2.1 Forthcoming and future work

In [Reference TubachTub24], 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 [Reference Ruimy and TubachRT24] 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 [Reference BeilinsonBei87], it is also the derived category of the abelian category of perverse sheaves $\mathrm{Perv}(X,\mathbb{Q})$ defined in [Reference Beilinson, Bernstein and DeligneBBD82]. 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 [Reference Hemo, Richarz and ScholbachHRS23].

We use the language of $\infty$ -categories as developed in [Reference LurieLur18a] 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 $\mathrm{map}_\mathcal{C}(-,-)$ .

3.1 Categories of étale motives

Ayoub [Reference AyoubAyo10, Reference AyoubAyo14a], Cisinski and Déglise [Reference Cisinski and DégliseCD19, Reference Cisinski and DégliseCD16] and Voevodsky [Reference VoevodskyVoe02] 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 [Reference RobaloRob15, 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 [Reference AyoubAyo14a, 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 [Reference RobaloRob14, 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 [Reference Cisinski and DégliseCD19, 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 [Reference Morel and VoevodskyMV99]. 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 [Reference AyoubAyo10] 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 [Reference BeilinsonBei87] and of the abelian category of constructible sheaves [Reference NoriNor02]. One can find in [Reference AyoubAyo14a] and [Reference Cisinski and DégliseCD16] 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 [Reference Richarz and ScholbachRS20, 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 [Reference Hemo, Richarz and ScholbachHRS23] 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 [Reference Ivorra and MorelIM22], 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 [Reference AyoubAyo24, 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 [Reference Ivorra and MorelIM22, 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 [Reference AyoubAyo07, 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 [Reference TerenziTer24] 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 [Reference SaitoSai90], 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.

4.1 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 [Reference AyoubAyo07]), 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 ([Reference Ivorra and MorelIM22, Theorem 5.1] and [Reference TerenziTer24, Main theorem]) or of mixed Hodge modules [Reference SaitoSai90, 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 [Reference Beilinson, Bernstein and DeligneBBD82, 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 [Reference Bunke, Cisinski, Kasprowski and WingesBCKW24, 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.

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.

Proof. For the first point, dualisable objects in $\mathrm{D}^b_c(X,\mathbb{Q})$ are exactly local systems by Ayoub [Reference AyoubAyo22, Lemma 1.24] in the analytic case, and by [Reference Hemo, Richarz and ScholbachHRS23, Proposition 7.6] together with [Reference Martini and WolfMW22, 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

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 [Reference NoriNor02, Proposition 2.2].

Moreover, Nori [Reference NoriNor02] 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 [Reference BarrettBar21, 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.

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 [Reference Beilinson, Bernstein and DeligneBBD82, 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.

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

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.

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.

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.

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.

Corollary 4.16 [Reference NoriNor02, 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.

4.4 Lisse motives enter the stage

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.

5.1 Categorical preliminaries

Recall the following fundamental definition due to Lurie.

The homotopy category $\mathrm{ho}(\mathcal{C})$ of a stable $\infty$ -category is always a triangulated category [Reference LurieLur17, 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.

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 [Reference Bunke, Cisinski, Kasprowski and WingesBCKW24, 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.

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 [Reference Mac LaneML98, Chapter XI, Section 1, Theorem 1] we know that any symmetric monoidal 1-category has all higher coherences. Written differently as in Lurie [Reference LurieLur17, 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 [Reference LurieLur17, 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 [Reference LurieLur17, Construction 2.4.3.1]). By [Reference HinichHin16, 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 unstraighteningFootnote ¹ and [Reference LurieLur17, 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.

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 [Reference Bunke, Cisinski, Kasprowski and WingesBCKW24, 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 [Reference CisinskiCis19, 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 [Reference LurieLur17, 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 [Reference LurieLur17, 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 [Reference Bunke, Cisinski, Kasprowski and WingesBCKW24, 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 [Reference HinichHin16, 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 [Reference Bunke, Cisinski, Kasprowski and WingesBCKW24, Proposition 7.4.11]. Thus, the map

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

factors through

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

By [Reference HinichHin16, 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.

We will use the following two theorems.

Recall that by [Reference LurieLur18a, Tag 01F1], we have the following proposition.

Proof. Indeed, denote by $\iota : A\to\mathcal{C}$ the canonical monomorphism of simplicial sets. Then by [Reference LurieLur18a, 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. [Reference CisinskiCis19, 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.

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 [Reference LurieLur18a, 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 [Reference LurieLur18a, Theorem 02T8], $\chi$ is a limit diagram if and only if the restriction morphism in (5.5) is an equivalence. In that case, by [Reference LurieLur18a, 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 [Reference LurieLur18a, 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 [Reference LurieLur18a, Definition 02KG]. By [Reference LurieLur18a, 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 [Reference LurieLur18a, 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 [Reference LurieLur18a, 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 [Reference LurieLur18a, 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 [Reference LurieLur11, Theorem 5.17].

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 [Reference Ivorra and MorelIM22] or the category of Saito’s mixed Hodge modules constructed in [Reference SaitoSai90]. 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 [Reference Hemo, Richarz and ScholbachHRS23], 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 [Reference LurieLur18b, 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 [Reference Hemo, Richarz and ScholbachHRS23, 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 $\mathcal{D}^b({\mathcal{M}_{\mathrm{ct}}}(-))$ .

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 [Reference SaitoSai90, (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 $f_\sharp$ .

Moreover, the projection formula for a smooth morphism holds in $ \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ .

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.

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 $\mathscr{{H om}}\,(-,-)$ .

Verdier duality also lifts.

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. Denote ${\mathcal{D}^b_\mathcal{M}}(X):=\mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))$ . By [Reference Cisinski and DégliseCD19, Corollary 3.3.38] (or see the proof of [Reference Bhatt and MathewBM21, 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 $\mathrm{D}^{\geqslant -m}_c(X)$ .

6.1 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 [Reference Drew and GallauerDG22] 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).

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, [Reference AokiAok23, 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 [Reference Richarz and ScholbachRS20, 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 [Reference Cisinski and DégliseCD19, 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 [Reference Cisinski and DégliseCD19, 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 [Reference Cisinski and DégliseCD19, 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.

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 [Reference LurieLur17, 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 [Reference LurieLur17, 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 [Reference CisinskiCis19, 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 [Reference LurieLur17, 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 [Reference Cisinski and DégliseCD19, 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 [Reference Cisinski and DégliseCD19, 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, [Reference Cisinski and DégliseCD19, 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.

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

Proof. The case of $\rho_\mathbb{Q}$ follows from [Reference AyoubAyo14a, 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 [Reference Cisinski and DégliseCD19, 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 [Reference LurieLur17, 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 [Reference AyoubAyo14b, Lemma 2.8], using that $\mathbb{Q}^+_Y\simeq f^*\mathbb{Q}^+_X$ .

Recall that by [Reference Ivorra and MorelIM22, Corollary 6.18], [Reference BondarkoBon11, Proposition 2.3.1] and [Reference HébertHéb11, 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 [Reference Cisinski and DégliseCD19, 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.

Now all that remains is $R_H : \mathcal{D}^b({\mathcal{M}_{\mathrm{perv}}}(X))\to\mathcal{D}^b(\mathrm{MHM}(X))$ . By [Reference Ivorra and MorelIM22, 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 [Reference Ivorra and MorelIM22, 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.