is an equivalence. The t-structure on DN<sub>gm</sub>(X, Λ) restricts to the full subcategory DM<sup>ét,sm0</sup><sub>c</sub>(X, Λ) and the heart of the restricted t-structure is equivalent to Rep<sup>A</sup>(π<sup>ét</sup><sub>1</sub>(X), Λ).

Proof. Using étale descent, continuity and Lemma 5.3.2, it suffices to prove the statements if X is of finite type over an algebraically closed field.

First note that the inclusion $\mathrm{DM}_{\mathrm{rig}}^{\mathrm{ét},0}(X, \Lambda) \to \mathrm{DN}_{\mathrm{rig}}^{0}(X, \Lambda)$ is an equivalence, where the right hand side consists of the objects of $\mathrm{DN}^{0}(X, \Lambda)$ which are dualisable in $\mathrm{DN}(X, \Lambda)$. Indeed, if $M$ belongs to $\mathrm{DN}_{\mathrm{rig}}^{0}(X, \Lambda)$ then its dual is also a 0-motive, because being a constructible 0-motive can be tested on points and all constructible 0-motives over a point are dualisable, with 0-motivic duals. But now, the t-structure of $\mathrm{DN}_{\mathrm{gm}}(X, \Lambda)$ restricts to $\mathrm{DM}_{\mathrm{rig}}^{\mathrm{ét},0}(X, \Lambda)$ because it can be restricted to both dualisable objects and 0-motives.

By Proposition 4.2.1 we have a pullback square

<figure><img src="image_17.png" /></figure>

where $\mathrm{Loc}(X^{\mathrm{an}}, \Lambda)$ is the abelian category of local systems on $X^{\mathrm{an}}$ with $\Lambda$ coefficients. By [71] the category $\mathrm{DM}_{\mathrm{rig}}^{\mathrm{ét},0}(X, \Lambda \otimes \mathbb{Q})^{\heartsuit}$ is equivalent (through $\rho_!$) to $\mathrm{Rep}^{A}(\pi_1^{\mathrm{ét}}(X), \Lambda \otimes_{\mathbb{Z}} \mathbb{Q})$. As the category $\mathrm{Rep}^{A}(\pi_1^{\mathrm{ét}}(X), \Lambda)$ also fits in the corner of this pullback diagram, this proves that the functor $\rho_!$ induces an equivalence $\mathrm{Rep}^{A}(\pi_1^{\mathrm{ét}}(X), \Lambda) \to \mathrm{DM}_{\mathrm{rig}}^{\mathrm{ét},0}(X, \Lambda)^{\heartsuit}$. Finally, by dévissage, because any object of $\mathrm{Rep}^{A}(\pi_1^{\mathrm{ét}}(X), \Lambda)$ defines an object of $\mathrm{DM}_{\mathrm{c}}^{\mathrm{ét,sm0}}(X, \Lambda)$, any rigid 0-motive is smooth Artin. $\square$

Using the framework of Nori motives, we can prove new cases of [64, Theorem 3.2.3.7] which is one of the main results of loc. cit. and is an analogue of the Artin Vanishing Theorem in the setting of Artin motives. We begin by the following lemma:

**Lemma 5.3.4.** Let $\Lambda$ be a regular ring and $X$ be a $\mathrm{qcqs}$ $\mathbb{Q}$-scheme. The ordinary and perverse t-structures on $\mathrm{DN}_{\mathrm{gm}}(X, \Lambda)$ restrict to the category $\mathrm{DN}_{\mathrm{c}}^{\mathrm{coh}}(X, \Lambda)$ defined as the thick subcategory of $\mathrm{DN}(X, \Lambda)$ generated by the $f_*\Lambda_Y$ where $f: Y \to X$ is proper.

Proof. By [71], the result holds if $\Lambda$ is a $\mathbb{Q}$-algebra. Moreover it suffices to prove the result over the spectrum of a field $k$. Because the map

$$ \mathrm{DN}_{\mathrm{c}}^{\mathrm{coh}}(k, \Lambda) \otimes_{\mathrm{Perf}_{\Lambda}} \mathrm{Perf}_{\Lambda \otimes_{\mathbb{Z}} \mathbb{Q}} \to \mathrm{DN}_{\mathrm{c}}^{\mathrm{coh}}(k, \Lambda \otimes_{\mathbb{Z}} \mathbb{Q}) $$

is an equivalence and if $M$ is in $\mathrm{DN}_{\mathrm{c}}^{\mathrm{coh}}(k, \Lambda)$ then $\mathrm{H}^i(M \otimes_{\mathbb{Z}} \mathbb{Q}) \simeq \mathrm{H}^i(M) \otimes_{\mathbb{Z}} \mathbb{Q}$ is cohomological, there is a cohomological motive $P$ in $\mathrm{DN}_{\mathrm{c}}^{\mathrm{coh}}(k, \Lambda)$ and a map $P \to \mathrm{H}^i(M)$ whose cone is torsion and dualisable, thus a constructible 0-motive, which are cohomological. The claim about the perverse t-structure is true by gluing. $\square$
Recall that a smooth Artin motive $M$ is $\mathbb{Q}$-constructible if $M \otimes_{\mathbb{Z}} \mathbb{Q}$ is constructible as a smooth Artin motive. We denote by $\mathrm{DM}_{\mathbb{Q}-c}^{\mathrm{et},sm0}(X, \Lambda)$ the category of $\mathbb{Q}$-constructible smooth Artin motives. We can now prove new cases of [64, Theorem 3.2.3.7].

**Proposition 5.3.5.** Let $S$ be an excellent scheme $\mathbb{Q}$-scheme, let $f: X \to S$ be an affine quasi-finite morphism and let $\Lambda$ be a regular ring. Assume that $X$ is regular. Then, the functor

$$f_!: \mathrm{DM}_{\mathbb{Q}-c}^{\mathrm{et},sm0}(X, \Lambda) \to \mathrm{DM}^{\mathrm{et},0}(S, \Lambda)$$

is t-exact with respect to the perverse homotopy t-structure (see [64, Definition 3.2.1.1]).

*Proof.* The same proof as in [64, Theorem 3.2.3.7] allows to reduce to the case when $\Lambda$ is a $\mathbb{Q}$-algebra. We now have to prove that $f_!: \mathrm{DN}_c^{sm0}(X, \Lambda) \to \mathrm{DN}^0(S, \Lambda)$ is t-exact with respect to the perverse homotopy t-structure (defined in the same fashion as in [64, Definition 3.2.1.1]).

The rest of the proof is an adaptation of the arguments of [65, Section 4.8]. We can write the functor

$$f_!: \mathrm{DM}_c^{\mathrm{et},sm0}(X, \Lambda) \to \mathrm{DM}^{\mathrm{et},0}(S, \Lambda)$$

as the composition

$$\mathrm{DM}_c^{\mathrm{et},sm0}(X, \Lambda) \simeq \mathrm{DN}_c^{sm0}(X, \Lambda) \xrightarrow{\iota_0} \mathrm{DN}_c^{\mathrm{coh}}(X, \Lambda) \xrightarrow{f_!} \mathrm{DN}_c^{\mathrm{coh}}(S, \Lambda) \xrightarrow{\omega^0} \mathrm{DN}_c^0(S, \Lambda) \simeq \mathrm{DM}_c^{\mathrm{et},0}(S, \Lambda),$$

where the Artin truncation functor

$$\omega^0: \mathrm{DN}^{\mathrm{coh}}(X, \Lambda) \to \mathrm{DN}^0(X, \Lambda)$$

defined as the right adjoint to the inclusion $\iota_0$ of Artin motives into cohomological motives, lands in $\mathrm{DN}_c^0(X, \Lambda)$ by [71]. Thus it suffices to check that all functors are t-exact. The functor $\omega^0$ is t-exact with respect to the perverse t-structure on the left hand side (which exists by the above Lemma 5.3.4) and the perverse homotopy t-structure on the right hand side: this is because an object of $\mathrm{DN}^0(X, \Lambda)$ is perverse homotopy t-negative if and only if it is perverse t-negative (this can be shown as in [65, Proposition 4.5.1]) and because the functor $\omega^0$ is perverse right t-exact as per [58, Lemma 3.2.6]. But as $\iota_0$ is t-exact when restricted to smooth Artin objects because it is t-exact for the ordinary t-structure, the result reduces to the usual Artin Vanishing Theorem. $\square$

## 5.4 Motivic intersection complex.

Let $\Lambda$ be a regular ring. Consider an excellent scheme $B$ of dimension 2 or less as well as a regular separated $B$-scheme of finite type $S$, endowed with a geometric and
⊗-invertible object $\omega_S$ in DN$(S, \Lambda)$ (note that $\omega_S = \Lambda_S$ works). We fix $X$ a finite type $S$-scheme.

From now on, we assume that $X$ is endowed with a bounded below dimension function $\delta$; by shifting, we can assume 0 to be its lower bound. Any non-increasing function $\bar{q}: \mathbb{N} \to \mathbb{Z}$, defines a perversity function $q = \bar{q} \circ \delta$ on $X$ along any stratification of $X$ by connected regular and locally closed strata (in the sense of [17, Section 2.1.1]). The construction given in Proposition 5.1.1 for $\bar{q} = -\text{Id}$ works whenever the condition

$$ (*) : \forall n \in \mathbb{N}, q(n) - q(n + 1) \in \{0, 1, 2\} $$

is satisfied. It then yields a $q$-perverse t-structure ${}^q\text{t}$ on DN$_{gm}(X, \Lambda)$, whose heart we will denote by $\mathcal{M}_{q-\text{perv}}(X, \Lambda)$ and whose cohomology functors are will be denoted by ${}^q\text{H}^n(-)$.

**Definition 5.4.1.** Let $\bar{q}: \mathbb{N} \to \mathbb{Z}$ be a non-increasing function satisfying condition $(*)$, let $q = \bar{q} \circ \delta$ and let $j: U \to X$ be an open immersion. The $q$-intermediate extension functor is the functor

$$ {}^q j_{1*}: \mathcal{M}_{q-\text{perv}}(U, \Lambda) \to \mathcal{M}_{q-\text{perv}}(X, \Lambda) $$

defined by

$$ {}^q j_{1*} M = \text{Im}({}^q\text{H}^0(j_* M) \to {}^q\text{H}^0(j_* M)). $$

The $q$-motivic intersection complex is the $q$-perverse Nori motive

$$ {}^q\text{IC}_{X,\Lambda} := {}^q j_{1*}(\Lambda[-q(U)])[q(U)]. $$

When $X$ is endowed with a dimension function $\delta$ and $q = -\delta$ is the self-dual perversity, we simply refer to those as the intermediate extension functor $j_{1*}$ and the motivic intersection complex $\text{IC}_{X,\Lambda}$.

**Remark 5.4.2.** In [73, 74] Jörg Wildeshaus constructed a motivic intersection complex in DM$(X, \mathbb{Q})$ using weights, in several cases (mainly when $X$ is a Shimura variety). In those cases, we expect the complex $\text{IC}_{X,\mathbb{Q}}$ to be the Nori realisation of Wildeshaus' motivic intersection complex up to a shift of $-\text{dim}(X)$. Indeed assuming the existence of the motivic t-structure, this would be true by uniqueness of Wildeshaus' complex [73, Theorem 3.1].

We fix a stratification $(X_i)$ of $X$ by regular and connected locally closed sub-schemes. Given an integer $n$, we set $U_n$ to be the union of those strata $X_i$ such that $\delta(X_i) \ge n$ and $j_n: U_n \to U_{n-1}$ to be the open immersion.

**Proposition 5.4.3.** Let $\bar{q}: \mathbb{N} \to \mathbb{Z}$ be a non-increasing function satisfying condition $(*)$ and let $q = \bar{q} \circ \delta$. Let $k \ge 0$ be an integer and let $j: U_k \to X = U_0$ be the open immersion. Then there is a canonical isomorphism of functors

$$ {}^q j_{1*} \simeq (\tau \le \bar{q}(0)-1(j_1)_*) \circ \dots \circ (\tau \le \bar{q}(k-2)-1(j_{k-1})_*) \circ (\tau \le \bar{q}(k-1)-1(j_k)_*) $$
on $\mathcal{M}_{q-\text{perv}}(U_k)$, with $\tau^{\le i}$ the truncations functors for the ordinary t-structure.

*Proof.* This is a consequence of [17, Proposition 1.4.23] (see the proof of [17, Proposition 2.1.11]). $\square$

**Corollary 5.4.4.** Let $\bar{q}, \bar{r}: \mathbb{N} \to \mathbb{Z}$ be non-increasing functions and $q = \bar{q} \circ \delta$ and $r = \bar{r} \circ \delta$. Let $k \ge 0$ be an integer and let $j: U_k \to X = U_0$ be the open immersion. There exists a canonical pairing in $\text{DN}_c(X, \Lambda)$

$$ {}^qj_{!*}(M[-q(U_k)]) \otimes {}^rj_{!*}(N[-r(U_k)]) \to {}^{q+r}j_{!*}(M \otimes N[-q(U_k) - r(U_k)]) $$

for $M, N \in \mathcal{M}_{\text{ord}}^{\text{rig}}(U_k)$ motivic local systems (that is, objects in the ordinary heart which are dualisable).

*Proof.* This is a direct consequence of the formula for the intermediate extension given in Proposition 5.4.3, knowing that the functors of the form $\gamma_*$ are lax-monoidal as right adjoints of monoidal functors and that for any $M$ and $N$, we have a map

$$ \tau^{\le n}(M) \otimes \tau^{\le m}(N) \to \tau^{\le n+m}(M \otimes N) $$

given by the right t-exactness of the tensor product and adjunction. $\square$

This yields a pairing

$$ \mathrm{IC}_{X, \Lambda}^q \otimes \mathrm{IC}_{X, \Lambda}^r \to \mathrm{IC}_{X, \Lambda}^{q+r} $$

in $\text{DN}(X, \Lambda)$. Now consider the top perversity $\bar{t} = -2\text{Id}_Z$. Gluing on the stratification as in [37, Section 5.1], whenever $\bar{q} \le \bar{t}$, there exists a map $\mathrm{IC}_{X, \Lambda}^q \to \omega_X[\delta(X)]$ where $\omega_X = \mathbb{D}_X(\Lambda_X)$ is the dualising complex. Whence a map

$$ \mathrm{IC}_{X, \Lambda} \otimes \mathrm{IC}_{X, \Lambda} \to \omega_X[\delta(X)]. \qquad (5.4.4.1) $$

When $\Lambda$ is a $\mathbb{Q}$-algebra, this pairing induces an equivalence

$$ \mathrm{IC}_{X, \Lambda} \xrightarrow{\sim} \mathbb{D}_X(\mathrm{IC}_{X, \Lambda})[-\delta(X)] \qquad (5.4.4.2) $$

by [37, Section 5.3]. For simplicity, assume now that $\Lambda = \mathbb{Z}$ is the ring of integers. Let $k$ be a field of characteristic zero.

**Lemma 5.4.5.** Let $K \in \text{DN}_c(k, \mathbb{Z})$ and $M \in \mathcal{M}(k, \mathbb{Z})$. For each $i \in \mathbb{Z}$ there exists an short exact sequence

$$ 0 \to \mathrm{H}^1(\mathrm{Hom}(\mathrm{H}^{-i+1}(K), M)) \to \mathrm{H}^i(\mathrm{Hom}(K, M)) \to \mathrm{H}^0(\mathrm{Hom}(\mathrm{H}^{-i}(K), M)) \to 0 $$

in $\mathcal{M}(k, \mathbb{Z})$.

*Proof.* This realises to the classical universal coefficient theorem of abelian groups (or $\mathbb{Z}_\ell$-modules if one takes $\ell$-adic realisations). $\square$
**Definition 5.4.6.** Assume that $X$ is a finite type $k$-scheme. The intersection homology complex of $X$ is the object

$$ \mathrm{IH}(X, \mathbb{Z}) := p_*\mathrm{IC}_{X,\mathbb{Z}} $$

in $\mathrm{DN}_c(k, \mathbb{Z})$ where $p: X \to \mathrm{Spec}(k)$ is the structural morphism. Denote by $\mathrm{IH}_i(X, \mathbb{Z}) = \mathrm{H}^{-i}(\mathrm{IH}(X, \mathbb{Z}))$ the $i$-th intersection homology motive. There is also a version with compact support

$$ \mathrm{IH}_c(X, \mathbb{Z}) := p_*\mathrm{IC}_{X,\mathbb{Z}} $$

and its homology motive $\mathrm{IH}_{i,c}(X, \mathbb{Z}) = \mathrm{H}^{-i}(\mathrm{IH}_c(X, \mathbb{Z}))$

When $\Lambda = \mathbb{Q}$, we obtain from (5.4.4.2) a perfect pairing

$$ \mathrm{IH}_{i,c}(X, \mathbb{Q}) \otimes_{\mathbb{Q}} \mathrm{IH}_{d-i}(X, \mathbb{Q}) \to \mathbb{Q}_k(-d), $$

where $d$ is the dimension of $X$.

We finish with a motivic lift of the results in [38, Section 3] providing a motivic Seifert linking pairing. The proof of the next proposition is the same as in loc. cit.

**Proposition 5.4.7.** For $i \in \mathbb{Z}$ denote by $T_i(X)$ (resp. by $T_{i,c}(X)$) the maximal torsion subobject of $\mathrm{IH}_i(X, \mathbb{Z})$ (resp. of $\mathrm{IH}_{i,c}(X, \mathbb{Z})$). There is a pairing

$$ T_{i,c}(X) \times T_{d-i-1}(X) \to \mathbb{Q}/\mathbb{Z}(-d) $$

in $\mathcal{M}(k, \mathbb{Z})$.

*Proof.* We apply Lemma 5.4.5 and obtain a short exact sequence

$$ 0 \to \mathrm{H}^1(\underline{\mathrm{Hom}}(\mathrm{IH}_{d-i-1}(X, \mathbb{Z}), \mathbb{Z})) \to \mathrm{H}^{-i}(\underline{\mathrm{Hom}}(\mathrm{IH}(X, \mathbb{Z})[d](d), \mathbb{Z})) \to \mathrm{H}^0(\underline{\mathrm{Hom}}(\mathrm{IH}_{d-i}(X, \mathbb{Z}), \mathbb{Z})) \to 0. $$

Using the exact triangle $\mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ and the fact that for any $M \in \mathcal{M}(k, \mathbb{Z})$ the object $\mathrm{H}^1(\underline{\mathrm{Hom}}(M, \mathbb{Q}))$ vanishes we obtain that

$$ \mathrm{H}^0(\underline{\mathrm{Hom}}(\mathrm{IH}_{d-i-1}(X, \mathbb{Z})(d), \mathbb{Q}/\mathbb{Z})) \xrightarrow{\sim} \mathrm{H}^1(\underline{\mathrm{Hom}}(\mathrm{IH}_{d-i-1}(X, \mathbb{Z})(d), \mathbb{Z})) $$

is an equivalence. Moreover because $\mathbb{Q}/\mathbb{Z}$ is torsion, the map

$$ \mathrm{H}^0(\underline{\mathrm{Hom}}(\mathrm{T}_{d-i-1}(X)(d), \mathbb{Q}/\mathbb{Z})) \to \mathrm{H}^0(\underline{\mathrm{Hom}}(\mathrm{IH}_{d-i-1}(X, \mathbb{Z})(d), \mathbb{Q}/\mathbb{Z})) $$

is also an equivalence (this is true after realisation). Also any map $M \to \mathrm{H}^0(\underline{\mathrm{Hom}}(\mathrm{IH}_{d-i}(X, \mathbb{Z})(d), \mathbb{Z}))$ with $M$ torsion vanishes because it can be checked after realisation. Thus the composed map $T_{i,c}(X) \to \mathrm{IH}_{i,c}(X, \mathbb{Z}) \to \mathrm{H}^{-i}(\underline{\mathrm{Hom}}(\mathrm{IH}(X, \mathbb{Z})[d](d), \mathbb{Z}))$ obtained from applying $\mathrm{H}^{-i}p_!$ to (5.4.4.1) factors through the inclusion

$$ \mathrm{H}^0(\underline{\mathrm{Hom}}(\mathrm{T}_{d-i-1}(X)(d), \mathbb{Q}/\mathbb{Z})) \to \mathrm{H}^{-i}(\underline{\mathrm{Hom}}(\mathrm{IH}(X, \mathbb{Z})[d](d), \mathbb{Z})), $$

yielding

$$ T_{i,c}(X) \to \mathrm{H}^0(\underline{\mathrm{Hom}}(\mathrm{T}_{d-i-1}(X)(d), \mathbb{Q}/\mathbb{Z})). $$

By adjunction (here we use that $\mathcal{M}(k, \mathbb{Z})$ has enough flat objects) this provides the desired pairing. $\square$
Remark 5.4.8. The above pairing is non-degenerate when $X$ is smooth. It is not non-degenerate in general, there is a topological obstruction introduced in [38] when $X$ is singular and proper.

# 6 Integral mixed Hodge modules.

## 6.1 Definition of mixed Hodge modules with integral coefficients

Let $X$ be a separated finite type $\mathbb{C}$-scheme. In [68], M. Saito constructed an abelian category $\mathrm{MHM}(X)$ of mixed Hodge modules over $X$, which is a relative version of mixed Hodge structures, modelled on perverse sheaves, in the sense that if $X = \mathrm{Spec}(\mathbb{C})$, we have that $\mathrm{MHM}(X) \simeq \mathrm{MHS}_{\mathbb{Q}}^P(\mathbb{C})$ is the category of polarisable mixed Hodge structure over $\mathbb{C}$ with rational coefficients (which consists of mixed Hodge structure such that the graded pieces of the weight filtration are polarisable [30, Définition 2.1.15], it is a full subcategory of Deligne’s mixed Hodge structures), and there is a faithful exact functor

$$ \mathrm{rat}: \mathrm{MHM}(X) \to \mathrm{Perv}(X^{\mathrm{an}}, \mathbb{Q}). $$

Moreover, M. Saito constructed the six operations on the bounded derived category $\mathrm{D}^b(\mathrm{MHM}(-))$, in a way compatible with the functor $\mathrm{rat}$. In [70] the second author proved that the construction of these six operations by M. Saito could be canonically lifted to the world of $\infty$-categories, with all possible coherences (this includes an extension to non separated schemes and morphisms). As a consequence, using the universal property of $\mathrm{DM}^{\mathrm{ét}}(-, \mathbb{Q})$, the second author obtained a realisation

$$ \rho_H: \mathrm{DM}^{\mathrm{ét}}(-, \mathbb{Q}) \to \mathrm{D}_H(-) $$

compatible with the six operations, where $\mathrm{D}_H(-)$ is the indization of $\mathrm{D}^b(\mathrm{MHM}(-))$. By [15] the abelian category of polarisable mixed Hodge structure with rational coefficients $\mathrm{MHS}_{\mathbb{Q}}^P(\mathbb{C})$ is of cohomological dimension 1. If $X$ is a complex variety of dimension $d$ with structural morphism $\pi_X$, as the cohomological amplitude of $(\pi_X)_{\mathrm{Hom}}(-, -)$ can be measured after applying the functor $\mathrm{rat}$, it is at most $2d + 1$; therefore the abelian category of $\mathrm{MHM}(X)$ is of cohomological dimension less that $2d + 2$. Thus we have in fact that $\mathrm{D}_H(X)$ is the unbounded derived category of ind-mixed Hodge modules $\mathrm{D}(\mathrm{Ind} \mathrm{MHM}(X))$ (because, for example by [26, Lemma 1.1.7] the functor $\mathrm{map}_D(M, -)$ commutes with colimits when $M$ an object of $\mathrm{MHM}(X)$).

In this section, we show that the methods we used for Nori motives apply for mixed Hodge modules, an that proofs are easier in this case.

**Definition 6.1.1.** The presentable $\infty$-category of mixed Hodge modules with integral
coefficients is the pullback

<figure><img src="image_18.png" /></figure>

where the pullback is taken in the $\infty$-category of functors $(\mathrm{Sch}_\mathbb{C}^{\mathrm{ft}})^{\mathrm{op}} \to \mathrm{Pr}_{\mathrm{St}}^{\mathrm{L},\omega}$.

Because all categories have a perverse t-structure and that the functors are t-exact, the $\infty$-category $\mathrm{D}_H(X, \mathbb{Z})$ affords a perverse t-structure, that restricts to compact objects, over any finite type $\mathbb{C}$-scheme $X$. Moreover, $\mathrm{D}_H(-, \mathbb{Z})$ affords a 6-functor formalism.

Recall that in [30, Définition 2.3.1] Deligne defines mixed Hodge structures that have an integral lattice. In fact, we have a pullback for the abelian categories of polarisable mixed Hodge structures

<figure><img src="image_19.png" /></figure>

with $\mathrm{Vect}_\mathbb{Q}^{\mathrm{fd}}$ the abelian category of finite-dimensional $\mathbb{Q}$-vector spaces. (note that the weight filtration only exists with rational coefficients by definition) so that by the universal property of the fiber product, we obtain a functor

$$ \mathrm{D}(\mathrm{Ind} \, \mathrm{MHS}_{\mathbb{Z}}^{P}(\mathbb{C})) \to \mathrm{D}_{\mathrm{H}}(\mathrm{Spec}(\mathbb{C})). $$

**Proposition 6.1.2.** *The above functor is an equivalence.*

*Proof.* This follows from the fact that the cohomological dimension of $\mathrm{MHS}_{\mathbb{Z}}^{P}$ is 1 by [15, Corollary 1.10 and Remark 2.4]. Indeed in that case by [26, Lemma 1.1.7] any object of $\mathrm{D}^b(\mathrm{MHS}_{\mathbb{Z}}^{P})$ defined a compact object of $\mathrm{D}(\mathrm{Ind} \, \mathrm{MHS}_{\mathbb{Z}}^{P})$. $\square$

There is a canonical realisation

$$ \rho_H: \mathrm{DM}^{\mathrm{et}}(-, \mathbb{Z}) \to \mathrm{D}_H(-, \mathbb{Z}) $$

that commutes with the 6 operations. Indeed, this functor is obtained from the universal property of the pullback, and as $\mathrm{D}_H(-, \mathbb{Z})$ is compactly generated all operations commute with rationalisation so that the proof consist in proving that the exchange maps are equivalence when rationalised (this follows from [27, Theorem 4.2.29]) and when taken mod $p$ for all primes $p$.
## 6.2 Mixed Hodge modules over $\mathbb{R}$-schemes.

In this section we extend the definition of mixed Hodge modules (with integral coefficients) to $\mathbb{R}$-schemes, using the same method as for Nori motives.

Let $X$ be a finite type $\mathbb{R}$-scheme. Then $X_{\mathbb{C}} := X \times_{\mathrm{Spec}(\mathbb{R})} \mathrm{Spec}(\mathbb{C})$ has a canonical $\mathbb{Z}/2\mathbb{Z}$ action given by the complex conjugation, and thus the $\infty$-category $\mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Z})$ is endowed with a $\mathbb{Z}/2\mathbb{Z}$ action. In fact this construction is functorial and we obtain a functor

$$(\mathrm{Sch}_{\mathbb{R}}^{\mathrm{ft}})^{\mathrm{op}} \to \mathrm{Fun}(\mathrm{BZ}/2\mathbb{Z}, \mathrm{CAlg}(\mathrm{Pr}^L))$$

that sends a finite type $\mathbb{R}$-scheme $X$ to $\mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Z})$.

Now consider the functor

$$(-)^{\mathbb{h}\mathbb{Z}/2\mathbb{Z}} : \mathrm{Fun}(\mathrm{BZ}/2\mathbb{Z}, \mathrm{CAlg}(\mathrm{Pr}^L)) \to \mathrm{CAlg}(\mathrm{Pr}^L)$$

that sends a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ with a $\mathbb{Z}/2\mathbb{Z}$ action to the homotopy fixed points $\mathcal{C}^{\mathbb{h}\mathbb{Z}/2\mathbb{Z}}$. Note that this functor takes a diagram $G: \mathrm{BZ}/2\mathbb{Z} \to \mathrm{CAlg}(\mathrm{Pr}^L)$ to its limit, thus has a left adjoint $F$ that takes an object $\mathcal{C} \in \mathrm{CAlg}(\mathrm{Pr}^L)$ to the constant diagram $\mathcal{C}$.

**Lemma 6.2.1.** If $X$ is a finite type $\mathbb{C}$-scheme, then the natural functor

$$p_1^* : \mathrm{D}_H(X, \mathbb{Z}) \to \mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Z})$$

induced by the first projection $p_1: X \times_{\mathrm{Spec}(\mathbb{R})} \mathrm{Spec}(\mathbb{C}) \to X$ identifies naturally the left hand side to the invariants of the right hand side. More precisely there is a canonical isomorphism between the functor $p_1^*$ and the counit

$$\eta: \mathrm{FG}(\mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Z})) \to \mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Z}).$$

*Proof.* The proof is entirely analogous to the proof of Proposition 1.4.2 in the case of a finite Galois extension: one remarks that $X_{\mathbb{C}} \simeq X \times_{\mathbb{R}} X$, and that the projection map $X_{\mathbb{C}} \to X$ is a étale cover, so that the Bousfield-Kan formula for the limit of the $\mathbb{Z}/2\mathbb{Z}$-invariants is exactly the limit for étale descent. $\square$

In particular the following definition does not introduces clash of notations:

**Definition 6.2.2.** The functoriality of the presentable $\infty$-category of mixed Hodge modules on finite type $\mathbb{R}$-schemes is the composition

$$\mathrm{D}_H(-, \mathbb{Z}) : (\mathrm{Sch}_{\mathbb{R}}^{\mathrm{ft}})^{\mathrm{op}} \to \mathrm{Fun}(\mathrm{BZ}/2\mathbb{Z}, \mathrm{CAlg}(\mathrm{Pr}^L)) \xrightarrow{(-)^{\mathbb{h}\mathbb{Z}/2\mathbb{Z}}} \mathrm{CAlg}(\mathrm{Pr}^L).$$

By the previous lemma, the restriction of this functor to finite type $\mathbb{C}$-schemes corresponds to ind-mixed Hodge modules as constructed by M. Saito. We may denote by $\mathrm{D}_H(-, \mathbb{Q}) := \mathrm{D}_H(-, \mathbb{Z}) \otimes \mathrm{Mod}_{\mathbb{Q}}$.
**Proposition 6.2.3.** Let $X$ be a finite type $\mathbb{R}$-scheme. Then $\mathrm{D}_H(X, \mathbb{Q})$ is compactly generated and its full subcategory of compact objects is canonically isomorphic to $\mathrm{D}^b(\mathrm{MHM}(X, \mathbb{Q}))$ where $\mathrm{MHM}(X, \mathbb{Q})$ consists of the $\mathbb{Z}/2\mathbb{Z}$-invariants of $\mathrm{MHM}(X, \mathbb{Q})$.

*Proof.* Because $\mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Q})$ has a t-structure that restricts to compact objects, it is true that $\mathrm{D}_H(X, \mathbb{Q})$ has a t-structure, that restricts to the invariants $\mathcal{D}(X)$ of the compact objects, seen as a full subcategory of $\mathrm{D}_H(X, \mathbb{Q})$. Now if $D$ belongs to $\mathcal{D}(X)$, and $M = \mathrm{colim}_i M_i$ is a filtered colimit of objects of $\mathrm{D}_H(X, \mathbb{Q})$, we have that

$$ \begin{align*} \mathrm{map}_{\mathrm{D}_H(X, \mathbb{Q})}(D, \mathrm{colim} \, M_i) &\simeq \mathrm{map}_{\mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Q})}(D|_{X_{\mathbb{C}}}, \mathrm{colim}_i (M_i)|_{X_{\mathbb{C}}})^{\mathbb{Z}/2\mathbb{Z}} \\ &\simeq \mathrm{R}\Gamma(\mathbb{R}_{\acute{e}t}, \mathrm{colim} \, \mathrm{map}_{\mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Q})}(D|_{X_{\mathbb{C}}}, (M_i)|_{X_{\mathbb{C}}})) \\ &\stackrel{(*)}{\simeq} \mathrm{colim}_i \mathrm{R}\Gamma(\mathbb{R}_{\acute{e}t}, \mathrm{map}_{\mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Q})}(D|_{X_{\mathbb{C}}}, (M_i)|_{X_{\mathbb{C}}})) \\ &\simeq \mathrm{colim}_i \mathrm{map}_{\mathrm{D}_H(X, \mathbb{Q})}(D, M_i) \end{align*} $$

where (*) follows from [26, Lemma 1.1.10]. Thus any object of $\mathcal{D}(X)$ is compact. Moreover, because the functor

$$ \mathrm{D}_H(X, \mathbb{Q}) \to \mathrm{D}_H(X_{\mathbb{C}}, \mathbb{Q}) $$

is conservative, the $\infty$-category $\mathcal{D}(X)$ generates $\mathrm{D}_H(X, \mathbb{Q})$ under colimits. Thus, the canonical functor $\mathrm{Ind} \, \mathcal{D}(X) \to \mathrm{D}_H(X, \mathbb{Z})$ is an equivalence.

Because $p_1^*$ is perverse t-exact, the functor $\mathrm{MHM}(-, \mathbb{Q})$ has descent along $p_1$. Moreover, because $\mathrm{D}^b(\mathrm{MHM}(-, \mathbb{Q})) \to \mathcal{D}(-)$ is conservative and commutes with $p_1^*$ as well as $(p_1)_*$ (which is also perverse t-exact), the same proof as **Proposition 2.1.8** gives that $\mathrm{D}^b(\mathrm{MHM}(-, \mathbb{Q}))$ has descent for the morphism $p_1$, but this implies that the functor $\mathrm{D}^b(\mathrm{MHM}(X, \mathbb{Q})) \to \mathcal{D}(X)$ is an equivalence, finishing the proof. $\square$

**Corollary 6.2.4.** The functor $\mathrm{D}_H(-, \mathbb{Z})$ is an étale hypersheaf on finite type $\mathbb{R}$-schemes. For any finite type $\mathbb{R}$-scheme $X$, the $\infty$-category $\mathrm{D}_H(X, \mathbb{Z})$ affords a t-structure that restricts to the subcategory $\mathrm{D}^b_{H,c}(X, \mathbb{Z})$ of constructible objects, which consists of those objects whose underlying ind-constructible sheaf is constructible. We denote by $\mathrm{MHM}(X, \mathbb{Z})$ the heart of the latter $\infty$-category.

*Remark 6.2.5.* Let $X$ be a smooth $\mathbb{R}$-scheme. Let $M \in \mathrm{MHM}(X, \mathbb{Z})$, then $M|_{X_{\mathbb{C}}}$ admits an underlying $\mathcal{D}$-module on $X_{\mathbb{C}}$ with an action of $\mathbb{Z}/2\mathbb{Z}$, so that in fact we have a functor

$$ \mathrm{MHM}(X, \mathbb{Z}) \to \mathcal{D}\text{-mod}(X) $$

by Galois descent of $\mathcal{D}$-modules.

**Proposition 6.2.6.** The heart of $\mathrm{D}^b_{H,c}(\mathrm{Spec}(\mathbb{R}), \mathbb{Z})$ is the category $\mathrm{MHS}^p_{\mathbb{Z}}(\mathbb{R})$ of polar-isable mixed Hodge modules over $\mathbb{R}$ with integral coefficients. Moreover, the canonical functor

$$ \mathrm{D}^b(\mathrm{MHS}^p_{\mathbb{Z}}(\mathbb{R})) \to \mathrm{D}^b_{H,c}(\mathrm{Spec}(\mathbb{R}), \mathbb{Z}) $$
is an equivalence.

*Proof.* We can identify the heart with the abelian category of polarisable mixed Hodge structures over $\mathbb{C}$ with integral coefficients, together with an action of $\mathbb{Z}/2\mathbb{Z}$, and this is exactly $\text{MHS}_{\mathbb{Z}}^p(\mathbb{R})$. Thus we have a functor

$$D^b(\text{MHS}_{\mathbb{Z}}^p(\mathbb{R})) \to \text{D}_{\text{H},c}^b(\text{Spec}(\mathbb{R}), \mathbb{Z}).$$

As in the proof of Proposition 6.2.3 the left hand side has descent along the morphism $\mathbb{R} \to \mathbb{C}$, so that the result follows from Proposition 6.1.2. $\square$

**Corollary 6.2.7.** Let $f: X \to \text{Spec} \mathbb{R}$ be a finite type map, then we recover the absolute Hodge cohomology groups of Beilinson ([15]) as

$$\text{R}\Gamma_{\mathcal{H}}(X, \mathbb{Z}(p)) \simeq \text{map}_{\text{D}_{\text{H}}(\text{Spec}(\mathbb{C}), \mathbb{Z})}(\mathbb{Z}, f_*\mathbb{Z}(p)) \simeq \text{map}_{\text{D}_{\text{H}}(X, \mathbb{Z})}(\mathbb{Z}, \mathbb{Z}(p)).$$

*Proof.* Indeed Saito proved the rational result for $X_{\mathbb{C}}$ in [68, Section 1.15], thus it suffices to check that Beilinson definition is indeed the $\mathbb{Z}/2\mathbb{Z}$-invariants of a pullback of the rational cohomology and the $\mathbb{Z}$-structure, but this is the case by [15, Section 4.1, Theorem 3.4, Section 7]. $\square$

We finish with a proposition computing the torsion of $\text{D}_{\text{H}}(X, \mathbb{Z})$.

**Proposition 6.2.8.** Let $X$ be a finite type $\mathbb{R}$-scheme. Then the functor

$$D(X_{\text{ét}}, \mathbb{Z}) \to \text{D}_{\text{H}}(X, \mathbb{Z})$$

obtained by taking Galois invariant of the "Artin object functor"

$$D((X_{\mathbb{C}})_{\text{ét}}, \mathbb{Z}) \to \text{DM}^{\text{ét}}(X_{\mathbb{C}}, \mathbb{Z}) \to \text{D}_{\text{H}}(X_{\mathbb{C}}, \mathbb{Z})$$

induces an equivalence

$$D(X_{\text{ét}}, \mathbb{Z}/n\mathbb{Z}) \simeq \text{D}_{\text{H}}(X, \mathbb{Z}) \otimes_{\mathbb{Z}} \text{Mod}_{\mathbb{Z}/n\mathbb{Z}}.$$

*Proof.* Because everything is obtained by taking Galois invariants, it suffices to prove the claim when $X$ is a finite type $\mathbb{C}$-scheme. In that case, the pullback definition provides us with a proof. $\square$

## 6.3 Motivic Hodge modules

As for Nori motives, this realisation functor factors through modules over the algebra $\mathcal{H}$ representing Hodge cohomology. Using the same method as [70, Section 4.3], we proved in [72, Section 2] that the canonical functor

$$\text{Mod}_{\mathcal{H} \otimes_{\mathbb{Z}} \mathbb{Q}}(\text{DM}^{\text{ét}}(-, \mathbb{Q})) \to \text{D}_{\text{H}}(-)$$
is fully faithful over finite type $\mathbb{C}$-schemes, and that the t-structure on $D_H(-)$ induces a t-structure on the image that we describe following Ayoub's ideas in [10, Theorem 1.98]: it is the localising subcategory generated by the $f_*\mathbb{Q}(n)$ with $f$ a proper morphism and $n$ an integer, so that we may call those mixed Hodge modules of geometric origin (this means that it is the smallest full subcategory stable under all colimits and finite limits that contains the $f_*\mathbb{Q}(n)$). This category of modules was first introduced by Drew in [32].

There is also an enriched version of this result, where instead of obtaining the localising subcategory generated by the $f_*\mathbb{Q}(n)$, we obtain the localising subcategory generated by the $f_*H(n)$ for any mixed Hodge structure $H$ (that we see as a constant mixed Hodge module), that we may call mixed Hodge modules of Hodge origin. Because this theory of enriched modules is more subtle that the geometric origin one, we will explain how to construct an integral version of the enriched category, and leave the case of geometric origin to the careful reader.

**Construction 6.3.1.** The functor $D_H(-, \mathbb{Z})$ naturally takes values in $D_H(\text{Spec}(\mathbb{R}), \mathbb{Z})$-linear presentable $\infty$-categories. Thus, the functor $\rho_H$ induces a $D_H(\text{Spec}(\mathbb{R}), \mathbb{Z})$-linear functor

$$ \rho_H: \text{DM}^{\text{ét}}(-, \mathbb{Z}) \otimes_{\text{Mod}_{\mathbb{Z}}} D_H(\text{Spec}(\mathbb{R}), \mathbb{Z}) \to D_H(-, \mathbb{Z}). $$

By the good properties of Lurie tensor product, it turns out that the left hand side is

$$ \text{DM}^{\text{ét}}(-, \mathbb{Z}) := \text{DM}^{\text{ét}}(-, D_H(\text{Spec}(\mathbb{R}), \mathbb{Z})) $$

the presentable $\infty$-category of étale motives with coefficients in mixed Hodge structures which has the same definition as étale motives with coefficient $\Lambda$, except that one replaces $\text{Mod}_{\Lambda}$ by $D_H(\text{Spec}(\mathbb{R}), \mathbb{Z})$. The right adjoint $\rho_*^H$ of $\rho_H$ is lax monoidal thus we get an object $\mathcal{H}_X = \rho_*^H(\mathbb{Z})$ of $\text{CAlg}(\text{DM}^{\text{ét}}(X, \mathbb{Z}))$ for all finite type $\mathbb{R}$-schemes $X$. There is a natural map

$$ \pi_X^* \mathcal{H}_{\text{Spec}(\mathbb{C})} \to \mathcal{H}_X $$

with $\pi_X: X \to \text{Spec}(\mathbb{R})$ the structural morphism, it will be shown to be an isomorphism. The presentable $\infty$-categories of integral motivic Hodge modules is the category

$$ \mathbf{DH}(X, \mathbb{Z}) := \text{Mod}_{\pi_X^*\mathcal{H}}(\text{DM}^{\text{ét}}(X, \mathbb{Z})). $$

As for Nori motives, it is also the base change of $\text{DM}^{\text{ét}}(-, \mathbb{Z})$ from $\text{DM}^{\text{ét}}(\text{Spec}(\mathbb{R}), \mathbb{Z})$-linear categories to $\mathbf{DH}(\text{Spec}(\mathbb{R}), \mathbb{Z})$-linear categories:

$$ \mathbf{DH}(-, \mathbb{Z}) \simeq \text{DM}^{\text{ét}}(-, \mathbb{Z}) \otimes_{\text{DM}^{\text{ét}}(\text{Spec}(\mathbb{R}), \mathbb{Z})} \mathbf{DH}(\text{Spec}(\mathbb{R}), \mathbb{Z}). $$

Thus as in Theorem 3.3.1, the functor $\mathbf{DH}(-, \mathbb{Z})$ affords all the six operations in such a way that the functor from $\text{DM}^{\text{ét}}(-, \mathbb{Z})$ and the canonical functor to $D_H(-, \mathbb{Z})$ commute with them, the composition of the two being $\rho_H$.
**Lemma 6.3.2.** Let $\iota$: Spec($\mathbb{C}$) $\to$ Spec($\mathbb{R}$) be the spectrum of the inclusion of $\mathbb{R}$ into $\mathbb{C}$. Then the canonical map $\iota^*\mathcal{H}_{\text{Spec}(\mathbb{R})} \to \mathcal{H}_{\text{Spec}(\mathbb{C})}$ is an equivalence.

*Proof.* Because $\iota$ is finite étale, the functor $\iota^* = i^!$, being a right adjoint, commutes with $\rho_H$. Thus $\iota^*\mathcal{H}_{\text{Spec}(\mathbb{R})} \simeq \rho_H(\iota^*1) \simeq \mathcal{H}_{\text{Spec}(\mathbb{C})}$. $\square$

**Theorem 6.3.3.** The canonical functor

$$ \underline{\rho_H}: \mathrm{DH}(-, \mathbb{Z}) \to \mathrm{D}_H(-, \mathbb{Z}) $$

factoring $\rho_H$ is fully faithful and its image is stable under truncations.

*Proof.* By étale descent and Lemma 6.3.2, it suffices to prove the claim over finite type $\mathbb{C}$-schemes. Because $\mathrm{D}_H(-, \mathbb{Z})$ is compactly generated, the right adjoint $\rho_*^H$ commutes with rationalisation, so that the algebra $\pi_X^*\mathcal{H}_{\text{Spec}(\mathbb{C})}$, tensored with $\mathbb{Q}$ is the algebra considered by Drew in [32] and the second author in [72]. This implies that $\underline{\rho_H} \otimes \mathrm{Mod}_{\mathbb{Q}}$ is fully faithful with image stable under truncations by the main theorem of [72, Section 2] (see also [10, Theorem 1.98]). When tensoring the square of Definition 6.1.1 by $\mathrm{Mod}_{\mathbb{Z}/n\mathbb{Z}}$, we obtain an equivalence

$$ \mathrm{D}_H(-, \mathbb{Z}) \otimes_{\mathrm{Mod}_\mathbb{Z}} \mathrm{Mod}_{\mathbb{Z}/n\mathbb{Z}} \xrightarrow{\sim} \mathrm{Ind}(\mathrm{D}_{\mathrm{cons}}(X^{\mathrm{an}}, \mathbb{Z}) \otimes_{\mathrm{Perf}_\mathbb{Z}} \mathrm{Perf}_{\mathbb{Z}/n\mathbb{Z}}), $$

but the right hand side coincides with the derived category of étale sheaves of $\mathbb{Z}/n\mathbb{Z}$-modules on $X$, thanks to Artin comparison theorem. By rigidity, this implies that the natural map

$$ \mathbb{Z}/n\mathbb{Z} \to \pi_X^* \mathcal{H}_{\text{Spec}(\mathbb{C})}/n $$

is an equivalence, and then that $\underline{\rho_H} \otimes_{\mathrm{Mod}_\mathbb{Z}} \mathrm{Mod}_{\mathbb{Z}/n\mathbb{Z}}$ is an equivalence. This finishes the proof of the fully faithfulness.

Now let $M$ be in the image of $\underline{\rho_H}$. Because the t-structure is compatible with filtered colimits (this is the case for the t-structure of all categories in the pullback defining $\mathrm{D}_H$), we have an exact triangle

$$ \tau^{\le 0} M \to \tau^{\le 0}(M \otimes_{\mathbb{Z}} \mathbb{Q}) \to \mathrm{colim}_n \tau^{\le 0}(M) \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}. $$

The middle term is in the image because we proved the theorem with rational coefficients in [72]. Thus it suffices to show that for all $n \in \mathbb{N}^*$, the object $\tau^{\le 0}(M) \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$ is in the image. This is the case because $\underline{\rho_H} \otimes_{\mathrm{Mod}_\mathbb{Z}} \mathrm{Mod}_{\mathbb{Z}/n\mathbb{Z}}$ is an equivalence. This proves that the image is stable under truncations, finishing the proof. $\square$

**Remark 6.3.4.** The fact that $\underline{\rho_H}$ is fully faithful implies readily that the map $\pi_X^*\mathcal{H}_{\text{Spec}(\mathbb{R})} \to \mathcal{H}_X$ is an equivalence for all $X$.

**Remark 6.3.5.** As explained at the beginning of this subsection, the above theorem also works for the non enriched version of the Hodge realisation of étale motives. We will call those Hodge modules of geometric origin, and denote them by $\mathrm{DH}(-, \mathbb{Z})$.
Remark 6.3.6. For X a finite type $\mathbb{R}$-scheme, the image of $\rho_{\mathbb{H}}$ can be described as the localising subcategory spanned by objects of the form $p_*H(n)$ for $p: Y \to X$ proper, and $H$ a polarisable integral mixed Hodge structure over $\mathbb{R}$ seen as a constant integral mixed Hodge module over $Y$.

## 6.4 Abelian categories of integral Hodge modules

By definition and by Theorem 6.3.3 for each finite type $\mathbb{R}$-scheme $X$ the three categories $D_{\mathbb{H}}(X, \mathbb{Z})$, $DH(X, \mathbb{Z})$ and $DH_c(X, \mathbb{Z})$ admit a perverse t-structure, that restrict to constructible object $D_{\mathbb{H},c}^b(X, \mathbb{Z})$, $DH_c(X, \mathbb{Z})$ and $DH_c(X, \mathbb{Z})$. As above, the two categories $DH_c(X, \mathbb{Z})$ and $DH_c(X, \mathbb{Z})$ are the thick categories spanned by the generators of Remark 6.3.6.

We will denote by $\mathrm{MHM}(X, \mathbb{Z})$, $\mathrm{MHM}_{\mathrm{geo}}(X, \mathbb{Z})$ and $\mathrm{MHM}_{\mathrm{Hdg}}(X, \mathbb{Z})$ the three hearts of the constructible objects.

In more concrete terms, a mixed Hodge module with integral coefficients is the data of three objects $(M, \mathcal{F}, S, \varphi)$ where $M \in \mathrm{MHM}(X_{\mathbb{C}}, \mathbb{Q})$ is a mixed Hodge module as constructed by M. Saito, $\mathcal{F} \in \mathrm{Perv}(X_{\mathbb{C}}^{\mathrm{an}}, \mathbb{Z})$ is a perverse sheaf, $S \in \mathrm{End}(M, \mathcal{F})$ is an involution and $\varphi: \mathcal{F} \otimes_{\mathbb{Z}} \mathbb{Q} \simeq \mathrm{rat}(M)$ is an isomorphism of perverse sheaves that commutes with $S$.

Because we have the 6 operations, we can also construct ordinary t-structures on the constructible objects by gluing along stratifications, doing the same construction as in the proof of Proposition 5.1.1 in reverse. We will denote their hearts by $\mathrm{MHM}^{\mathrm{ord}}(X, \mathbb{Z})$, $\mathrm{MHM}_{\mathrm{geo}}^{\mathrm{ord}}(X, \mathbb{Z})$ and $\mathrm{MHM}_{\mathrm{Hdg}}^{\mathrm{ord}}(X, \mathbb{Z})$ respectively. Using the same proof as Theorem 4.3.2, we obtain:

**Proposition 6.4.1.** *The natural functors*

$$D^b(\mathrm{MHM}^{\mathrm{ord}}(X, \mathbb{Z})) \to D_{\mathbb{H},c}^b(X, \mathbb{Z}),$$

$$D^b(\mathrm{MHM}_{\mathrm{geo}}^{\mathrm{ord}}(X, \mathbb{Z})) \to \mathrm{DH}_c(X, \mathbb{Z})$$

and

$$D^b(\mathrm{MHM}_{\mathrm{Hdg}}^{\mathrm{ord}}(X, \mathbb{Z})) \to \mathbf{DH}_c(X, \mathbb{Z})$$

are equivalences.

**Corollary 6.4.2.** *Over finite type $\mathbb{C}$-schemes, the natural functors*

$$D(\mathrm{Ind} \mathrm{MHM}^{\mathrm{ord}}(X, \mathbb{Z})) \to D_{\mathbb{H}}(X, \mathbb{Z}),$$

$$D(\mathrm{Ind} \mathrm{MHM}_{\mathrm{geo}}^{\mathrm{ord}}(X, \mathbb{Z})) \to \mathrm{DH}(X, \mathbb{Z})$$

and

$$D(\mathrm{Ind} \mathrm{MHM}_{\mathrm{Hdg}}^{\mathrm{ord}}(X, \mathbb{Z})) \to \mathbf{DH}(X, \mathbb{Z})$$

are equivalences.
**Proof.** Indeed the hearts of the compact objects have finite cohomological dimension (by the same argument as for the perverse hearts), so that the corollary follows by taking indization of the theorem. □

**Remark 6.4.3.** The above result is false for ordinary motives over $\mathbb{C}$-schemes. Indeed, assuming that the functor $\mathrm{DM}^{\mathrm{ét}}(X, \Lambda) \to \mathrm{DN}(X, \Lambda)$ is an equivalence, Ayoub’s counterexample [9, Lemma 2.4] would imply that the functor $\mathrm{DN}(X, \Lambda) \to \mathrm{D}(X^{\mathrm{an}}, \Lambda)$ is not conservative, but the functor $\mathrm{D}(\mathrm{Ind} \mathcal{M}_{\mathrm{ord}}(X, \Lambda)) \to \mathrm{D}(X^{\mathrm{an}}, \Lambda)$ is conservative. This is the only obstruction, see Proposition 3.5.3.

**Remark 6.4.4.** As for Nori motives, we suspect that the functors

$$ \mathrm{D}^b(\mathrm{MHM}_{\mathrm{geo}}(X, \mathbb{Z})) \to \mathrm{DH}_c(X, \mathbb{Z}) $$

and

$$ \mathrm{D}^b(\mathrm{MHM}_{\mathrm{Hdg}}(X, \mathbb{Z})) \to \mathbf{DH}_c(X, \mathbb{Z}) $$

are equivalences, because in some sense, objects of geometric origin have a tendency to be flat, or at least, resolved by flat objects.

We finish the paper by linking our notion of mixed Hodge modules to the classical notion of variation of mixed Hodge structure. We say that an integral mixed Hodge module $M \in \mathrm{MHM}(X, \mathbb{Z})$ is lisse if its underlying perverse sheaf $\mathrm{int}(M)$ is locally constant. We denote by $\mathrm{MHM}_{\mathrm{lis}}(X, \mathbb{Z})$ the category of lisse Mixed Hodge modules on a finite-type $\mathbb{C}$-scheme. Note that the pullback defining $\mathrm{D}_{\mathrm{H}}(X, \mathbb{Z})$ implies that the perverse t-structure induces a t-structure on the subcategory of dualisable objects of $\mathrm{D}_{\mathrm{H}}(X, \mathbb{Z})$; the heart of this t-structure is precisely $\mathrm{MHM}_{\mathrm{lis}}(X, \mathbb{Z})$.

**Proposition 6.4.5.** Let $X$ be a finite-type $\mathbb{C}$-scheme and let $\mathrm{VMHS}_{\mathrm{ad}}(X, \mathbb{Z})$ be the category of admissible variations of mixed Hodge structures, where a variations of mixed Hodge structure in the sense of [31, 1.8.14] is admissible if its rationalisation is admissible in the sense of [67, 2.1]. Then, we have an equivalence of categories

$$ \mathrm{MHM}_{\mathrm{lis}}(X, \mathbb{Z}) \to \mathrm{VMHS}_{\mathrm{ad}}(X, \mathbb{Z}). $$

**Proof.** We have a pullback square:

<figure><img src="image_20.png" /></figure>

where $\mathrm{Loc}(X, \Lambda)$ is the category of locally constant sheaves of $\Lambda$-modules with values in a finitely generated $\Lambda$-modules.
By [67, Theorem 2.2], we have an equivalence of categories

$$ \mathrm{MHM}_{\mathrm{lis}}(X, \mathbb{Q}) \to \mathrm{VMHS}_{\mathrm{ad}}(X, \mathbb{Q}). $$

But by definition, a VMHS with integral coefficients is exactly a VMHS with rational coefficients admitting an integral lattice. Hence, we also have a pullback diagram:

$$ \begin{array}{ccc} \mathrm{VMHS}_{\mathrm{ad}}(X, \mathbb{Z}) & \to & \mathrm{Loc}(X^{\mathrm{an}}, \mathbb{Z}) \\ \downarrow & & \downarrow \\ \mathrm{VMHS}_{\mathrm{ad}}(X, \mathbb{Q}) & \to & \mathrm{Loc}(X^{\mathrm{an}}, \mathbb{Q}), \end{array} $$

and the result follows.

<figure><img src="image_21.png" /></figure>

Remark 6.4.6. One could go further and define the usual operations of mixed Hodge modules, but with integral coefficients. For example if $f: X \to \mathbb{A}_{\mathbb{C}}^1$ is a map of finite type $\mathbb{C}$-schemes, one can define nearby cycles $\Psi_f$ associated to $f$ as the functor

$$ \Psi_f: \mathrm{D}_{\mathrm{H},c}^b(X_\eta, \mathbb{Z}) \to \mathrm{D}_{\mathrm{H},c}^b(X_s, \mathbb{Z}), $$

with $X_\eta = f^{-1}(\mathbb{G}_m)$ and $X_s = f^{-1}(\{0\})$, defined by the pullback property:

<figure><img src="image_22.png" /></figure>

so that all known properties about $\Psi_f$ on $\mathrm{D}_c^b(-, \mathbb{Z})$ remain true for the one for $\mathrm{D}_{\mathrm{H},c}^b(-, \mathbb{Z})$, for example t-exactness or compatibility with duality. Similarly, one can define vanishing cycles, monodromic mixed Hodge modules and have a Thom-Sebastiani formula by pullback. Taking $\mathbb{Z}/2\mathbb{Z}$-invariants, we also have a version over finite type $\mathbb{R}$-schemes.

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Raphaël Ruimy raphael.ruimy@unimi.it

Dipartimento di Matematica "F. Enriques" - Universita degli Studi di Milano,
via Cesare Saldini 50, I-20133 Milano, Italy

Swann Tubach swann.tubach@ens-lyon.fr

E.N.S Lyon, UMPA,
46 Allée d'Italie, 69364 Lyon Cedex 07, France