm{Sch}_k^\mathrm{op}\to\mathrm{CAlg}(\mathrm{Cat}_\infty)$ . By the last part of Lemma 6.23 we have a functor $\Delta^1\times\mathrm{Var}_k^\mathrm{op}\to\mathrm{SymMono}_1$ sending, for $f\colon Y\to X$ a map of quasi-projective varieties, the map $(0<1,f)$ to ${\mathcal{M}_{\mathrm{ct}}}(Y)\xrightarrow{f^*}{\mathcal{M}_{\mathrm{ct}}}(X)\xrightarrow{\gamma_X}\mathcal{CM}(X)$ , which is also isomorphic to ${\mathcal{M}_{\mathrm{ct}}}(Y)\xrightarrow{\gamma_Y}\mathcal{CM}(Y)\xrightarrow{f^*}\mathcal{CM}(X)$ . Thus, by Proposition 5.6 there is a natural transformation

\[\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))\to\mathcal{D}^b(\mathcal{CM}(-))\]

of functors $\mathrm{Var}_k^\mathrm{op}\to \mathrm{CAlg}(\mathrm{Cat}_\infty)$ . Composing the two, we obtain a natural transformation

$$\delta\colon\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))\Rightarrow \mathcal{DM}^{\mathrm{\acute{e}t}}_c$$

of functors with values in symmetric monoidal stable $\infty$ -categories. The restriction to ${\mathcal{M}_{\mathrm{ct}}}(X)$ of $\delta_X$ is, by definition, the restriction to ${\mathcal{M}_{\mathrm{ct}}}(X)$ of $\gamma_X$ . Thus, $\delta_X\simeq \gamma_X$ by Theorem 5.3.

To prove that $\delta$ is a morphism of coefficient systems, it suffices to prove that if $f\colon Y\to X$ is a smooth morphism of varieties, the natural transformation $f_\sharp\circ\delta_Y\to\delta_X\circ f_\sharp$ is an equivalence. For this, we may apply the conservative functor

\[\rho_\ell\colon\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\to\mathrm{D}^b_c(X_{\mathrm{\acute{e}t}},\mathbb{Q}_\ell),\]

so that the exchange morphism above becomes

\[f_\sharp\circ \rho_\ell\circ \delta_Y\to \rho_\ell\circ\delta_X\circ f_\sharp.\]

Now remark that for S a variety, the composition $\mathrm{Nor}^*_S\circ\delta_S$ is the identity functor and the functor $\mathrm{Nor}^*$ commutes with the $\ell$ -adic realisation, thus we have an isomorphism

\[\rho_\ell\circ\delta_S\simeq R_\ell\circ\mathrm{Nor}^*\circ\delta_S\simeq R_\ell.\]

As the functor $R_\ell$ commutes with $f_\sharp$ , the proof is finished.

Proof of Theorem 6.22. Because both $\mathcal{DM}^\mathrm{\acute{e}t}_c$ and $\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(-))$ are Zariski hypersheaves and $\mathrm{Nor}^*$ is a morphism of sheaves, we may assume that X is quasi-projective.

Let $\delta_X$ be the functor of Lemma 6.24. We know that $\mathrm{Nor}^*_X\circ\delta_X\simeq\mathrm{Id}_{\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))}.$ We can now show that $\delta_X\circ\mathrm{Nor}^*_X\simeq\mathrm{Id}_{\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)}$ . By Lemma 6.24 and Theorem 6.5 the functor $\delta_X\circ\mathrm{Nor}^*_X$ is a morphism of coefficient systems. In particular, its composition with the canonical symmetric monoidal functor

\[M_X\colon \mathrm{Sm}_X\to\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)\]

is isomorphic to the canonical symmetric monoidal functor $M_X$ . Using the universal property of $\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)$ proved by Robalo in [Rob15, Corollary 2.29] (or rather, its étale and $\mathbb{Q}$ -linear counterpart which can be deduced from it) this implies that

$$\delta_X\circ\mathrm{Nor}^*_X\simeq\mathrm{Id}_{\mathcal{DM}^{\mathrm{\acute{e}t}}_c(X)}.$$

This finishes the proof as now we have that $\mathrm{Nor}^*_X$ and $\delta_X$ are inverses of each other.