\[\DN(X,\Lambda)^\heart \to \left(\DN(X_{\bar{k}},\Lambda)^\heart\right)^{\mathrm{hGal}(\bar{k}/k)}.\]

A classical question when dealing with motivic t-structure is to know whether a reasonable t-structure on the big category of motives restricts to the small category of constructible motives. This is one of the purpose of the next section.

§.§ The ordinary t-structure on geometric Nori motives

In this section, we show how the constructions on the big categories induce a t-structure on geometric Nori motives and extend the definition to quite general $\Q$-schemes.

Let $\Lambda$ be a commutative ring. Let $k$ be an algebraically closed subfield of $\C$ and let $X$ be a finite type $k$-scheme.
Then, we have a cartesian square induced by the square of <Ref>

\[\begin{tikzcd}
\DNgm(X,\Lambda) \arrow[r] \arrow[d]
& \dgman(X^\mathrm{an},\Lambda) \arrow[d] \\
\DNgm(X,\Lambda\otimes\Q) \arrow[r]
& \dgman(X^\mathrm{an},\Lambda\otimes\Q)
\end{tikzcd}\]

Using <cit.>, it suffices to show that we have a pullback square after passing to the $ \In$-categories. When $\Lambda=\Z$, this is <Ref> because <Ref> ensures that $\DN(X,\Z)$ and $\DN(X,\Q)$ are compactly generated with compact objects $\DNgm(X,\Z)$ and $\DNgm(X,\Q)$ respectively. As tensoring with $\Mod_\Lambda$ preserves limits because it is compactly generated hence a dualizable object of $\mathrm{Pr}^{L,\otimes}$, the general case follows by taking $\Lambda$-modules in <Ref>.

Let $\Lambda$ be a regular ring. Let $k$ be a subfield of $\C$ and let $X$ be a finite type $k$-scheme. The ordinary t-structure on $\DN(X,\Lambda)$ defined in <Ref> induces a t-structure on $\DNgm(X,\Lambda)$ such that the refined Betti realisation functor
\[R_\mathrm{B}\colon \DNgm(X,\Lambda) \to \dgman(X^\mathrm{an},\Lambda)\]
is conservative and t-exact. In particular, letting $\mcal_\mathrm{ord}(X,\Lambda)$ be the heart of this t-structure, we get a faithful exact functor
\[\mcal_\mathrm{ord}(X,\Lambda)\to \Sh_\mathrm{gm}(X^\mathrm{an},\Lambda).\]

When $k$ is algebraically closed, the result follows from the pullback square of <Ref> above, from <Ref> and from the same statement with rational coefficients. Note that here, the assumption that $\Lambda$ is regular is needed to ensure that $\dgman(X^\mathrm{an},\Lambda)$ is endowed with a t-structure.

In the general case, let $f\colon X_{\bar{k}} \to X$ be the canonical map. Using the regularity of $\Lambda$ again, <Ref> ensures that $\DNgm(-,\Lambda)$ is an étale sheaf; therefore a motive $M$ in $\DN(X,\Z)$ is geometric if and only if its image through the functor $f^*$ is geometric. As the functor $f^*$ is also t-exact by definition of the t-structure, this ensures that $\tau^{\leqslant 0}(M)$ is geometric when $M$ is. Hence, the subcategory $\DNgm(X,\Z)$ of $\DN(X,\Z)$ is stable under truncation and the result follows.

Any t-exactness property which is valid for the ordinary t-structure on constructible sheaves is valid for the ordinary t-structure on Nori motives. In particular, pullback functors are t-exact and so is the tensor product with respect to both variable.

If $X$ is a qcqs finite-dimensional $\Q$-scheme, we can write it as a filtered colimit of finite type $\Q$-schemes. Continuity (<Ref>) and <Ref> yield a t-structure on $\DMgm(X,\Lambda)$ because the functors of type $f^*$ are t-exact. This t-structure doesn't depend on the way $X$ is written as a colimit, again because the $f^*$ are t-exact. Hence, the following definition makes sense:

Let $\Lambda$ be a regular ring. Let $X$ be a qcqs finite-dimensional $\Q$-scheme. The ordinary t-structure on $\DNgm(X,\Lambda)$ is the only t-structure which coincides with the already defined ordinary t-structure over schemes of finite type over a subfield of $\C$ and such that the functors of type $f^*$ are t-exact for any morphism $f$.
We denote its heart by $\mcal_\mathrm{ord}(X,\Lambda)$.

Assume that $\Lambda$ is of localisation of a ring integral and flat over $\Z$. Then the functor

\[\DNgm(X,\Lambda)\xrightarrow{R_\ell} \dconspro(X_\proet,\Lambda_\ell) \subseteq \D(X_\proet,\Lambda_\ell)\]

is t-exact.

By definition of the t-structure, since $R_\ell$ is compatible with pullbacks, we can assume that $X$ is a finite type $\Q$-scheme. If $x$ is a point of $X$, denote by $i_x$ its inclusion in $X$. By <cit.>, the family of the $i_x^*$ is conservative. As it is also t-exact, we can reduce to the case where $X=\Spec(k)$ with $k$ a subfield of $\C$. The pullback $(\Spec(\C)\to \Spec(k))^*$ is conservative and t-exact because of <Ref>. Hence, we can assume that $X=\Spec(\C)$. In that case, the commutative diagram given in the proof of <cit.> yields a commutative diagram:

\[\begin{tikzcd}
\DNgm(X,\Lambda) \ar[r,"R_\mathrm{B}"] \ar[rd,swap,"R_\ell"] &\Perf_\Lambda \ar[d,"-\otimes_\Lambda \Lambda_\ell"] \\
& \Perf_{\Lambda_\ell}
\end{tikzcd}\]

As the functor $-\otimes_\Lambda \Lambda_\ell$ is t-exact, this implies the result.

As the $\ell$-adic realisation is also conservative by <Ref>, any t-exactness property of the ordinary t-structure on $\ell$-adic sheaves transfers to the ordinary t-structure on Nori motives.

Let $k$ be a subfield of $\C$ and let $X$ be a $k$-variety.
Arapura considered in [5] an abelian category of motivic sheaves modelled on constructible sheaves. By construction, his category
$\mcal_{\mathrm{Arapura}}(X,\Z)$ has an unique property, so that there exists a faithful exact functor $\mcal_{\mathrm{Arapura}}(X,\Z)\to\mcal_\mathrm{ord}(X,\Z)$ compatible with the Betti realisations. We do not know much about this functor, except that it is an equivalence for $X=\Spec(k)$.

§.§ The derived category of ordinary Nori motives

We now prove that $\DNgm$ is the derived category of ordinary Nori motives following Nori [54]. To that end, we first need:

The functor $\D^b(\mcal_\mathrm{ord}(-,\Lambda))$ is a finitary étale hypersheaf on qcqs finite-dimensional  schemes.

Because for $f$ an étale map, the functors $f^*$ and $f_!$ are t-exact, they exist on $\D^b(\mcal_\mathrm{ord}(-,\Lambda))$ as an adjunction, compatible with the realisation functor
\[\D^b(\mcal_\mathrm{ord}(-,\Lambda))\to \DNgm(-,\Lambda)\]
which is conservative. Thus, because $\DNgm(-,\Lambda)$ is an étale hypersheaf, the functor $\mcal_\mathrm{ord}(-,\Lambda)$ is an étale sheaf: it is a sub presheaf of $\DNgm(-,\Lambda)$ and the cohomological amplitude with respect to the ordinary t-structure can be tested étale-locally. Hence, the same proof as <Ref> gives the descent result. The finitarity result follows from <Ref> and the finitarity property of $\DNgm$.

Let $\Lambda$ be a regular ring which is flat over $\Z$. Let $X$ be a qcqs finite-dimensional $\Q$-scheme. Then, the canonical functor

\[\D^b(\mcal_\mathrm{ord}(X,\Lambda))\to \DNgm(X,\Lambda)\]
is an equivalence.

By <Ref>, continuity and Zariski hyperdescent for $\DNgm(-,\Lambda)$, we can assume that $X$ is of finite type over $\Q$ and separated.
Note that by <cit.>, to prove the theorem, it is sufficient to prove that for any pair of Nori motives $M,N$ in $\mcal_\mathrm{ord}(X,\Lambda)$ and any integer $i>0$, the map

\[\mathrm{Ext}^i_{\mcal_\mathrm{ord}(X,\Lambda)}(M,N)=\Hom_{\D^b(\mcal_\mathrm{ord}(X,\Lambda))}(M,N[i])\to \Hom_{\DNgm(X,\Lambda)}(M,N[i])\]
is an isomorphism, which, given the characterisation of the $\mathrm{Ext}^i$, is equivalent to the functor
\[\mathrm{E}_M^i:=\Hom_{\DNgm(X,\Lambda)}(M,(-)[i])\colon \mcal_{\mathrm{ord}}(X,\Lambda)\to\mathrm{Ab}\]
being effaceable for each $i>0$ and each $M$ in $\mcal_\mathrm{ord}(X,\Lambda)$. More explicitly, this means that for every Nori motive $N$ in $\mcal_\mathrm{ord}(X,\Lambda)$, there exists a monomorphism $f\colon N\to P$ whose image under our functor $\mathrm{E}^i_M$ vanishes.

The proof of this fact is then very similar to the proof the second author gave, following Nori, in the case of Nori motives with rational coefficients (see [54, 65]). Until the end of this proof, we will call admissible an Nori motive $M$ in $\mcal_{\mathrm{ord}}(X,\Lambda)$ such that for all $i>0$ the functor $\mathrm{E}_M^i$ is effaceable.

By remarking that all the proofs that lead to <cit.> work verbatim with integral coefficients (because we only use the four operations $f^*,f_*,f^!,f_!$, geometric arguments and the vanishing <cit.> works with integral coefficients because Nori proved it in this generality in <cit.>), we know that the constant object $\Lambda_X$ is admissible on $X$. To be more precise, we first prove by induction on $n$ the admissibility of the constant object on $\A^n_\Q$ (the case $n=0$ is true because we already know the theorem in that case by <Ref>, while the induction uses well chosen projection to $\A^{n-1}$ and the already cited <cit.>). Then, using Noether normalisation one can prove that the result is true for any affine scheme. Finally, because our scheme is separated, we can do an induction on the numbers of affine open subsets needed to cover $X$ and deduce that $\Lambda_X$ is indeed admissible on $X$

The next step is to prove that any torsion-free lisse sheaf $L$ on $X$ is admissible. This follows from the fact that because $L$ is dualizable, the functor $-\otimes L$ is t-exact, because $L$ is torsion-free (this can be checked after realisation and then on stalks so that this reduces to the same statement about modules) and left adjoint of a t-exact functor; thus by <cit.> the object $\Lambda\otimes L = L$ is admissible. Using <cit.> again, the functor $f_!$ for $f$ étale preserves admissibility: the functors $f^*$ and $f_!$ are t-exact because they are t-exact after taking the Betti realisation. Moreover by <cit.>, if one has a short exact sequence
\[0\to M'\to M\to M'' \to 0\]
with $M'$ and $M''$ admissible, then $M$ is admissible. Now if $L$ is a lisse motive, then we have the short exact sequence

\[0\to L_\mathrm{tors} \to L\to L_\mathrm{fr}\to 0\]
with $L_\mathrm{tors}$ the biggest torsion subobject and $L_\mathrm{fr}$ torsion-free. Note that $L_\mathrm{tors}$ and $L_\mathrm{fr}$ are also lisse because they are after taking the Betti realisation. Hence $L_\mathrm{fr}$ is admissible.

We claim that $L_\mathrm{tors}$ is also admissible. As it is torsion, it is in fact of the form $\iota(F)$ for a lisse étale sheaf on $X$: it is of the form $\iota(F)$ because of <Ref> and $F$ is lisse because the functor $\D(X_\et,\Lambda)\to\D(X^\mathrm{an},\Lambda)$ is conservative and the image of $F$ in $\DN(X,\Lambda)$ is dualizable, and thus its image in $\D(X^\mathrm{an},\Lambda)$ is also dualizable which yields the dualisability of $F$ by conservativity. Choose some $f\colon U\to X$ étale such that $f^*F = N$ is a constant $\Lambda$-module. We can choose a surjection $\Lambda^n\to N$, so that by t-exactness of $f_!$, we have a composition of surjections $ f_!\Lambda^n\twoheadrightarrow  f_!f^*F\twoheadrightarrow F$. Let $K$ be the kernel of this composed surjection. It is also a lisse and torsion-free étale sheaf: the ring $\Lambda$ is flat over $\Z$ which is a Dedeking domain, so that a subobject of a torsion-free object is again torsion-free. Thus, in $\D^b(\mcal_\mathrm{ord}(X,\Lambda))$ we have an exact triangle

\[\iota(K)\to \iota(f_!\Lambda^n)\to L_\mathrm{tors}.\]
Because $\iota(K)$ and $\iota(f_!\Lambda^n)$ are admissible, for any motive $P\in\mcal_\mathrm{ord}(X,\Lambda)$ the maps

\[\map_{D^b(\mcal(X,\Lambda))}(\iota(K),P)\to \map_{\DNgm(X,\Lambda)}(\iota(K),P)\]

\[\map_{D^b(\mcal(X,\Lambda))}(\iota(f_!\Lambda^n),P)\to \map_{\DNgm(X,\Lambda)}(\iota(f_!\Lambda^n),P)\]
are equivalences. Thus the analogous map for $L_\mathrm{tors}$ is an equivalence so that $L_\mathrm{tors}$ is also admissible. This implies that $L$ is admissible. Now, the same proof as <cit.> (which is the same proof as <cit.>) gives by dévissage that any motive is admissible, finishing the proof.

§.§ Applications: motivic Leray spectral sequence; arc descent

The existence of ordinary integral Nori motives implies that the Leray spectral sequence, with integral coefficients, is motivic; this is a generalisation of <cit.> and [4].

Let $X$ be a finite type $k$-scheme with $k$ a subfield of $\C$.
Let $\fcal$ be an object of the abelian category $\Sh_\mathrm{cons}(X^\an,\Z)$ of constructible sheaves of abelian groups over $X$. Assume that $\fcal$ is motivic, meaning that it is the image of some integral Nori motive under the Betti realisation. Let
\[X\xrightarrow{f}Y\xrightarrow{g}Z\]
be morphisms of finite type $\C$-schemes. Then the Leray spectral sequence

\[\mathrm{E}_2^{p,q}=\mathrm{R}^pg_*\mathrm{R}^qf_*\fcal\Rightarrow \mathrm{R}^{p+q}(g\circ f)_*\fcal\]
is the image of a spectral sequence in $\mcal_\mathrm{ord}(Z)$.

This is a consequence of the existence of the t-structure and the functoriality of pushforwards.

As an application of the existence of the ordinary t-structure, we prove that Nori motives afford arc-descent.

Let $\Lambda$ be a regular ring. The functor
satisfies arc-hyperdescent over finite-dimensional  qcqs $\Q$-schemes.

\[\DNgm(X,\Lambda) = \bigcup_n \DNgm(X,\Lambda)^{[-n,n]}\]
we see that to prove that $\DNgm(-,\Lambda)$ is an arc-hypersheaf, it suffices to show that $\DNgm(-,\Lambda)^{[-n,n]}$ is an arc-hypersheaf (hence even an arc-sheaf), where the truncation is taken for the ordinary t-structure.

The presheaf $\DNgm(-,\Lambda)^{[-n,n]}$ takes values in $\mathrm{Cat}_{2n+1}$ which is compactly generated by cotruncated objects (see <cit.>). We consider
\[\fcal\colon (\Sch_\Q^\mathrm{qcqs})^\op\to \mathrm{Cat}_{2n+1}\]
its left Kan extension to $(\Sch_\Q^\mathrm{qcqs})^\op$ the category of non necessarily finite-dimensional  qcqs $\Q$-schemes. By <Ref> we have that $\fcal(X)= \DNgm(X,\Lambda)^{[-n,n]}$ for $X$ of finite dimension. We may now apply the main theorem of [19]: a finitary $v$-sheaf taking values in an $\infty$-category which is compactly generated by cotruncated objects is an arc-sheaf if and only if it is excisive. By definition, $\fcal$ is finitary.

First, we want to prove that $\fcal$ is a $v$-sheaf. Given a $v$-covering $f\colon Y\to X$, we can write it as $\lim_i (f_i\colon Y_i\to X_i)$ with $X_i$ and $Y_i$ of finite type over $\Q$. Moreover, as the sheaf condition for $f$ and $\fcal$ is finite because it is a totalisation in an $\infty$-category which is compactly generated by cotruncated objects, it is compatible with the filtered colimit of the continuity. Hence, it suffices to show that $\DNgm(-,\Lambda)^{[-n,n]}$ is a $v$-sheaf when restricted to finite type $\Q$-schemes, but for Noetherian schemes the $v$-topology and the $h$-topology agree, so that this is follows from <Ref> and <Ref> (in fact in our setting there is a simpler proof thanks to the t-structure: one can do as in <cit.>, in which they reduce $v$-descent to descent along proper surjective map thanks to étale hyperdescent, and then the proper base change implies the result, with a combination of Barr-Beck-Lurie theorem and the finitary property. This proof essentially dates back to Deligne in [29]).

Next, we prove that $\fcal$ satisfies excision. Because any Milnor square can be approximated by Milnor squares of finite-dimensional  schemes (<cit.>), it suffices to prove that $\DNgm(X,\Lambda)^{[-n,n]}$ satisfies excision on finite type $\Q$-schemes. We first prove that this is true for the whole bounded category $\DNgm(X,\Lambda)$. The proof is entirely analogous to the proof of Milnor excision for $\DMgm$ in <cit.>: the proof is cut in two, the rationalised part which follows from [33], and the torsion part that follows from [19]. To prove that $\DNgm(-,\Lambda)^{[-n,n]}$ satisfies excision as well, it suffices to show that given a Milnor square

\[\begin{tikzcd}W \ar[d,"g"]\ar[r,"k"] & Y \ar[d,"f"] \\
Z \ar[r,"i"] & X
\end{tikzcd},\]
if $M$ belongs to $\DNgm(X,\Lambda)$ and is such that $f^*M$ and $i^*M$ are in cohomological degrees $[-n,n]$ then in fact $M$ is also in degrees $[-n,n]$. This is true because the map $Z\cup Y\to X$ is surjective hence the hypothesis imply that for any point $x$ of $X$, the motive $x^*M$ in $\DNgm(X,\Lambda)$ is concentrated in degrees $[-n,n]$, so that $M$ is also concentrated in degrees $[-n,n]$.

Conjecturally, the category $\DMgm(X,\Lambda)$ has an ordinary t-structure after rationalisation. In that case the functor $\DMgm(X,\Lambda)\to\DNgm(X,\Lambda)$ is an equivalence because it is an equivalence rationalised and mod $n$ for any nonzero integer $n$, so that $\DMgm(-,\Lambda)$ is a finitary arc-hypersheaf and the category $\DMgm(X,\Lambda)$ has an ordinary t-structure without needing to rationalise. In fact, it would suffice to prove that $\DMgm(\Spec(\Q),\Q)$ has a motivic t-structure.

§ PERVERSE NORI MOTIVES.

§.§ The perverse t-structure

In this subsection, we define the perverse t-structure on Nori motives. The definition is very similar to that of the perverse t-structure on constructible complexes of $\Z/\ell^n\Z$-modules given in <cit.>. We also prove that the derived category of perverse Nori motives is $\DNgm$, which is in the spirit of [15].
Let $\Lambda$ be regular ring which is a localisation of an integral and flat ring over $\Z$ and let $X$ be an excellent finite-dimensional $\Q$-scheme endowed with a dimension function $\delta$[See <cit.> for the definition of a dimension function.].
We let
\[{}^p\DNgm(X,\Lambda)^{\leqslant 0} :=\{M \in \DNgm(X,\Lambda) \mid \forall x \in X, i_x^*(M)\leqslant -\delta(x)\},\]

\[{}^p\DNgm(X,\Lambda)^{\geqslant 0} :=\{M \in \DNgm(X,\Lambda) \mid \forall x \in X, i_x^!(M)\geqslant -\delta(x)\}.\]

Then, the pair $({}^p\DNgm(X,\Lambda)^{\leqslant 0}, {}^p\DNgm(X,\Lambda)^{\geqslant 0})$ defines a t-structure on $\DNgm(X,\Lambda)$ such that the $\ell$-adic realisation functor
\[R_\ell\colon \DNgm(X,\Lambda)\to \dconspro(X_\proet,\Lambda_\ell)\]
is t-exact when the right hand side is endowed with its perverse t-structure.

The proof is the same as in <cit.> but we include a sketch for the reader's convenience.
Consider couples $(\scal,\lcal)$ where where $\scal$ is a finite stratification of $X$ by locally closed connected regular subschemes, and $\lcal$ is the data, for each stratum $Z$ of $\scal$ of a finite set $\lcal(Z)$ of dualizable Nori motives on $Z$. We denote by $\DN_{(\scal,\lcal)}(X,\Lambda)$ the full subcategory of of $\DNgm(X,\Lambda)$ made of $(\scal,\lcal)$-constructible Nori motives, that is the motives $M$ such that, for each stratum $Z$ of $\scal$ and each integer $n$, $\HH^n(M|_{Z})$ is isomorphic to an iterated extension of elements of $\lcal(Z)$.

Any such couple can be refined into a couple $(\scal,\lcal)$ such that if $Z$ is a stratum in $\scal$ and $N$ belongs to $\lcal(Z)$, letting $j\colon Z\to X$ be the inclusion, the Nori motive $j_*(N)$ is $(\scal,\lcal)$-constructible.

This assumption ensures that we can defined a t-structure on $\DN_{(\scal,\lcal)}(X,\Lambda)$ by gluing (in the sense of<cit.>)the t-structure on each stratum $Z$ by $-\delta(Z)$. If $(\scal,\lcal)$ refines $(\scal',\lcal')$, the t-structure on $\DN_{(\scal,\lcal)}(X,\Lambda)$ induces the t-structure on $\DN_{(\scal',\lcal')}(X,\Lambda)$; this is a consequence of the absolute purity property for Nori motives:

Let $\Lambda$ be a localisation of a ring regular and flat and integral over $\Z$, let $i\colon Z\to X$ be a closed immersion everywhere of codimension $c$ between regular schemes and let $M$ be a dualizable object of $\DN(Z,\Lambda)$, then the natural map
\[i^*(M)(-c)[-2c] \to i^!(M)\]
is an equivalence.

This follows from the same statement for $\ell$-adic sheaves <cit.> and from the conservativity of the $\ell$-adic realisation <Ref> and its compatibility with the six functors over quasi-excellent schemes <Ref>.

To finish the proof of the proposition, note that
\[\DNgm(X,\Lambda)=\colim_{(\scal,\lcal)}\ \DN_{(\scal,\lcal)}(X,\Lambda)\]
and therefore we get a t-structure on $\DNgm(X,\Lambda)$ by <Ref>. The t-exactness of the realisation follows from the fact that the perverse t-structure is defined in the same way by gluing along stratifications in <cit.>. The characterisation of the positive and negative parts of the t-structure then follows from the t-exactness of the realisation combined with its conservativity and the analogous characterisation of the positive and negative parts of the perverse t-structure on $\dconspro(X_\proet,\Lambda_\ell)$ which is <cit.>.

Along the way we proved that the perverse t-structure on the subcategory $\DN_\mathrm{lisse}(X,\Lambda)$ of $\DNgm(X,\Lambda)$ made of dualizable objects is the ordinary t-structure shifted by $-\delta(X)$.

This result also implies that any t-exactness property of the perverse t-structure on $\ell$-adic sheaves is true for the perverse t-structure on Nori motives. In particular, the Lefschetz Hyperplane Theorem (also known as Artin Vanishing) is true for Nori motives.

Let $\Lambda$ be a localisation of a regular ring integral and flat over $\Z$ and let $X$ be an excellent finite-dimensional $\Q$-scheme endowed with a dimension function. The abelian category of perverse Nori motives is the heart of the perverse t-structure on $\DNgm(X,\Lambda)$. We denote it by $\mcal_\mathrm{perv}(X,\Lambda)$.

One of the main feats of the perverse t-structure with rational coefficients is that it is preserved by Verdier duality. This is famously not the case with integral coefficients (note for instance that $\map_{\D(\Z)}(\Z/n\Z,\Z)=\Z/n\Z[-1]$). However, if $\ell$ is a prime number there is a t-structure on $\dconspro(X_\proet,\Z_\ell)$ that is exchanged with the perverse t-structure by duality (see <cit.>).

We can mimic its definition to get a t-structure on $\DNgm(X,\Lambda)$ as follows; assume that $\Lambda$ is a Dedekind domain and define
\[{}^{\mathrm{ord}^+}\DNgm(X,\Lambda)^{\leqslant 0}:=\{ M \in \DNgm(X,\Lambda) \mid M\leqslant_\mathrm{ord} 1 \text{ and } \HH^1(M)\otimes_\Z \Q=0\}\]

\[{}^{\mathrm{ord}^+}\DNgm(X,\Lambda)^{\geqslant 0}:=\{ M \in \DNgm(X,\Lambda) \mid M\geqslant_\mathrm{ord} 0 \text{ and } \HH^0(M) \text{ is torsion-free} \}\]

This is a t-structure on $\DNgm(X,\Lambda)$; then by gluing as before this t-structure along stratifications (using the same shifts) we get a t-structure $({}^{p^+}\DNgm(X,\Lambda)^{\leqslant 0},{}^{p^+}\DNgm(X,\Lambda)^{\geqslant 0})$ whose heart we denote by $\mcal_\mathrm{perv}^+(X,\Lambda)$.

Because the six functors are compatible with the $\ell$-adic realisation and because same statement is true after realisation by <cit.>, the Verdier duality functor $\mathbb{D}_X$ of <Ref> exchanges the subcategory ${}^{p^+}\DNgm(X,\Lambda)^{\leqslant 0}$ (resp. ${}^{p^+}\DNgm(X,\Lambda)^{\geqslant 0}$) with ${}^{p}\DNgm(X,\Lambda)^{\geqslant 0}$ (resp. ${}^{p^+}\DNgm(X,\Lambda)^{\leqslant 0}$) and therefore it exchanges $\mcal_\mathrm{perv}(X,\Lambda)$ with $\mcal_\mathrm{perv}^+(X,\Lambda)$.

Using <cit.>, we can also define the intermediate extension functor for Nori motives: more precisely letting $j\colon U\to X$ be an open immersion where $X$ is as above, we define $j_{!*}=\im(\HHp^0j_!\to \HHp^0j_*)$. When defined, its images through the $\ell$-adic and Betti realisations are the usual intermediate extension functors because both realisation are t-exact and compatible with the six operations.

§.§ Universal motives and the derived category of perverse Nori motives

By the universal property of the bounded derived category, there is a canonical realisation functor

\begin{equation}
\label{realMperv}
\D^b(\mcal_\mathrm{perv}(X,\Lambda))\to \DNgm(X,\Lambda).
\end{equation}

This functor is an equivalence after tensorisation by $\Perf_\Q$. We have the following rigidity result to compute its torsion:

Let $\Lambda$ be a localisation of a regular ring integral and flat over $\Z$ and let $X$ be an excellent finite-dimensional $\Q$-scheme endowed with a dimension function $\delta$.
Let $n$ be a prime number.
Consider the composition

\[\iota_n\colon \mathrm{Perv}(X_\et,\Lambda/n)\subseteq\dconset(X_\et,\Lambda/n) \xrightarrow{f}\dconset(X_\et,\Lambda)_{tors}\xrightarrow{\iota} \DNgm(X,\Lambda)_{tors}\]
where $\mathrm{Perv}(X_\et,\Lambda/n)$ is the category of étale perverse sheaves and $f$ is the forgetful functor. It lands in $\mcal_\mathrm{perv}(X,\Lambda)[n]$ and induces an equivalence
\[\mathrm{Perv}(X_\et,\Lambda/n)\xrightarrow{\sim} \mcal_\mathrm{perv}(X,\Lambda)[n].\]

By <Ref> the last functor is an equivalence. Moreover as $\DNgm(X,\Lambda)_{tors}$ is the kernel of the perverse t-exact functor $-\otimes_\Lambda(\Lambda\otimes_\Z\Q)$, it is endowed with a perverse t-structure, whose heart is the abelian category of torsion perverse motives.

First note that the forgetful functor $f$ is t-exact: indeed, the heart of the perverse t-structure on $\dconset(X_\et,\Lambda)$ consists of those complexes $K$ such there is a stratification of $X$ by nil-regular subschemes such that on each stratum $S$ we have $K|_S \simeq L[-\delta(X)]$ for a lisse sheaf $L$; the image through the forgetful functor of a lisse sheaf of $\Lambda/n$-modules is a lisse sheaf of $\Lambda$-modules, whence the functor $f$ is indeed t-exact.

This implies that the functor $\iota_n$ lands in $\mcal_\mathrm{perv}(\Lambda)[n]$. Furthermore, the restriction of the forgetful functor to the heart affords a left adjoint, namely $\HHp^0(-\otimes_\Lambda \Lambda/n)$; it is in fact the underived tensor product with $\Lambda/n$ that sends a perverse sheaf $K$ to the cokernel of the map $K\xrightarrow{\times n} K$. With this, it is clear that  if $M$ belongs to $\mathrm{Perv}(X_\et,\Lambda/n)$, we have
\[\HH^0(f(M)\otimes_\Lambda\Lambda /n)=M\]
and for any object of $\dconset(X_\et,\Lambda)_{tors}^\heartsuit$ which is of $n$-torsion, we have
\[f(\HH^0(N\otimes_\Lambda\Lambda/n))=N.\]
Whence, the functor $\iota_n$ indeed induces an equivalence.

Unfortunately, we do not know if the abelian category $\mcal_\mathrm{perv}$ has enough torsion-free sheaves. If it was true, we would use <Ref> to conclude that the functor <ref> is fully faithful thanks to <Ref> and Beilinson's Theorem on the derived category of perverse sheaves [15]; this would then imply that this functor is an equivalence. We have a partial answer in that direction, that requires to introduce the universal category of perverse motives. Let $k$ be a subfield of the complex numbers (a more general field can be considered, but the following results for those fields can be reduced to subfields of $\C$ by the continuity properties of perverse categories considered along fields).

Let $\Lambda$ be a localisation of a regular ring integral and flat over $\Z$ and and let $X$ be a finite type $k$-scheme. Consider the cohomological functor

\[\HHp^0\circ\rho_\mathrm{B}\colon\DMgm(X,\Lambda)\to \mathrm{Perv}(X^\mathrm{an},\Lambda)\]
where $\mathrm{Perv}(X^\mathrm{an},\Lambda)$ is the category of analytic perverse sheaves and $\HHp^0$ is the $0$-th perverse cohomology functor.
As in <Ref>, we can consider the universal abelian factorisation $\mcal_\mathrm{univ}(X,\Lambda)$ of this functor through a faithful exact functor $r_\mathrm{B}\colon\mcal_\mathrm{univ}(X,\Lambda)\to\mathrm{Perv}(X^\mathrm{an},\Lambda)$. We thus have a universal homological functor

\[\HH_\mathrm{univ}\colon \DMgm(X,\Lambda)\to\mcal_\mathrm{univ}(X,\Lambda).\]

Because the Betti realisation of perverse Nori motives is faithful exact (as the realisation from $\DNgm(X,\Lambda)$ is conservative), we actually have a finer factorisation:

\[\begin{tikzcd}
{\DMgm(X,\Lambda)} & {\mcal_\mathrm{univ}(X,\Lambda)} & {\mcal_\mathrm{perv}(X,\Lambda)}
\arrow["{\mathrm{comp}}", from=1-2, to=1-3]
\arrow["{\HH_\mathrm{univ}}", from=1-1, to=1-2]
\arrow["{\HHp^0\circ\rho_\mathrm{N}}"', from=1-1, to=1-3,bend right=20]
\end{tikzcd}\]

with $\mathrm{comp}$ a faithful exact functor.

Let $\Lambda$ be a localisation of a regular ring integral and flat over $\Z$ and and let $X$ be a finite type $k$-scheme. The functor

\[\mathrm{comp}\otimes\Q\colon \mcal_\mathrm{univ}(X,\Lambda)\otimes\Q\to\mcal_\mathrm{perv}(X,\Lambda)\otimes\Q\]
is an equivalence.

The definition of $\mcal_\mathrm{perv}(X,\Lambda\otimes\Q)$ is $\mcal_\mathrm{univ}(X,\Lambda\otimes\Q)$. Thus we have to show that $\mcal_\mathrm{univ}(X,\Lambda)\otimes\Q$ is the universal category for $\HHp^0\circ\rho_\mathrm{B}\otimes\Q$. This follows easily from the construction of the universal category together with the fact that the rationalisation of the abelian hull of an additive category if the abelian hull of the rationalisation.

For each nonzero integer $n$, we have an Artin motive functor

\[\iota\colon\dconset(X_\et,\Lambda/n)\to \DMgm(X,\Lambda)\]
such that the composition with the realisation is perverse t-exact and conservative. Thus it induces a faithful exact functor

\[i\colon \mathrm{Perv}(X_\et,\Lambda/n)\to \mcal_\mathrm{univ}(X,\Lambda)[n].\]

The following result is an adaptation of a proof due to Nori ([34]) in the case of fields.

Let $\Lambda$ be a localisation of a regular ring integral and flat over $\Z$ and and let $X$ be a finite type $k$-scheme. Let $n$ be an integer. The functor

\[i\colon\mathrm{Perv}(X_\et,\Lambda/n)\to\mcal_{\mathrm{univ}}(X,\Lambda)[n]\]
is an equivalence of quasi-inverse the functor

\[\mcal_{\mathrm{univ}}(X,\Lambda)[n] \xrightarrow{\mathrm{comp}} \mcal_{\mathrm{perv}}(X,\Lambda)[n] \simeq \mathrm{Perv}(X_\et,\Lambda/n).\]

We begin by noting three things:

* The composite functor $\mathrm{comp}\circ i$ lands in $\mcal_\mathrm{perv}(X,\Lambda)[n] =\mathrm{Perv}(X_\et,\Lambda/n)$ and induces the identity of the latter, because we have that $\mathrm{comp}\circ i = \HHp^0\circ\rho_\mathrm{N}\circ \iota$.
* The functors $\mathrm{comp}$ and $i$ are faithful; thus it suffices to prove that $i$ is essentially surjective. Indeed, if $i$ is essentially surjective, this implies that $\mathrm{comp}$ is fully faithful, thus $i$ is also fully faithful.

* The image of $i$ is stable under subquotients. Indeed, if $M$ belongs to $\mcal_\mathrm{perv}(X,\Lambda)[n]$ and is a subobject of something of the form $i(N)$, then $\mathrm{comp}(M)$ is a subobject of $\mathrm{comp}\circ i (N)=N$. This implies that $i\circ\mathrm{comp}(M)$ and $M$ are both subobjects of $i(N)$ that are isomorphic after applying the faithful functor $\mathrm{comp}$ thus they are isomorphic. For the quotients, one uses that if there is a surjection $i(N)\to M$ then the kernel is in the image of $i$ thus the cokernel also.

Note also that if $N$ belongs to $\DMgm(X,\Lambda)$, we can write $N/n = \iota(K)$ for some $K$ in $\dconset(X_\et,\Lambda/n)$, thus
\[\HH_\mathrm{univ}(N/n) =\HH_\mathrm{univ}(\iota(K)) \overset{(*)}{=} \HH_\mathrm{univ}(\iota(\HHp^0(K)))=i(\HHp^0(K))\]

is in the image of $i$, where $(*)$ follows from the perverse t-exactness of the functor

\[\dconset(X_\et,\Lambda/n)\to\DNgm(X,\Lambda).\]

It suffices to show that any object of the form $M/nM$ is in the image of $i$, because any $n$-torsion object is of this form.  Thus, let $M$ be in $\mcal_\mathrm{univ}(X,\Lambda)$ and let $\HH_\mathrm{univ}(N)\to M$ be a surjective map (this exists by definition of the universal category see <Ref>). As there is an exact triangle

\[N\xrightarrow{\times n} N\to N/n\]
in $\DMgm(X,\Lambda)$, we have a short exact sequence

\[\HH_\mathrm{univ}(N)\xrightarrow{\times n}\HH_\mathrm{univ}(N)\to \HH_\mathrm{univ}(N/n).\]
Denote by t the image of the map $\HH_\mathrm{univ}(N)\to\HH_\mathrm{univ}(N/n)$. Because $\HH_\mathrm{univ}(N/n)$ is in the image of $i$ we see that t is also in the image of $i$ as the latter is closed under subquotients. Moreover we have the following commutative diagram with exact rows:

\[\begin{tikzcd}
{\HH_\mathrm{univ}(N)} & {\HH_\mathrm{univ}(N)} & T & 0 \\
M & M & {M/nM} & 0
\arrow["{\times n}", from=1-1, to=1-2]
\arrow[from=1-1, to=2-1]
\arrow[from=1-2, to=1-3]
\arrow[from=1-2, to=2-2]
\arrow[from=1-3, to=1-4]
\arrow[dashed, from=1-3, to=2-3]
\arrow["{\times n}"', from=2-1, to=2-2]
\arrow[from=2-2, to=2-3]
\arrow[from=2-3, to=2-4]
\end{tikzcd}\]
where the map $T\to M/nM$ exists because the composite of the map $M\to M/nM$ with the map $\HH_\mathrm{univ}(N)\to M$ vanishes on the kernel of the quotient map $\HH_\mathrm{univ}(N)\to T$. As $\HH_\mathrm{univ}(N)\to M$ is surjective, this is also the case of the map $T\to M/nM$ thus $M/nM$ is in the image, finishing the proof.

Thus the functor

\[\mathrm{comp}\colon \mcal_{\mathrm{univ}}(X,\Lambda)\to \mcal_{\mathrm{perv}}(X,\Lambda)\]
is an equivalence after being tensored by $\Q$ and induces an equivalence on $n$-torsion objects for all $n$. To conclude that it is an equivalence, one could compute some $\mathrm{Ext}^1$ in $\mcal_{\mathrm{perv}}(X,\Lambda)$, but this seems to be beyond the scope of our current technology. However, the following conjecture seems more reachable:

Let $X$ be an algebraic variety over $\Q$. Then the abelian category $\mcal_\mathrm{univ}(X,\Lambda)$ has enough torsion-free object: for every $M$ in $\mcal_\mathrm{univ}(X,\Lambda)$ there exists an epimorphism $N\twoheadrightarrow M$ in $\mcal_\mathrm{univ}(X,\Lambda)$ with $N$ a torsion-free object.

This conjecture would follow from the equivalence
\[\mcal_\mathrm{pairs}(X,\Lambda)\xrightarrow{\sim} \mcal_\mathrm{univ}(X,\Lambda),\]
$\mcal_\mathrm{pairs}(X,\Lambda)$ is the universal category associated to a diagram of very good pairs as in Nori's original construction, as considered by F. Ivorra in [49] (he was considering rational motives but his construction remains valid for integral sheaves). This equivalence would follows from the fact that Ivorra's realisation functor, the main construction in [48] is the $\HHp^0$ of our functor $\rho_\mathrm{N}$. Note that in [48] the author was only considering smooth varieties but it is not hard to see that a slight modification of his functor (namely, replace the constant sheaf $\Q[-\dim X]$ on $X$ by $\pi_X^!\Q_{\Spec(\Q)}$) would give a functor for any variety. Such a conjecture is related to <cit.>, and might be doable with the current technology.

We now explain how <Ref> implies that both $\mcal_\mathrm{univ}(X,\Lambda)\simeq \mcal_\mathrm{perv}(X,\Lambda)$ for all $X$ but also that $\D^b(\mcal_\mathrm{perv}(X,\Lambda))\simeq\DNgm(X,\Lambda)$.

Let $\Lambda$ be a localisation of a regular ring integral and flat over $\Z$. Assume that <Ref> holds. Then for every field $k$ of characteristic zero and $X$ a finite type $k$-scheme, the functor $\mathrm{comp}\colon\mcal_\mathrm{univ}(X,\Lambda)\to\mcal_\mathrm{perv}(X,\Lambda)$ is an equivalence, and the functor

\[\D^b(\mcal_\mathrm{univ}(X,\Lambda))\to \DNgm(X,\Lambda)\]
is an equivalence.

By continuity along change of fields, it suffices to prove the theorem when $X$ is of finite type over $\Q$.
It suffices to prove that the functor

\[\D^b(\mcal_\mathrm{univ}(X,\Lambda))\to\DNgm(X,\Lambda)\]
induced by universal property from the functor

\[\mcal_\mathrm{univ}(X,\Lambda)\to \mcal_\mathrm{perv}(X,\Lambda)\to \DNgm(X,\Lambda)\]
is fully faithful. Indeed, once it is fully faithful, it will be an equivalence by dévissage because the essential image is then stable under extension. Indeed if $M\in\mcal_\mathrm{perv}(X,\Lambda)$, then there is an short exact sequence
\[0\to M_\mathrm{tors}\to M\to M_\mathrm{fr}\to 0\]
in $\mcal_\mathrm{perv}(X,\Lambda)$ with $M_\mathrm{tors}$ of torsion and $M_\mathrm{fr}$ torsion-free. By <Ref> and <Ref> the object $M_\mathrm{tors}$ is in the image. Moreover by <Ref> there exists $P\in\mcal_\mathrm{univ}(X,\Lambda)$ without torsion and a map $P\to M_\mathrm{fr}$ which becomes an equivalence after tensorisation by $\Q$. In particular, this map has to be injective, and the cokernel is torsion, thus in the image. By fully faithfulness, the object $M_\mathrm{fr}$ is in the image, whence $M$ is in the image.

The functor is an equivalence after tensorisation by $\Q$, and if we tensor the mapping spectra by $\otimes_\Z\Z/p\Z$, using <Ref> (we have enough torsion-free objects by hypothesis!) together with [15] and rigidity, we conclude. Note that this also proves that $\mathrm{comp}$ is an equivalence.

§ INTEGRAL MIXED HODGE MODULES.

§.§ Definition of mixed Hodge modules with integral coefficients

Let $X$ be a separated finite type $\C$-scheme. In [62], 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 $X= \Spec(\C)$, we have that $\mathrm{MHM}(X)\simeq \mathrm{MHS}^P_\Q(\C)$ is the category of polarisable mixed Hodge structure over $\C$ with rational coefficients (which consists of mixed Hodge structure such that the graded pieces of the weight filtration are polarisable <cit.>, it is a full subcategory of Deligne's mixed Hodge structures), and there is a faithful exact functor
\[\mathrm{rat}\colon \mathrm{MHM}(X)\to\mathrm{Perv}(X^\mathrm{an},\Q).\]

Moreover, M. Saito constructed the six operations on the bounded derived category $\D^b(\mathrm{MHM}(-))$, in a way compatible with the functor $\mathrm{rat}$.
In [65] 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 $\DM(-,\Q)$, the second author obtained a realisation

\[\rho_\mathrm{H}\colon\DM(-,\Q)\to\D_\mathrm{H}(-)\]
compatible with the six operations, where $\D_\mathrm{H}(-)$ is the indization of $\D^b(\mathrm{MHM}(-))$. By [14] the abelian category of polarisable mixed Hodge structure with rational coefficients $\mathrm{MHS}_\Q^P(\C)$ is of cohomological dimension 1. If $X$ is a complex variety of dimension $d$ with structural morphism $\pi_X$, as the cohomological amplitude of $(\pi_X)_*\underline{\Hom}(-,-)$ 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 $\D_\mathrm{H}(X)$ is the unbounded derived category of ind-mixed Hodge modules $\D(\In\mathrm{MHM}(X))$ (because, for example by <cit.> the functor $\map_{\D}(M,-)$ 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.

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

\[ \begin{tikzcd}
\D_\mathrm{H}(-,\Z) \arrow[r] \arrow[d]
\arrow[dr, phantom, very near start, "{ \lrcorner }"]
& \D_\mathrm{H}(-) \arrow[d,"\mathrm{rat}"] \\
\In\dconsan((-)^\mathrm{an},\Z) \arrow[r,"-\otimes \Q",swap]
& \In \dconsan((-)^\mathrm{an},\Q)
\end{tikzcd}\]
where the pullback is taken in the $\infty$-category of functors $(\Sch_\C^\mathrm{ft})^\op\to\mathrm{Pr}^{\mathrm{L},\omega}_\St$.

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

Recall that in <cit.> 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

\[ \begin{tikzcd}
\mathrm{MHS}^P_\Z(\C) \arrow[r] \arrow[d,"\mathrm{int}"]
\arrow[dr, phantom, very near start, "{ \lrcorner }"]
& \mathrm{MHS}^P_\Q(\C) \arrow[d] \\
\mathrm{Ab}^\mathrm{ft} \arrow[r]
& \mathrm{Vect}_\Q^\mathrm{fd}
\end{tikzcd}\]
with $\mathrm{Vect}_\Q^\mathrm{fd}$ the abelian category of finite-dimensional  $\Q$-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

\[\D(\In\mathrm{MHS}_\Z^P(\C))\to\D_\mathrm{H}(\Spec(\C)).\]

The above functor is an equivalence.

This follows from the fact that the cohomological dimension of $\mathrm{MHS}_\Z^P$ is $1$ by <cit.>. Indeed in that case by <cit.> any object of $\D^b(\mathrm{MHS}_\Z^P)$ defined a compact object of $\D(\In\mathrm{MHS}_\Z^P)$.

There is a canonical realisation

\[\rho_\mathrm{H}\colon \DM(-,\Z)\to\D_\mathrm{H}(-,\Z)\]
that commutes with the $6$ operations. Indeed, this functor is obtained from the universal property of the pullback, and as $\D_\mathrm{H}(-,\Z)$ 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 <cit.>) and when taken mod $p$ for all primes $p$.

§.§ Mixed Hodge modules over R-schemes.

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

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

\[(\Sch^\mathrm{ft}_\R)^\op\to\mathrm{Fun}(\mathrm{B}\Z/2\Z,\CAlg(\PrL))\]
that sends a finite type $\R$-scheme $X$ to $\D_\mathrm{H}(X_\C,\Z)$.

Now consider the functor
\[(-)^{\mathrm{h}\Z/2\Z}\colon\mathrm{Fun}(\mathrm{B}\Z/2\Z,\CAlg(\PrL))\to \CAlg(\PrL)\]
that sends a presentably symmetric monoidal $\infty$-category $\ccal$ with a $\Z/2\Z$ action to the homotopy fixed points $\ccal^{\mathrm{h}\Z/2\Z}$. Note that this functor take a diagram $G\colon \mathrm{B}\Z/2\Z\to \CAlg(\PrL)$ to its limit, thus has a left adjoint $F$ that takes an object $\ccal\in\CAlg(\PrL)$ to the constant diagram $\underline{\ccal}$.

If $X$ is a finite type $\C$-scheme, then the natural functor
\[p_1^*\colon\D_\mathrm{H}(X,\Z)\to\D_\mathrm{H}(X_\C,\Z)\]
induced by the first projection $p_1\colon X\times_{\Spec(\R)}\Spec(\C)\to X$ 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

\[\eta\colon FG(\D_\mathrm{H}(X_\C,\Z))\to \D_\mathrm{H}(X_\C,\Z).\]

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

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

The functoriality of the presentable $\infty$-category of mixed Hodge modules on finite type $\R$-schemes is the composition

\[\D_\mathrm{H}(-,\Z)\colon (\Sch_\R^\mathrm{ft})^\op\to \mathrm{Fun}(\mathrm{B}\Z/2\Z,\CAlg(\PrL)) \xrightarrow{(-)^{\mathrm{h}\Z/2\Z}} \CAlg(\PrL).\]

By the previous lemma, the resriction of this functor to finite type $\C$-schemes corresponds to ind-mixed Hodge modules as constructed by M. Saito.
We may denote by $\D_\mathrm{H}(-,\Q):=\D_\mathrm{H}(-,\Z)\otimes\Mod_\Q$.

Let $X$ be a finite type $\R$-scheme. Then $\D_\mathrm{H}(X,\Q)$ is compactly generated and its full subcategory of compact objects is canonically isomorphic to $\D^b(\mathrm{MHM}(X,\Q))$ where $\mathrm{MHM}(X,\Q)$ consists of the $\Z/2\Z$-invariants of $\mathrm{MHM}(X_\C,\Q)$.

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

\begin{align*}
\map_{D_\mathrm{H}(X,\Q)}(D,\colim M_i) &\simeq \map_{D_\mathrm{H}(X_\C,\Q)}(D_{\mid X_\C},\colim_i (M_i)_{\mid X_\C})^{\mathrm{h}\Z/2\Z} \\
& \simeq \mathrm{R}\Gamma(\R_\et,\colim\map_{D_\mathrm{H}(X_\C,\Q)}(D_{\mid X_\C}, (M_i)_{\mid X_\C}) ) \\
& \overset{(*)}{\simeq} \colim_i\mathrm{R}\Gamma(\R_\et,\map_{D_\mathrm{H}(X_\C,\Q)}(D_{\mid X_\C}, (M_i)_{\mid X_\C}) )\\
& \simeq\colim_i \map_{\D_\mathrm{H}(X,\Q)}(D,M_i)
\end{align*}
where $(*)$ follows from <cit.>. Thus any object of $\dcal(X)$ is compact. Moreover, because the functor

\[\D_\mathrm{H}(X,\Q)\to\D_\mathrm{H}(X_\C,\Q)\]
is conservative, the $\infty$-category $\dcal(X)$ generates $\D_\mathrm{H}(X,\Q)$ under colimits. Thus, the canonical functor $\In\dcal(X)\to\D_\mathrm{H}(X,\Z)$ is an equivalence.

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

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

Let $X$ be a smooth $\R$-scheme.
Let $M\in\mathrm{MHM}(X,\Z)$, then $M_{\mid X_\C}$ admits an underlying $\dcal$-module on $X_\C$ with an action of $\Z/2\Z$, so that in fact we have a functor
\[\mathrm{MHM}(X,\Z)\to \dcal\text{-}\mathrm{mod}(X)\]
by Galois descent of $\dcal$-modules.

The heart of $\D^b_\mathrm{H,c}(\Spec(\R),\Z)$ is the category $\mathrm{MHS}_\Z^p(\R)$ of polarisable mixed Hodge modules over $\R$ with integral coefficients. Moreover, the canonical functor

\[\D^b(\mathrm{MHS}_\Z^p(\R))\to\D^b_\mathrm{H,c}(\Spec(\R),\Z)\]

is an equivalence.

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

\[\D^b(\mathrm{MHS}_\Z^p(\R))\to\D^b_\mathrm{H,c}(\Spec(\R),\Z).\]

As in the proof of <Ref> the left hand side has descent along the morphism $\R\to\C$, so that the result follows from <Ref>.

Let $f\colon X\to \Spec \R$ be a finite type map, then we recover the absolute Hodge cohomology groups of Beilinson ([14]) as

\[\mathrm{R}\Gamma_\hcal(X,\Z(p))\simeq\map_{\D_\mathrm{H}(\Spec(\C),\Z)}(\Z,f_*\Z(p))\simeq \map_{\D_\mathrm{H}(X,\Z)}(\Z,\Z(p)).\]

Indeed Saito proved the rational result for $X_\C$ in <cit.>, thus it suffices to check that Beilinson definition is indeed the $\Z/2\Z$-invariants of a pullback of the rational cohomology and the $\Z$-structure, but this is the case by <cit.>.

We finish with a proposition computing the torsion of $\D_\mathrm{H}(X,\Z)$.

Let $X$ be a finite type $\R$-scheme. Then the functor

\[\D(X_\et,\Z)\to\D_\mathrm{H}(X,\Z)\]

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

\[\D((X_\C)_\et,\Z)\to\DM(X_\C,\Z)\to\D_\mathrm{H}(X_\C,\Z)\]

induces an equivalence

\[\D(X_\et,\Z/n\Z)\simeq\D_\mathrm{H}(X,\Z)\otimes_\Z\Mod_{\Z/n\Z}.\]

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

§.§ Motivic Hodge modules

As for Nori motives, this realisation functor factors through modules over the algebra $\hcal$ representing Hodge cohomology. Using the same method as <cit.>, we proved in <cit.> that the canonical functor

\[\Mod_{\hcal\otimes_\Z\Q}(\DM(-,\Q))\to \D_\mathrm{H}(-)\]
is fully faithful over finite type $\C$-schemes, and that the t-structure on $\D_\mathrm{H}(-)$ induces a t-structure on the image that we describe following Ayoub's ideas in <cit.>: it is the localising subcategory generated by the $f_*\Q(n)$ 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 $f_*\Q(n)$). This category of modules was first introduced by Drew in [31].

There is also an enriched version of this result, where instead of obtaining the localising subcategory generated by the $f_*\Q(n)$, 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.

The functor $\D_\mathrm{H}(-,\Z)$ naturally takes values in $\D_{H}(\Spec(\R),\Z)$-linear presentable $\infty$-categories. Thus, the functor $\rho_\mathrm{H}$ induces a $\D_{H}(\Spec(\R),\Z)$-linear functor

\[\bm{\rho_\mathrm{H}}\colon \DM(-,\Z)\otimes_{\Mod_\Z}\D_{H}(\Spec(\R),\Z) \to \D_\mathrm{H}(-,\Z).\]

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

\[\bm{\DM}(-,\Z):=\DM(-,\D_{H}(\Spec(\R),\Z))\]

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 $\Mod_\Lambda$ by $\D_{H}(\Spec(\R),\Z)$. The right adjoint $\bm{\rho^\mathrm{H}_*}$ of $\bm{\rho_\mathrm{H}}$ is lax monoidal thus we get an object $\bm{\hcal}_X=\bm{\rho^\mathrm{H}_*}(\Z)$ of $\mathrm{CAlg}(\bm{\DM}(X,\Z))$ for all finite type $\R$-schemes $X$.
There is a natural map

\[\pi_X^*\bm{\hcal}_{\Spec(\C)}\to\bm{\hcal}_X\]
with $\pi_X\colon X\to\Spec(\R)$ the structural morphism, it will be shown to be an isomorphism.
The presentable $\infty$-categories of integral motivic Hodge modules is the category

\[\bm{\mathrm{DH}}(X,\Z):=\mathrm{Mod}_{\pi_X^*\bm{\hcal}}(\bm{\DM}(X,\Z)).\]

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

\[\bm{\mathrm{DH}}(-,\Z)\simeq \bm{\DM}(-,\Z)\otimes_{\bm{\DM}(\Spec(\R),\Z)}\bm{\mathrm{DH}}(\Spec(\R),\Z).\]
Thus as in <Ref>, the functor
$\bm{\mathrm{DH}}(-,\Z)$ affords all the six operations in such a way that the functor from $\bm{\DM}(-,\Z)$ and the canonical functor to $\D_\mathrm{H}(-,\Z)$ commute with them, the composition of the two being $\bm{\rho_\mathrm{H}}$.

Let $\iota\colon\Spec(\C)\to\Spec(\R)$ be the spectrum of the inclusion of $\R$ into $\C$. Then the canonical map
$\iota^*\bm{\hcal}_{\Spec(\R)}\to\bm{\hcal}_{\Spec(\C)}$ is an equivalence.

Becuase $\iota$ is finite étale, the functor $\iota^*=i^!$, being a right adjoint, commutes with $\bm{\rho_\mathrm{H}}$. Thus $\iota^*\bm{\hcal}_{\Spec(\R)} \simeq \bm{\rho_\mathrm{H}}(\iota^*\un)\simeq\bm{\hcal}_{\Spec(\C)}$.

The canonical functor
\[\bm{\underline{\rho_\mathrm{H}}}\colon\bm{\mathrm{DH}}(-,\Z)\to\D_\mathrm{H}(-,\Z)\]
factoring $\bm{\rho_\mathrm{H}}$ is fully faithful and its image is stable under truncations.

By étale descent and <Ref>, it suffices to prove the claim over finite type $\C$-schemes.
Because $\D_\mathrm{H}(-,\Z)$ is compactly generated, the right adjoint $\bm{\rho_*^\mathrm{H}}$ commutes with rationalisation, so that the algebra $\pi_X^*\bm{\hcal}_{\Spec(\C)}$, tensored with $\Q$ is the algebra considered by Drew in [31] and the second author in [66]. This implies that $\bm{\underline{\rho_\mathrm{H}}}\otimes\mathrm{Mod}_\Q$ is fully faithful with image stable under truncations by the main theorem of <cit.> (see also<cit.>). When tensoring the square of <Ref> by $\mathrm{Mod}_{\Z/n\Z}$, we obtain an equivalence

\[\D_\mathrm{H}(-,\Z)\otimes_{\Mod_\Z}\Mod_{\Z/n\Z} \xrightarrow{\sim} \In(\dconsan(X^\mathrm{an},\Z)\otimes_{\Perf_\Z}\Perf_{\Z/n\Z}),\]
but the right hand side coincides with the derived category of étale sheaves of $\Z/n\Z$-modules on $X$, thanks to Artin comparison theorem. By rigidity, this implies that the natural map
\[\Z/n\Z\to \pi_X^*\bm{\hcal}_{\Spec(\C)}/n\]
is an equivalence, and then that $\bm{\underline{\rho_\mathrm{H}}}\otimes_{\Mod_\Z}\Mod_{\Z/n\Z}$ is an equivalence. This finishes the proof of the fully faithfulness.

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

\[\tau^{\leqslant 0}M\to \tau^{\leqslant 0}(M\otimes\Q) \to \colim_n \tau^{\leqslant 0}(M)\otimes_\Z \Z/n\Z.\]

The middle term is in the image because we proved the theorem with rational coefficients in [66]. Thus it suffices to show that for all $n\in\N^*$, the object $\tau^{\leqslant 0}(M)\otimes_\Z\Z/n\Z$ is in the image. This is the case because $\bm{\underline{\rho_\mathrm{H}}}\otimes_{\Mod_\Z}\Mod_{\Z/n\Z}$ is an equivalence. This proves that the image is stable under truncations, finishing the proof.

The fact that $\bm{\underline{\rho}_\mathrm{H}}$ is fully faithful implies readily that the map $\pi_X^*\bm{\hcal}_{\Spec(\R)}\to\bm{\hcal}_X$ is an equivalence for all $X$.

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 $\mathrm{DH}(-,\Z)$.

For $X$ a finite type $\R$-scheme, the image of $\bm{\rho_\mathrm{H}}$ can be
described as the localising subcategory
spanned by objects of the form $p_*H(n)$ for $p\colon Y\to X$ proper, and $H$ a polarisable integral mixed Hodge structure over $\R$ seen as a constant integral mixed Hodge module over $Y$.

§.§ Abelian categories of integral Hodge modules

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

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

In more concrete terms, a  mixed Hodge module with integral coefficients is the data of three objects $(M,\fcal,S,\varphi)$ where $M\in\mathrm{MHM}(X_\C,\Q)$ is a mixed Hodge module as constructed by M. Saito, $\fcal\in\mathrm{Perv}(X_\C^\mathrm{an},\Z)$ is a perverse sheaf, $S\in\operatorname{End}(M,\fcal)$ is an involution and $\varphi\colon \fcal\otimes\Q\simeq \mathrm{rat}(M)$ 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 <Ref> in reverse. We will denote their hearts by $\mathrm{MHM}^\mathrm{ord}(X,\Z)$, $\mathrm{MHM}_\mathrm{geo}^\mathrm{ord}(X,\Z)$ and $\mathrm{MHM}_\mathrm{Hdg}^\mathrm{ord}(X,\Z)$ respectively. Using the same proof as <Ref>, we obtain:

The natural functors

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

\[\D^b(\mathrm{MHM}_\mathrm{geo}^\mathrm{ord}(X,\Z))\to\mathrm{DH}_c(X,\Z)\]
\[\D^b(\mathrm{MHM}_\mathrm{Hdg}^\mathrm{ord}(X,\Z))\to\bm{\mathrm{DH}}_c(X,\Z)\]
are equivalences.

Over finite type $\C$-schemes, the natural functors

\[\D(\In\mathrm{MHM}^\mathrm{ord}(X,\Z))\to \D_\mathrm{H}(X,\Z),\]

\[\D(\In\mathrm{MHM}_\mathrm{geo}^\mathrm{ord}(X,\Z))\to\mathrm{DH}(X,\Z)\]
\[\D(\In\mathrm{MHM}_\mathrm{Hdg}^\mathrm{ord}(X,\Z))\to\bm{\mathrm{DH}}(X,\Z)\]
are equivalences.

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.

The above result is false for ordinary motives over $\C$-schemes. Indeed, assuming that the functor $\DM(X,\Lambda)\to\DN(X,\Lambda)$ is an equivalence, Ayoub's counterexample <cit.> would imply that the functor $\DN(X,\Lambda)\to\D(X^\mathrm{an},\Lambda)$ is not conservative, but the functor $\D(\In\mcal_\mathrm{ord}(X,\Lambda))\to\D(X^\mathrm{an},\Lambda)$ is conservative. This is the only obstruction, see <Ref>.

As for Nori motives, we suspect that the functors

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

\[\D^b(\mathrm{MHM}_\mathrm{Hdg}(X,\Z))\to\bm{\mathrm{DH}}_c(X,\Z)\]
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 $M \in \mathrm{MHM}(X,\Z)$ is lisse if its underlying perverse sheaf $\mathrm{int}(M)$ is locally constant. We denote by $\mathrm{MHM}_\mathrm{lis}(X,\Z)$ the category of lisse Mixed Hodge modules on a finite-type $\C$-scheme. Note that the pullback defining $\D_\mathrm{H}(X,\Z)$ implies that the perverse t-structure induces a t-structure on the subcategory of dualizable objects of $\D_\mathrm{H}(X,\Z)$; the heart of this t-structure is precisely $\mathrm{MHM}_\mathrm{lis}(X,\Z)$.

Let $X$ be a finite-type $\C$-scheme and let $\mathrm{VMHS}_\mathrm{ad}(X,\Z)$ be the category of admissible variations of mixed Hodge structures, where a variations of mixed Hodge structure in the sense of <cit.> is admissible if its rationalisation is admissible in the sense of <cit.>. Then, we have an equivalence of categories
\[\mathrm{MHM}_\mathrm{lis}(X,\Z)\to \mathrm{VMHS}_\mathrm{ad}(X,\Z).\]

We have a pullback square:

\[\begin{tikzcd}
\mathrm{MHM}_\mathrm{lis}(X,\Z)\ar[r]\ar[d] & \mathrm{Loc}(X^\an,\Z) \ar[d]\\
\mathrm{MHM}_\mathrm{lis}(X,\Q) \ar[r]& \mathrm{Loc}(X^\an,\Q)
\end{tikzcd}\]
where $\mathrm{Loc}(X,\Lambda)$ is the category of locally constant sheaves of $\Lambda$-modules with values a in finitely generated $\Lambda$-modules.

By <cit.>, we have an equivalence of categories
\[\mathrm{MHM}_\mathrm{lis}(X,\Q)\to \mathrm{VMHS}_\mathrm{ad}(X,\Q).\]

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:

\[\begin{tikzcd}
\mathrm{VMHS}_\mathrm{ad}(X,\Z)\ar[r]\ar[d] & \mathrm{Loc}(X^\an,\Z)\ar[d] \\
\mathrm{VMHS}_\mathrm{ad}(X,\Q) \ar[r]& \mathrm{Loc}(X^\an,\Q),
\end{tikzcd}\]

and the result follows.

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

\[ \Psi_f\colon \D^b_\mathrm{H,c}(X_\eta,\Z)\to\D^b_\mathrm{H,c}(X_s,\Z),\]

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

\[\begin{tikzcd}
{\D_\mathrm{H,c}(X_\eta,\Z)} \\
& {\D_\mathrm{H,c}(X_s,\Z)} & {\D^b_c(X_s,\Z)} \\
& {\D_\mathrm{H,c}(X_s,\Q)} & {\D^b_c(X_s,\Z)}
\arrow["{\Psi_f}"{description}, dashed, from=1-1, to=2-2]
\arrow["{\Psi_f(\mathrm{int}(-))}", from=1-1, to=2-3,bend left=20]
\arrow["{\Psi_f(-\otimes\Q)}"', from=1-1, to=3-2,bend right=20]
\arrow[from=2-2, to=2-3]
\arrow[from=2-2, to=3-2]
\arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=2-2, to=3-3]
\arrow[from=2-3, to=3-3]
\arrow[from=3-2, to=3-3]
\end{tikzcd},\]
so that all known properties about $\Psi_f$ on $\D^b_c(-,\Z)$ remain true for the one for $\D^b_{\mathrm{H,c}}(-,\Z)$, 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 $\Z/2\Z$-invariants, we also have a version over finite type $\R$-schemes.

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