Notation 3.1.2. As the reader begins to guess, both situations, motivic Hodge modules and enriched motivic Hodge modules, are parallel. Thus from now on except for the important results we will only deal with motivic Hodge modules, the proofs in the enriched case being the same. This is only to avoid doubling the size of this article and the extensive use of bold.

Proof. The proof is the same as the proof of [Reference TubachTub25, Lemma 4.14] and is originally due to Cisinski and Déglise. We recall it here. First over $\operatorname {Spec}\mathbb {C}$ the functor is fully faithful by [Reference DrewDre18, Lemma 4.11], because $\operatorname {\mathrm {DM}_{\mathrm {\acute {e}t}}}(\mathbb {C})$ is rigid and $\rho _{\mathrm {H}}$ preserves compact objects. Over a general base it suffices to prove that the functor is fully faithful on compact objects $\mathrm {DH}^\omega $ . Using that

where $p\colon X\to \operatorname {Spec}(\mathbb {C})$ is the structural morphism, and that the same formula holds in $\mathrm {DH}$ , we see that the lemma would follow from the commutation of $\underline {\rho _{\mathrm {H}}}$ with pushforwards and internal homomorphisms. Because the image of the compact preserving functor $-\otimes \mathcal {H}\colon \operatorname {\mathrm {DM}_{\mathrm {\acute {e}t}}}\to \mathrm {DH}$ generates $\mathrm {DH}$ , it suffices even to prove that $\underline {\rho _{\mathrm {H}}}$ commutes with pushforwards and internal homomorphisms when restricted to the essential image of $-\otimes \mathcal {H}\colon (\operatorname {\mathrm {DM}_{\mathrm {\acute {e}t}}})^\omega \to \mathrm {DH}^\omega $ . This finishes the proof as $\rho _{\mathrm {H}}=\underline {\rho _{\mathrm {H}}}\circ (-\otimes \mathcal {H})$ and $-\otimes \mathcal {H}$ commute with the six operations.

Proof. This results from the commutation with the operations and the fact that $\mathrm {DH}(X)$ (resp. $\mathbf {DH}(X)$ ) is generated under colimits by the $p_*\pi _X^*K(m)$ for $K\in \mathrm {Mod}_{\mathbb {Q}}$ (resp. $K\in \mathrm {D}_{\mathrm {H}}(\mathbb {C}))$ , thanks to [Reference AyoubAyo07a, Lemme 2.2.23].

Proof. Note that by [Reference LurieLur09, 4.4.5.21, 5.5.7.8], the colimit is in fact computed in $\mathrm {Pr}^L_\omega $ , the $\infty $ -category of compactly generated $\infty $ -categories, and we can rewrite it as

$$\begin{align*}\mathrm{Ind}\Big(\mathrm{colim}_{\eta\in U}\mathrm{DH}(U)^\omega\to \mathrm{colim}_{\eta\in U}(\mathrm{D}_{H,\mathrm{geo}}(U))^\omega\Big)\end{align*}$$

where now the colimit inside the $\mathrm {Ind}(-)$ is taken in $\mathrm {Cat}_\infty $ . By [Reference Haine, Ramzi and SteinebrunnerHRS25, Proposition 2.1] the functor between compact objects is fully faithful, using Lemma 3.2.1. It remains to show that it is essentially surjective. It suffices to reach compact objects.

Denote by

$$ \begin{align*}\mathcal{D}:=\mathrm{colim}_{\eta\in U}(\mathrm{D}_{H,\mathrm{geo}}(U)^\omega)\end{align*} $$

the colimit of the compact objects, computed in $\mathrm {Cat}_\infty $ .

For each U, the compact objects of $\mathrm {D}_{H,\mathrm {geo}}(U)$ have a standard t-structure and its heart is the category defined in the same way as $\mathrm {MHM}_{\mathrm {geo}}^{\mathrm {std}}(U)$ , but allowing only direct factors instead of all filtered colimits. As all transitions in the diagram are t-exact, the colimit $\mathcal {D}$ of the compact objects (which are the compact objects of the colimit) has a bounded t-structure. Moreover, the category $\mathcal {D}$ is generated under finite colimits, finite limits, extensions and truncations by images in the colimit of objects of the form $g_*\pi _Z^*K(n)$ where $g\colon Z\to U$ is a proper morphism and $n\in \mathbb {Z}$ is an integer.

By continuity of motives [Reference Elmanto, Hoyois, Iwasa and KellyEHIK21, Lemma 5.1 (ii)] and [Reference LurieLurHA, Theorem 4.8.5.11], the left-hand side of equation (3.2.4.1) is canonically equivalent to $\mathrm {DH}(\eta ):=\mathrm {Mod}_{u^*\mathcal {H}_{\mathbb {C}}}(\operatorname {\mathrm {DM}_{\mathrm {\acute {e}t}}}(\eta ))$ , where $u:\eta \to \operatorname {Spec} \mathbb {C}$ is the structural morphism. Moreover, by [Reference AyoubAyo07a, Lemma 2.2.27], the $\infty $ -category $\mathrm {DH}(\eta )$ is generated under colimits by the $f_*\pi _Y^*K(n)$ where $f:Y\to \eta $ is a smooth and projective morphism, $n\in \mathbb {Z}$ and $K\in \mathrm {Mod}_{\mathbb {Q}}$ . By spreading out ([Reference GrothendieckGro66, 8.10.5(xiii)], [Reference GrothendieckGro67, 17.7.8(ii)]) and proper base change, such an object is the restriction to $\eta $ of a $g_*\pi _{Z}^*K(n)$ with $g:Z\to U$ a smooth projective morphism, and U an open subset of X.

Now consider an object of the form $g_*\pi _Z^*K(n)$ in $\mathcal {D}$ , where $g\colon Z\to U$ is a proper morphism and $n\in \mathbb {Z}$ is an integer. This object is the image by equation (3.2.4.1) of $(g_*\pi _Z^*K(n))_\eta \in \mathrm {DH}(\eta )$ . By the above paragraph, there exists an open subset V of U, a finite category I and a functor $F\colon I\to \mathrm {DH}(V)$ such that for each $i\in I$ , the object $F(i)$ is of the form $(f_i)_*\pi _{Z_i}^*K(n_i)$ for $f_i\colon Z_i\to U$ a smooth and projective morphism and $n_i\in \mathbb {Z}$ , and such that in $\mathrm {DH}(\eta )$ , we have

$$\begin{align*}(g_*\pi_Z^*K(n))_\eta \simeq (\mathrm{colim}_I F(i))_\eta .\end{align*}$$

In particular, in $\mathcal {D}$ , the object $g_*\pi _Z^*K(n)$ is isomorphic to the image in the colimit over all open subsets, of the finite colimit $\mathrm {colim}_I F'(i)$ , where $F'$ is the composition of the functor F with equation (3.2.4.1), which has the same formula for $F'(i)$ .

This gives that the category $\mathcal {D}$ is generated under finite colimits, finite limits, direct factors, extensions and truncations by images in the colimit of objects of the form $g_*\pi _Z^*K(n)$ with $g:Z\to U$ smooth and projective. We may also assume U to be smooth.

It turns out that to prove that the functor of the proposition is essentially surjective, it suffices to check that it reaches all objects of the heart of the compact objects. Thanks to [Reference AyoubAyo22, Lemma 1.6.22], we see that it suffices to show that all subquotients of the image in the colimit of all ${\ ^{\mathrm {std}}\mathrm {H}}^n(f_*\pi _Y^*K)$ , for $K\in \mathrm {Mod}_{\mathbb {Q}}$ compact and $f:Y\to U$ projective and smooth, are in the image. Now by dévissage we can assume K to be pure, so that by the conservation of weights under pushforwards by proper maps, and because U is smooth hence Y is also smooth so that $\pi _Y^*K$ is pure, the object $f_*\pi _Y^*K$ is pure. Using the decomposition theorem for pure complexes, we see that there is a decomposition

$$\begin{align*}f_*\pi_Y^*K\simeq \bigoplus_n{\ ^{\mathrm{p}}\mathrm{H}}^n(f_*\pi_Y^*K)[-n].\end{align*}$$

In particular, each ${\ ^{\mathrm {p}}\mathrm {H}}^n(f_*\pi _Y^*K)$ lies in the image. As f is smooth and proper and $\pi _Y^*K$ is a dualisable object coming from $\operatorname {Spec}(\mathbb {C})$ , we see that (as in [Reference RiouRio05, Theorem 2.2], the dual is $f_*\pi _Y^*K^\vee (d)[2d]$ where d is the relative dimension of f) the complex $f_*\pi _Y^*K$ is dualisable, so that ${\ ^{\mathrm {p}}\mathrm {H}}^n(f_*\pi _Y^*K)$ are local systems. Because duality datum consists of finitely many maps, and because compact objects in the colimit are dualisable, we see that any subquotient of the image of ${\ ^{\mathrm {p}}\mathrm {H}}^n(f_*\pi _Y^*K)$ in the colimit will lift as a subquotient in the category of local systems, of the restriction of ${\ ^{\mathrm {p}}\mathrm {H}}^n(f_*\pi _Y^*K)$ to a smaller open subset. Moreover, as ${\ ^{\mathrm {p}}\mathrm {H}}^n(f_*\pi _Y^*K)$ is lisse we have ${\ ^{\mathrm {p}}\mathrm {H}}^n(f_*\pi _Y^*K[d])[-d]\simeq {\ ^{\mathrm {std}}\mathrm {H}}^{n}(f_*\pi _Y^*K)$ where $d=\dim U$ (this is because over a smooth scheme, and for lisse complexes, the standard t-structure is the perverse t-structure, shifted by the dimension [Reference Beilinson, Bernstein and DeligneBBD82, proof of Proposition 2.1.3]). This proves that any subquotient of the image of ${\ ^{\mathrm {std}}\mathrm {H}}^n(f_*\pi _Y^*K)$ in the colimit is in fact a subquotient in perverse sheaves of a suitable restriction of ${\ ^{\mathrm {p}}\mathrm {H}}^n(f_*\pi _Y^*K[d])$ to an open subset of U. As this object is a pure perverse sheaf, any subquotient is in fact a direct factor, thus is in the image, and the proof is finished.

Proof. This is now a simple Noetherian induction: if $K\in \mathrm {D}_{H,\mathrm {geo}}(X)$ is compact, we may find a nonempty irreducible open subset U of X with generic point $\eta $ and Proposition 3.2.4 ensures that up to reducing the size of U, the restriction $K_{\mid U}$ of K to U is in the image, but the localisation sequence

$$ \begin{align*}j_!K_{\mid U}\to K\to i_*i^*K\end{align*} $$

with $j:U\to X$ and $i:Z\to X$ the complement, together with the full faithfulness and a Noetherian induction, give that K is in the image.

Proof. The proofs of [Reference KhanKha19, Appendix A] hold for any motivic coefficient system, so that 1,2,3 and 4 are [Reference KhanKha19, Theorem A.5], 5. is [Reference KhanKha19, Theorem A.9], 6. is [Reference KhanKha19, Proposition A.10] and 7. is [Reference KhanKha19, Theorem A.13]. The last point 8. follows from [Reference Liu and ZhengLZ17, Proposition 4.3.5].

Proof. We have to first deal with the case where $\mathfrak {X}$ is an algebraic space. In this case, the map $\pi $ can be chosen to be étale, and as the diagonal of $\mathfrak {X}$ is representable by schemes, the limit diagram

$$\begin{align*}\mathrm{D}_{\mathrm{H}}(\mathfrak{X})\to\lim_\Delta(X^{n/X})\end{align*}$$

has t-exact transitions whence the limit $\mathrm {D}_{\mathrm {H}}(\mathfrak {X})$ has a perverse t-structure. The left separation and compatibility with colimits can be checked after applying $\pi ^*$ , hence hold. The right completeness is checked exactly as in the case of stacks, which we deal with below.

We can assume that $\mathfrak {X}$ is a connected algebraic stack. Choose a presentation $\pi :X\to \mathfrak {X}$ which is smooth of relative dimension d for some integer d. Then all the projections

$$ \begin{align*}p_n:X\times_{\mathfrak{X}} X \times_{\mathfrak{X}} \cdots \times_{\mathfrak{X}} X \to X\times_{\mathfrak{X}} \cdots \times_{\mathfrak{X}} X\end{align*} $$

are also smooth of relative dimension d, so that each $p_n^*[d]$ is t-exact for the perverse t-structure. As equation (4.1.0.1) can be done with the shifts $p_n^*[d]$ , this creates a t-structure on the limit of the $\check {\text {C}}$ ech nerve of $\pi $ such that $\pi ^*$ is t-exact. It does not depend on $\pi $ as one can see by taking another presentation $p:Y\to \mathfrak {X}$ and then $Y\times _{\mathfrak {X}} X\to \mathfrak {X}$ . The t-structure obviously restricts to $\mathrm {D}^b_{\mathrm {H},c}(\mathfrak {X})$ . For the right completeness, we want to show that the natural functor

$$ \begin{align*}\mathrm{D}_{\mathrm{H}}(\mathfrak{X})\to \lim\Big(\cdots \xrightarrow{\tau^{\geqslant 1}} \mathrm{D}_{\mathrm{H}}^{\geqslant 1}(\mathfrak{X}) \xrightarrow{\tau^{\geqslant 2}}\mathrm{D}_{\mathrm{H}}^{\geqslant 2}(\mathfrak{X})\xrightarrow{\tau^{\geqslant 3}}\mathrm{D}_{\mathrm{H}}^{\geqslant 3}(\mathfrak{X})\xrightarrow{\tau^{\geqslant 4}}\cdots\Big)\end{align*} $$

is an equivalence. As (shifts of) pullbacks by smooth maps are t-exact, this can be checked locally on $\mathfrak {X}$ , hence holds.

Proof. The proof is exactly the same as for Proposition 4.1.4, except that we do not need to shift the pullback functors.

Proof. Both conditions on K are local on $\mathfrak {X}$ , hence we can assume that $\mathfrak {X}$ is a finite type $\mathbb {C}$ -scheme X. If K is bounded the result is trivial, and if not, we can reduce to the bounded case by noting the following: $\tau ^{p\leqslant n}$ preserves $\mathrm {D}_{\mathrm {H}}(X)^{\mathrm {std}\leqslant n}$ , $\tau ^{\mathrm {std}\geqslant n}$ preserves $\mathrm {D}_{\mathrm {H}}(X)^{p\geqslant 0}$ , the restriction of $\tau ^{p\geqslant n}$ to $\mathrm {D}_{\mathrm {H}}(X)^{\mathrm {std}\geqslant n}$ is the identity and the restriction of $\tau ^{\mathrm {std}\leqslant n}$ to $\mathrm {D}_{\mathrm {H}}(X)^{p\leqslant n}$ is the identity. Here $\tau ^{\mathrm {std}}$ and $\tau ^{p}$ are the truncation functors for the standard and perverse t-structures, respectively.

Proof. The proof is essentially the same as the proofs of [Reference Liu and ZhengLZ17, 6.4.4 and 6.4.5]. The point 1. is easy. We show how to obtain 2. by hand, for example: Take $\pi :X\to \mathfrak {X}$ a presentation of $\mathfrak {X}$ and form the pullback

Then to show that each ${\ ^{\mathrm {std}}\mathrm {H}}^n(f_!K)$ are constructible, we may check it locally, whence as $\pi ^*$ is t-exact it suffices to show that ${\ ^{\mathrm {std}}\mathrm {H}}^n(\pi ^*f_!K)$ is constructible. By base change, it suffices to show that ${\ ^{\mathrm {std}}\mathrm {H}}^n(g_!q^*K)$ is constructible: we reduced to the case where $\mathfrak {X}$ is a scheme. Now, choose a presentation $r:Z\to \mathfrak {Z}$ of $\mathfrak {Z}$ . Descent for $!$ -functoriality implies that $q^*K = \mathrm {colim}_\Delta (g_n)_!r_n^!q^*K$ where $(g_n)$ is the composition of the $\check {\text {C}}$ ech nerve $(r_n)$ of r with g. The spectral sequence induced by this geometric realisation reads

$$ \begin{align*}E_1^{p,q}={\ ^{\mathrm{std}}\mathrm{H}}^{-q}((g_p)_!r_p^!q^*K)\Rightarrow {\ ^{\mathrm{std}}\mathrm{H}}^{p-q}(g_!q^*K).\end{align*} $$

Now the each $r_p^!q^*K$ is cohomologically constructible by purity (7. of Theorem 4.1.1) and each $g_p$ is a morphism of algebraic spaces, of relative dimension the sum of the relative dimensions of g and of r. In particular, all $g_p$ have the same cohomological amplitude thus our spectral sequence is concentrated in a shifted quadrant: it vanishes if $p<0$ and if q is smaller than some bound depending on the cohomological amplitude of $g_!$ and of K. This implies that the spectral sequence converges and we have that $\mathrm {H}^n(g_!q^*K)$ is constructible for all n. In the case where f is representable by algebraic spaces, we can reduce to the case of schemes where it follows from finite cohomological amplitude. The case of $f_*$ is similar.

Proof. Once the left-hand side has been identified with a $\mathrm {Hom}$ -group in $\mathrm {DM}$ by [Reference Khan and RaviKR24, Example 12.17] (or [Reference Khan and RaviKR24, Corollary 6.5] for the singular case), this is a consequence of the functoriality of the realisation functor.

Proof. The smooth projection formula implies that the canonical map

is an equivalence. Now, using the purity isomorphism twice $g^*\simeq g^!(-d)[-2d]$ we see that the right-hand side is equivalent to , and because $g^!\omega _{\mathfrak {X}}\simeq \omega _{\mathfrak {Y}}$ , the proof is finished.

Proof. First note that duality preserves $\mathrm {D}^b_{\mathrm {H},c}$ because this can be checked locally. There is a canonical map

$$ \begin{align*}\mathrm{Id}\to \mathbb{D}_{\mathfrak{X}}\circ\mathbb{D}_{\mathfrak{X}}.\end{align*} $$

For a given $M\in \mathrm {D}^b_{\mathrm {H},c}(\mathfrak {X})$ and a presentation $g:\mathfrak {X}\to X$ , it suffices to check that the map

$$ \begin{align*}g^*M\to g^*(\mathbb{D}_{\mathfrak{X}}\circ\mathbb{D}_{\mathfrak{X}}(M))\end{align*} $$

is an equivalence. But now by Lemma 4.2.2 the above map is an equivalence because it is the map

$$ \begin{align*}g^*M\to \mathbb{D}_X\circ\mathbb{D}_X(g^*M).\end{align*} $$

The remaining assertions follow from the autoduality over schemes ([Reference SaitoSai90b, Theorem 0.1]).

Proof. By the previous assertion, the functor $\mathbb {D}_{\mathfrak {X}}$ is perverse t-exact, and an isomorphism on the heart. As the perverse t-structure is left separated when restricted to $\mathrm {D}_{\mathrm {H},c}^+(\mathfrak {X})$ , this implies the claim.

Proof. By Theorem 4.1.1 we know the result if f is representable by algebraic spaces. Note that if f is a proper morphism of algebraic stacks, then the formation of $f_*$ is compatible with base change on $\mathrm {D}^+_{\mathrm {H,c}}$ . Indeed, using [Reference OlssonOls05, Theorem 1.1] the proof is the same as [Reference Laumon and Moret-BaillyLMB00, Théorème 18.5.1]: one uses the cohomological descent spectral sequence to reduce to the case of representable morphisms. Thus, by choosing a smooth atlas of the target of f, we can reduce to the case where $\mathfrak {X}$ is a scheme and $\mathfrak {Y}$ is a Deligne-Mumford stack. Now by [Reference Laumon and Moret-BaillyLMB00, Théorème 16.6], there is a finite covering $Y\to \mathfrak {Y}$ with Y a scheme, thus [Reference KhanKha19, Theorem A.7.] applies.

Proof. As both functors on stacks are defined using the right Kan extension on the category of correspondences (see the proof of [Reference MannMan22, Proposition A.5.16]), it is clear that the map $\mathrm {D}_{\mathrm {H}}\to \mathrm {D}_B$ commutes with tensor products, $*$ -pullbacks and $!$ -pushforwards. Point 4. is a direct consequence of 3. and Proposition 4.2.5. For and Verdier duality one can reduce to schemes directly using that $\mathrm {rat}$ commutes with twists and that both functors and $\mathbb {D}_{\mathfrak {X}}$ commute with $\pi ^*$ for $\pi $ a smooth morphism, up to a twist and a shift (by purity).

Now on constructibles, Verdier duality is an equivalence and swaps $*$ and $!$ ’s; thus points $1$ and $2$ follow from $2$ and $3$ of the above list.

Construction 4.3.1. Let $w\in \mathbb {Z}$ and X be a finite type $\mathbb {C}$ -scheme. Recall that following [Reference MorelMor08] we can construct a weight-t-structure on $\mathrm {D}^b_{\mathrm {H},c}(X)$ by setting

$$ \begin{align*}\mathrm{D}^b_{\mathrm{H},c}(X)^{\omega\leqslant w}:=\{K\in \mathrm{D}^b_{\mathrm{H},c}(X)\mid \forall n\in\mathbb{Z}, \forall m>w,\ \mathrm{gr}_m^W({\ ^{\mathrm{p}}\mathrm{H}}^n(K))=0\}\end{align*} $$

and

$$ \begin{align*}\mathrm{D}^b_{\mathrm{H},c}(X)^{\omega\geqslant w}:=\{K\in \mathrm{D}^b_{\mathrm{H},c}(X)\mid \forall n\in\mathbb{Z}, \forall m<w,\ \mathrm{gr}_m^W({\ ^{\mathrm{p}}\mathrm{H}}^n(K))=0\},\end{align*} $$

where the $\mathrm {gr}_m^W$ are the graded pieces of the weight filtration on mixed Hodge modules ([Reference SaitoSai89, Proposition 1.5]). The pair $(\mathrm {D}^b_{\mathrm {H},c}(X)^{\omega \leqslant w},\mathrm {D}^b_{\mathrm {H},c}(X)^{\omega \geqslant w+1})$ form a t-structure (of trivial heart) on $\mathrm {D}^b_{\mathrm {H},c}(X)$ , such that the truncation functors $\omega _{\leqslant w}$ and $\omega _{\geqslant w+1}$ are exact functors, t-exact for the perverse t-structure ([Reference MorelMor08, Proposition 3.1.1.]). They give a filtration $(\omega _{\leqslant w}K)_w$ on each complex K, that gives back the weight filtration on the heart. For any smooth map $f:Y\to X$ of finite type $\mathbb {C}$ -schemes, and any integer d, the functor $f^*[d]$ is t-exact for the weight t-structures (indeed, if d is the relative dimension of f we see that $f^*[d]$ is perverse t-exact and preserves weights, thus is t-exact for the weight-t-structure. The shift functor being weight-t-exact, this proves the claim). This induces a weight t-structure on $\mathrm {D}_{\mathrm {H}}(X)$ by indization and it has a similar description.

Proof. Once again assume that $\mathfrak {X}$ is connected, choose $\pi :X\to \mathfrak {X}$ a presentation of $\mathfrak {X}$ , of relative dimension d. Then if $f_n$ is a part of the $\check {\text {C}}$ ech nerve of $\pi $ , the functors $f_n^*[d]$ are t-exact for the weight t-structures, thus this induces a weight t-structure on the limit as in equation (4.1.0.1) (as in the proof of Proposition 4.1.4 we have to first deal with algebraic spaces but everything works the same). Of course, it restricts to $\mathrm {D}^b_{\mathrm {H},c}(\mathfrak {X})$ , and then to $\mathrm {D}_{\mathrm {H},c}(\mathfrak {X})$ . The perverse t-exactness of the weight truncation follows from the fact that each $f_n^*[d]$ is also perverse t-exact so that this follows from the same property over schemes.

Proof. By definition, if $\pi :X\to \mathfrak {X}$ is a smooth presentation, the functor $\pi ^*$ is conservative and detects weights. Thus, all results follow from the usual results over schemes (see [Reference SaitoSai89, Propositions 1.7 and 1.9]).

Proof. By choosing a smooth presentation $\pi $ of $\mathfrak {Y}$ , using proper base change, and the fact that $\pi ^*[d]$ is conservative, perverse t-exact and weight-t-exact, it detects punctual weights, so we may assume that $\mathfrak {Y}=Y$ is a separated variety. First note that for any stack $\mathfrak {X}$ , we have the formula

$$\begin{align*}\mathrm{D}_{\mathrm{H}}(\mathfrak{X})_{\leqslant w}=\{K\in \mathrm{D}_{\mathrm{H}}(\mathfrak{X})\mid \forall i\in\mathbb{Z}, {\ ^{\mathrm{p}}\mathrm{H}}^i(\omega^{w+i+1}K)=0\}.\end{align*}$$

In particular, as the functors $\omega ^{w+i+1}$ and ${\ ^{\mathrm {p}}\mathrm {H}}^i$ commute with filtered colimits, we see that $\mathrm {D}_{\mathrm {H}}(\mathfrak {X})_{\leqslant w}$ is stable under filtered colimits in $\mathrm {D}_{\mathrm {H}}(\mathfrak {X})$ .

By [Reference KreschKre99, Proposition 3.5.2 and Proposition 3.5.9] and the fact that $\mathfrak {X}$ is of finite type over k, there exists a finite stratification $\mathfrak {X}_i$ of $\mathfrak {X}$ by locally closed substacks such that each $\mathfrak {X}_i\simeq [X_i/G_i]$ is a global quotient stack of a quasi-projective variety $X_i$ by a smooth connected algebraic group $G_i$ acting linearly on $X_i$ . As $\mathrm {D}_{\mathrm {H}}(Y)_{\leqslant w}$ is closed under extensions, using the localisation triangle we see that we may assume that $\mathfrak {X}\simeq [X/G]$ is a global quotient stack with X quasi-projective.

By [Reference TotaroTot99, Remark 1.4] we may find a specific increasing sequence $(U_n)$ of representations of G (hence vector bundles on $BG$ ), called a Borel resolution, that helps compute the invariants of $[X/G]$ as follows: Let $X\overset {G}{\times }U_n := [X/G]\times _{BG} U_n$ , then for any $\mathbb {A}^1$ -invariant étale sheaf $F\colon \mathrm {Sm}^{\mathrm {op}}_{[X/G]}\to \mathcal {C}$ on the category of schemes that are smooth over $[X/G]$ with values in an $\infty $ -category $\mathcal {C}$ admitting all small limits, the canonical map

$$\begin{align*}F^\lhd([X/G])\to \lim_n F^\lhd(X\overset{G}{\times}U_n)\end{align*}$$

is an equivalence, where $F^\lhd $ is the right Kan extension of F to algebraic stacks. This is [Reference Khan and RaviKR24, Theorem 3.6]. Note that by [Reference Khan and RaviKR24, Remark 3.4], each $X\overset {G}{\times }U_n$ is a quasi-projective variety.

Let $K\in \mathrm {D}_{\mathrm {H}}([X/G])_{\leqslant w}$ . We apply the above to the functor F, opposite to the functor that sends a smooth map $g\colon Y\to [X/G]$ , with Y a variety, to $\pi ^Y_!g^!K$ . By $!$ -descent for the étale topology and $\mathbb {A}^1$ -invariance of $\mathrm {D}_{\mathrm {H}}$ this functor F satisfies the conditions stated in the previous paragraph. Moreover, its right Kan extension to stacks over $[X/G]$ has the same formula, again by $!$ -descent. In particular, we have that the map

$$\begin{align*}\mathrm{colim}_n (fp_n)_!p_n^!K\to f_!K\end{align*}$$

is an equivalence, where $p_n\colon X\times ^G U_n\to [X/G]$ is the projection. Note that this is a filtered colimit, so that it suffices to prove that each $({fp}_n)_!p_n^!K$ is of punctual weights $\leqslant w$ . But as $p_n$ is smooth, the weights of $p_n^!K$ are the weights of K, and the map $fp_n$ is a map of schemes, so that we have reduced to the case of schemes, where this is true by [Reference SaitoSai89, Proposition 1.7].

Proof. By duality, this follows from the case of $f_!$ .

Proof. We want to use [Reference BondarkoBon10, Theorem 4.3.2] to construct a weight structure from its heart of pure objects of weight 0. Thus we want to prove that pure objects of weight $0$ generated $\mathrm {D}^b_{\mathrm {H},c}(\mathfrak {X})$ as a thick subcategory, and that it is negative (for us, semi-simple will be enough). Thus if $K,L\in \mathrm {D}_{H}^b(\mathfrak {X})$ are pure of weight zero, we want to show that $\mathrm {Hom}_{\mathrm {D}_{H}^b(\mathfrak {X})}(K,L[1])$ vanishes. Let $f:\mathfrak {X}\to \operatorname {Spec}\mathbb {C}$ be the structural morphism. As we have

by Corollary 4.3.7 and the usual orthogonality in the derived category of mixed Hodge structures, it suffices to show that has weights $\geqslant 0$ , which results from Proposition 4.3.5.

Now objects of weight $0$ generate the $\infty $ -category under finite colimits. Indeed by dévissage it suffices to show that objects of the perverse heart which are pure are in the $\infty $ -category generated by the complexes of weight $0$ , but now this is only a shift. The last sentence of the corollary is now formal, see [Reference Beilinson, Bernstein and DeligneBBD82, Théorème 5.4.5] and [Reference Beilinson, Bernstein and DeligneBBD82, Théorème 5.3.8].

Proof. This is the combination of [Reference Cass, van den Hove and ScholbachCHS24, Corollary 2.20 and Lemma 2.17], which we reproduce for the reader’s convenience. First we remark that the functor $p_*$ indeed preserves Hodge Tate objects because $p_*\mathbb {Q}(n)\simeq \mathbb {Q}(n)\oplus \mathbb {Q}(n+1)[1]$ . Now, it is clear that $p_*$ preserves colimits as its left adjoint $p^*$ preserves compact objects, and it is also conservative because if $p_*M=0$ then for each $n\in \mathbb {Z}$ the mapping spectra $\mathrm {Map}_{\mathrm {D}_{HT}({\mathbb {G}_m}_{\mathbb {C}})}(\mathbb {Q}(n),M)\simeq \mathrm {Map}_{\mathrm {D}_{HT}(\mathbb {C})}(\mathbb {Q}(n),p_*M)$ vanishes, so that $M=0$ . Thus, by Barr-Beck’s theorem, the functor $p_*$ upgrades to an equivalence

$$ \begin{align*}\mathrm{D}_{\mathrm{H}}({\mathbb{G}_m}_{\mathbb{C}})\xrightarrow{\sim} \mathrm{Mod}_{p_*\mathbb{Q}}(\mathrm{D}_{\mathrm{H}}(\mathbb{C})).\end{align*} $$

There is a unique colimit preserving symmetric monoidal functor

$$ \begin{align*}i\colon\mathrm{Mod}_{\mathbb{Q}}^{\mathbb{Z}}\to \mathrm{D}_{HT}(\mathbb{C})\end{align*} $$

that sends $\mathbb {Q}(1)$ to $\mathbb {Q}(1)$ . The image under the Hodge realisation of [Reference Cass, van den Hove and ScholbachCHS24, Lemma 2.18] tells us that $p_*\mathbb {Q}$ is, as a commutative algebra object, the object $i(\Lambda )$ where $\Lambda $ is the split square zero extension $\Lambda :=\mathbb {Q}\oplus \mathbb {Q}(-1)[-1]$ . Thus we have that

$$ \begin{align*}\mathrm{D}_{HT}({\mathbb{G}_m}_{\mathbb{C}})\simeq \mathrm{Mod}_{p_*\mathbb{Q}}(\mathrm{D}_{HT}(\mathbb{C}))\simeq \mathrm{D}_{HT}(\mathbb{C})\otimes_{\mathrm{Mod}^{\mathbb{Z}}_{\mathbb{Q}}}\mathrm{Mod}_{\Lambda}(\mathrm{Mod}_{\mathbb{Q}}^{\mathbb{Z}}).\end{align*} $$

and the equivalence $\mathrm {Mod}_{\Lambda }(\mathrm {Mod}_{\mathbb {Q}}^{\mathbb {Z}})\simeq \operatorname {\mathrm {Nilp}}$ of [Reference Cass, van den Hove and ScholbachCHS24, Lemma 2.9] finishes the proof.

Proof. The proof is verbatim the same as their proof in [Reference Cass, van den Hove and ScholbachCHS24, Section 3.3]. The only thing to check is that over stacks, where our categories are not compactly generated (but over schemes everything is compactly generated hence, for example, mixed Hodge Tate categories are rigid), all functors in the play indeed preserve colimits. Only $j_*$ could be a problem, but the preservation of colimits can be checked on a presentation and it thus follows from smooth base change. Also, $\operatorname {\mathrm {Nilp}}\mathrm {D}_{Hs}$ stays a h-sheaf because $\operatorname {\mathrm {Nilp}}$ is compactly generated, hence the functor $\otimes _{\mathrm {Mod}_{\mathbb {Q}}^{\mathbb {Z}}}\operatorname {\mathrm {Nilp}}$ commutes with limits.

Proof. The functor

$$ \begin{align*}\mathrm{rat}\colon\mathrm{D}_{\mathrm{H}}\to\mathrm{D}_B\end{align*} $$

commutes with $\Upsilon _f$ by [Reference Cass, van den Hove and ScholbachCHS24, Lemma 3.18] (note that in the proof only is needed right adjointability for $j^*$ for j an open immersion which is in particular representable, so that we may use Proposition 4.2.6). On top of this, it detects constructibility and is conservative on constructible objects. Moreover, by taking a presentation $\pi :X\to \mathfrak {X}$ of $\mathfrak {X}$ , it suffices to prove the proposition for $\Upsilon _f$ with $f:X\to \mathbb {A}^1_{\mathbb {C}}$ a function on a scheme. Thus the proof of the proposition reduces to the case of a function of a scheme, on analytic sheaves. The comparison [Reference Cass, van den Hove and ScholbachCHS24, Example 3.19] of $\Upsilon $ in this case, with Beilinson’s unipotent nearby cycles functor [Reference BeilinsonBei87a] imply the result, because for analytic sheaves the functor $\Upsilon _f[-1]$ is perverse t-exact and commutes with the external tensor product.

Proof. This is [Reference Cass, van den Hove and ScholbachCHS24, Proposition 3.9], shifted by $-1$ .

Proof. As both functors are t-exact it suffices to construct an isomorphism on $\mathrm {MHM}(U)$ . By [Reference SaitoSai90a, Proposition 1.3] for $K\in \mathrm {MHM}(U)$ there is a canonical isomorphism

$$ \begin{align*}\mathrm{colim}_n{\ ^{\mathrm{p}}\mathrm{H}}^{-1}i^*j_*(K\otimes f^*E_n)\xrightarrow{\sim} \psi_{f,1}K,\end{align*} $$

where $E_n = \mathbb {Q}\oplus \cdots \mathbb {Q}(-n)\in \mathrm {D}_{\mathrm {H}}({\mathbb {G}_m})$ . The proof of the comparison with Beilinson construction [Reference Cass, van den Hove and ScholbachCHS24, Example 3.19] gives in fact that

$$ \begin{align*}\Upsilon_f(K)\simeq \mathrm{colim}_n i^*j_*(K\otimes f^*E_n),\end{align*} $$

so that the compatibility with colimits of the t-structure gives the result.

Proof. The map is defined as follows: $\Upsilon _f$ is lax monoidal, hence there is a canonical map

that we can compose with the evaluation map

to obtain a map

By adjunction, this provides a map

Now, the commutation of $\Upsilon $ with $!$ -pushforwards implies that there is a exchange map

$$\begin{align*}\Upsilon_f\circ f_\eta^!\to f_\sigma^!\circ\Upsilon_{\mathrm{Id}},\end{align*}$$

and by [Reference Cass, van den Hove and ScholbachCHS24, Lemma 3.12] (to which we apply the Hodge realisation), we have that $\Upsilon _{\mathrm {Id}}(\mathbb {Q})\simeq \mathbb {Q}$ . This gives a map

in which on the right-hand side we recognise (because $f_\sigma $ is the structural map of $\mathfrak {Z}$ ) the functor $\mathbb {D}_{\mathfrak {Z}}\circ \Upsilon _f$ . The dualising object on $\mathfrak {U}$ is $\omega _{\mathfrak {U}} = f_\eta ^!\pi _{{\mathbb {G}_m}}^!\mathbb {Q} \simeq f_\eta ^!\mathbb {Q}(1)[2]$ , with $\pi _{\mathbb {G}_m}$ the structural map of ${\mathbb {G}_m}$ . Using that $\Upsilon _f$ commutes with Tate twists (this follows easily from the projection formula and the fact that it commutes with $p_!$ , for $p\colon \mathfrak {X}\times \mathbb {P}^1\to \mathfrak {X}$ ), we finally have a map

$$\begin{align*}\Upsilon_f\circ\mathbb{D}_{\mathfrak{U}}(-) \to \mathbb{D}_{\mathfrak{Z}}\circ\Upsilon_f(-)(1)[2].\end{align*}$$

This comparison map is an isomorphism on $\mathrm {D}^b_{\mathrm {H}}$ because this can be checked locally hence on finite type $\mathbb {C}$ -schemes, where it has been proven by Saito (alternatively, one can apply the functor to analytic sheaves and use the result of Beilinson).

Proof. First, because the t-structure is compatible with colimits and each $e_n^*$ is perverse t-exact, it is clear that $\Psi _f[-1]$ is perverse t-exact on $\mathrm {D}(\mathrm {Ind}\mathrm {MHM}(U))$ . Thus, by dévissage it suffices to construct an isomorphism $\Psi _f[-1]\simeq \Psi _f^S$ of exact functors on the heart $\mathrm {MHM}(U)$ .

By definition ([Reference SaitoSai06, Proposition 5.7]) together with the comparison of Proposition 4.4.6, for $M\in \mathrm {MHM}(U)$ we have

$$ \begin{align*}\Psi_f^S(M)=\Upsilon_f(M)[-1]=\Upsilon_f(M\otimes f_n^*(\pi_n)_*\mathbb{Q})[-1]\end{align*} $$

for n divisible enough, in other terms,

$$ \begin{align*}\Psi_f^S = \mathrm{colim}_n\Upsilon_f(-\otimes f_n^*(\pi_n)_*\mathbb{Q})[-1].\end{align*} $$

For each $n\in \mathbb {N}^*$ because $e_n$ is proper we have $(i_n)_*\Upsilon _{f_n}\simeq \Upsilon _f\circ (e_n)_*$ and now the proper projection formula and proper base change give $(e_n)_*\simeq (-\otimes f_n^*(\pi _n)_*\mathbb {Q})$ , finishing the proof.

Proof. Thanks to the same property for $\Upsilon $ the functor $\Psi $ commutes with smooth pullback, hence all those properties are local for the smooth topology and can be checked on a separated finite type $\mathbb {C}$ -scheme, where they follow from the results on the underlying perverse sheaves and the comparison with Saito’s functor. The last assertion follows from the result on the bounded category and t-exactness.

Proof. This can be checked on a smooth atlas, thus on schemes where this holds for in constructible sheaves $\mathrm {D}^b_c(-,\mathbb {Q})$ .

Proof. First, if G acts freely on X (for the definition of a free action, see [Reference AcharAch13, Definition 6.4]), then $[X/G]$ is a scheme and the result is trivial. Then exactly for any n-acyclic map $U\to X$ such that the action of G on U is free and $a\leqslant b$ integers such that $b-a<n$ , the same proof as [Reference AcharAch13, Lemma 8.1] implies that the functor

$$ \begin{align*}\mathrm{D}_{\mathrm{H}}^{[a,b]}([X/G])\to\mathrm{D}_{\mathrm{H}}^{[a,b]}([U/G])\end{align*} $$

is fully faithful. Thus, the functor

$$ \begin{align*}\mathrm{ho}\mathrm{D}^b_{\mathrm{H},c}([X/G])\to\mathrm{D}_G(X)\end{align*} $$

is fully faithful. It is not hard to check that the heart of $\mathrm {D}^b_{\mathrm {H},c}([X/G])$ is also the abelian category of G-equivariant mixed Hodge modules (because they satisfy descent), so that this finishes the proof.

Proof. First note that the counit map

$$ \begin{align*}(b_i)_!(b_i)^!\mathbb{Q}_{\mathfrak{X}}\to\mathbb{Q}_{\mathfrak{X}}\end{align*} $$

is an equivalence by $\mathbb {A}^1$ -invariance, because $b_i$ is a vector bundle on $\mathfrak {X}$ . As $b_i$ is smooth of relative dimension $ir$ , purity gives that the canonical map

$$ \begin{align*}(q_i)_!\mathbb{Q}_{\mathfrak{V}_i}\{-ir\}\to p_!\mathbb{Q}_{\mathfrak{X}}\end{align*} $$

is an equivalence for each $i\in \mathbb {N}$ . Thus it suffices to prove that the map

$$ \begin{align*}\mathrm{colim}_i (a_i)_!\mathbb{Q}_{\mathfrak{U}_i}\{-ir\} \to\mathrm{colim}_i (b_i)_!\mathbb{Q}_{\mathfrak{V}_i}\{-ir\}\end{align*} $$

on $\mathfrak {X}$ induced by the counits $(\iota _i)_!\iota _i^*\to \mathrm {Id}$ is an equivalence, because applying $p_!$ would produce the sought isomorphism.

Denote by $\pi :X\to \mathfrak {X}$ the projection. Recall that $\pi ^*$ is conservative, proper base change ensures that it suffices to deal with the analogous situation over X

where every thing has been pulled back to X, and the map is now

$$ \begin{align*}\mathrm{colim}_i (\alpha_i)_!\mathbb{Q}_{{U}_i}\{-ir\} \to\mathrm{colim}_i (\beta_i)_!\mathbb{Q}_{{V}_i}\{-ir\}.\end{align*} $$

Everything is now a scheme. The family of pullbacks $(x^*)_{x\in X}$ is conservative on $\operatorname {\mathrm {DM}_{\mathrm {\acute {e}t}}}(X,\mathbb {Q})$ by localisation, thus by proper base change we reduce further to the case where $X = \operatorname {Spec} k$ is the spectrum of a field, and each $V_i$ is a smooth k-scheme (in fact, a vector space). Furthermore, the codimension of $W_i:= V_i\setminus U_i$ in $V_i$ goes to $\infty $ when $i\to \infty $ , and the smoothness of $\alpha _i$ and $\beta _i$ ensure that

$$ \begin{align*}(\alpha_i)_!\alpha_i^*\mathbb{Q}\{-ir\} \simeq (\alpha_i)_\sharp\alpha_i^*\mathbb{Q}\simeq M(U_i)\in\operatorname{\mathrm{DM}_{\mathrm{\acute{e}t}}}(k,\mathbb{Q}),\end{align*} $$

and the same for $(\beta _i)_!\mathbb {Q}\{-ir\}=M(V_i)$ , where for Y a smooth k-scheme, the object $M(Y)$ is the motive of Y. We are looking at

$$ \begin{align*}\mathrm{colim}_i M(U_i)\to\mathrm{colim}_i M(V_i)\end{align*} $$

in $\operatorname {\mathrm {DM}_{\mathrm {\acute {e}t}}}(k,\mathbb {Q})$ , with $\mathrm {codim}_{V_i}(V_i\setminus U_i)\underset {i\to \infty }{\longrightarrow }\infty $ . By [Reference Hoskins and Pepin LehalleurHPL21, Proposition 2.13] this is an equivalence.

Proof. The Hodge realisation

$$ \begin{align*}\rho_{\mathrm{H}}\colon\operatorname{\mathrm{DM}_{\mathrm{\acute{e}t}}}\to\mathrm{D}_{\mathrm{H}}\end{align*} $$

extends naturally to stacks, in a way that commutes with the operations and colimits, thus the first statement is just the Hodge realisation applied to Proposition 4.5.2. The second statement follows from the fact that the t-structure on $\mathrm {D}_{\mathrm {H}}(\mathcal {M})$ is compatible with filtered colimits and that each ${\ ^{\mathrm {p}}\mathrm {H}}^n(p_!\mathbb {Q}_{\mathfrak {X}})$ is constructible thus Noetherian.