Notation 1.1.1. Given a small $\infty $ -category $\mathcal {C}$ , we denote by $\mathcal {P}(\mathcal {C})$ the $\infty $ -category of $\mathcal {S}$ -valued presheaves on $\mathcal {C}$ . (As usual, $\mathcal {S}$ is the $\infty $ -category of spaces, aka., homotopy types.) If $\mathcal {C}$ is endowed with a Grothendieck topology $\tau $ , we denote by $\mathcal {P}^{(\wedge )}_{\tau }(\mathcal {C})$ the full sub- $\infty $ -category of $\mathcal {P}(\mathcal {C})$ spanned by the $\tau $ -(hyper)sheaves. More generally, if $\mathcal {D}$ is another $\infty $ -category, we denote by $\mathrm {Psh}(\mathcal {C};\mathcal {D})$ the $\infty $ -category of $\mathcal {D}$ -valued presheaves on $\mathcal {C}$ and by $\mathrm {Shv}_{\tau }^{(\wedge )}(\mathcal {C};\mathcal {D})$ its full sub- $\infty $ -category spanned by $\tau $ -(hyper)sheaves. When $\mathcal {D}=\mathrm {Mod}_{\Lambda }$ is the $\infty $ -category of $\Lambda $ -modules for some commutative ring spectrum $\Lambda \in \mathrm {CAlg}$ , we write $\mathrm {Psh}(\mathcal {C};\Lambda )$ and $\mathrm {Shv}_{\tau }^{(\wedge )}(\mathcal {C};\Lambda )$ instead. If M is a $\Lambda $ -module, we denote by $M_{\mathrm {cst}}$ the associated constant presheaf and (hyper)sheaf on $\mathcal {C}$ . There are obvious functors

(1.1) $$ \begin{align} \Lambda(-):\mathcal{P}(\mathcal{C}) \to \mathrm{Psh}(\mathcal{C};\Lambda) \qquad \text{and} \qquad \Lambda_{\tau}(-):\mathcal{P}_{\tau}^{(\wedge)}(\mathcal{C}) \to \mathrm{Shv}_{\tau}^{(\wedge)}(\mathcal{C};\Lambda), \end{align} $$

induced by the unique colimit-preserving functor $\mathcal {S}\to \mathrm {Mod}_{\Lambda }$ sending the one-point space to $\Lambda $ . We also denote by $\Lambda (-)$ and $\Lambda _{\tau }(-)$ the precompositions of the functors in (1.1) with the Yoneda embedding and its (hyper)sheafified version (which fails to be an embedding in general).

Notation 1.1.8. Let X be a smooth S-scheme. We set

$$ \begin{align*}\mathrm{M}^{\mathrm{eff}}(X)=\mathrm{L}_{\mathbb{A}^1}(\Lambda_{\acute{\mathrm{e}}\mathrm{t}}(X)) \qquad \text{and} \qquad \mathrm{M}(X)= \Sigma_{\mathrm{T}}^{\infty}\mathrm{M}^{\mathrm{eff}}(X)= \Sigma_{\mathrm{T}}^{\infty}\mathrm{L}_{\mathbb{A}^1}(\Lambda_{\acute{\mathrm{e}}\mathrm{t}}(X)).\end{align*} $$

These are objects of $\mathrm {MSh}^{\mathrm {eff}}(S;\Lambda )$ and $\mathrm {MSh}(S;\Lambda )$ , respectively. The object $\mathrm {M}(X)$ is called the homological motivic sheaf associated to X.

Notation 1.1.9. We denote by $\Lambda $ (or $\Lambda _S$ ) the monoidal unit of $\mathrm {MSh}(S;\Lambda )^{\otimes }$ . For $n\in \mathbb {N}$ , we denote by $\Lambda (n)$ the image of $\mathrm {T}^{\otimes n}_S[-n]$ by $\Sigma _{\mathrm {T}}^{\infty }$ , and by $\Lambda (-n)$ the $\otimes $ -inverse of $\Lambda (n)$ . For $n\in \mathbb {Z}$ , we denote by $M\mapsto M(n)$ the Tate twist given by tensoring with $\Lambda (n)$ .

Proof. See [Reference Ayoub, Gallauer and VezzaniAGV22, Lemma 3.1.6].

Notation 1.1.11. Let S be a scheme and $\Lambda \in \mathrm {CAlg}$ a commutative ring spectrum. We denote by $\mathrm {MSh}^{(\mathrm {eff})}(S;\Lambda )^{\varpi }$ the smallest full sub- $\infty $ -category of $\mathrm {MSh}^{(\mathrm {eff})}(S;\Lambda )$ containing the motivic sheaves $\mathrm {M}^{(\mathrm {eff})}(X)$ , with $X\in {\mathrm {Sm}}_S$ , and stable under finite limits, finite colimits, direct summands and negative Tate twists when applicable. In [Reference AyoubAyo07a, Definition 2.2.3], objects in $\mathrm {MSh}(S;\Lambda )^{\varpi }$ were called constructible, but we will avoid this terminology in this paper. Instead, we will call them motivic sheaves of finite generation.

Proof. See [Reference Ayoub, Gallauer and VezzaniAGV22, Proposition 2.4.22 & Remark 2.4.23].

Proof. By Lemma 2.1.16 below, the exponent characteristic of k is automatically inverted in $\mathrm {MSh}(k;\Lambda )$ . Thus, we may assume that the exponent characteristic of k is invertible in $\pi _0\Lambda $ . The result then follows from [Reference Elmanto and KhanEK20, Theorem 3.2.1], which is based on [Reference Bondarko and DégliseBD17]; see [Reference AyoubAyo26, Lemma 1.12]. If k has characteristic zero, one can use instead [Reference RiouRio05].

Proof. We refer to [Reference RobaloRob15] for the construction of such a functor.

Notation 1.1.15. Let $f:Y\to X$ be a morphism in $\mathrm {Sch}_S$ . The image of f by the functor (1.3) is the symmetric monoidal functor

$$ \begin{align*}f^*:\mathrm{MSh}^{(\mathrm{eff})}(X;\Lambda)^{\otimes}\to \mathrm{MSh}^{(\mathrm{eff})}(Y;\Lambda)^{\otimes}\end{align*} $$

whose underlying functor, also denoted by $f^*$ , is called the inverse image functor. The latter has a right adjoint $f_*$ , called the direct image functor.

Proof. All the axioms in Definition 1.1.16 are direct consequences of the construction, except for axiom (4), which is a reformulation of the Morel–Voevodsky localisation theorem [Reference Morel and VoevodskyMV99, §3.2, Theorem 2.21]. For a fully detailed exposition, see [Reference AyoubAyo07b, §4.5.2 & 4.5.3]. For a modern treatment of the Morel–Voevodsky localisation theorem, see [Reference HoyoisHoy18, §1]. For the second assertion, we note that the assumptions on S ensure that every S-scheme of finite presentation is $\Lambda $ -good. Proposition 1.1.12 therefore implies that, for every $X\in \mathrm {Sch}_S$ , the $\infty $ -category $\mathrm {MSh}(X;\Lambda )$ is compactly generated by the motivic sheaves $\mathrm {M}(Y)$ , with $Y\in {\mathrm {Sm}}_X$ , together with their desuspensions and negative twists. As tensor products and inverse image functors preserve this class of generators, the result follows.

Proof. Let $f:Y \to X$ be a morphism in $\mathrm {Sch}_S$ . We need to show that $f_*:\mathrm {MSh}(Y;\Lambda ) \to \mathrm {MSh}(X;\Lambda )$ is colimit-preserving. By [Reference RobaloRob15, Corollary 2.22], it would be enough to show that the functors

$$ \begin{align*}f_*:\mathrm{MSh}^{\mathrm{eff}}(Y;\Lambda) \to \mathrm{MSh}^{\mathrm{eff}}(X;\Lambda) \quad \text{and} \quad \Omega^1_{\mathrm{T}}:\mathrm{MSh}^{\mathrm{eff}}(Z;\Lambda) \to \mathrm{MSh}^{\mathrm{eff}}(Z;\Lambda),\end{align*} $$

with $Z\in \{X,Y\}$ , are colimit-preserving. The case of $\Omega ^1_{\mathrm {T}}$ follows from the case of the direct image functor along the projection $\mathbb {A}^1_Z\smallsetminus 0_Z\to Z$ . Thus, we only need to consider the case of $f_*$ . It is enough to show that the non- $\mathbb {A}^1$ -localised direct image functor

(1.4) $$ \begin{align} f_*:\mathrm{Shv}^{\wedge}_{\acute{\mathrm{e}}\mathrm{t}}({\mathrm{Sm}}_Y;\Lambda) \to \mathrm{Shv}^{\wedge}_{\acute{\mathrm{e}}\mathrm{t}}({\mathrm{Sm}}_X;\Lambda) \end{align} $$

is colimit-preserving. (Indeed, this functor takes morphisms of the form $\Lambda _{\acute {\mathrm {e}}\mathrm {t}}(\mathbb {A}^1_V) \to \Lambda _{\acute {\mathrm {e}}\mathrm {t}}(V)$ , for $V\in {\mathrm {Sm}}_Y$ , to $\mathbb {A}^1$ -equivalences, as can be shown using the explicit $\mathbb {A}^1$ -homotopy

$$ \begin{align*}\Lambda_{\acute{\mathrm{e}}\mathrm{t}}(\mathbb{A}^1_X)\otimes f_*\Lambda_{\acute{\mathrm{e}}\mathrm{t}}(\mathbb{A}^1_V) \to f_*\Lambda_{\acute{\mathrm{e}}\mathrm{t}}(\mathbb{A}^2_V) \xrightarrow{m} f_*\Lambda_{\acute{\mathrm{e}}\mathrm{t}}(\mathbb{A}^1_V),\end{align*} $$

where $m:\mathbb {A}^2\to \mathbb {A}^1$ is the multiplication.)

To show that the functor (1.4) is colimit-preserving, we need to show that, for every $U\in {\mathrm {Sm}}_X$ , the direct image functor for the small étale sites, associated to the projection $Y\times _X U \to U$ , is colimit-preserving. Replacing X and Y by U and $Y\times _X U$ , we are led to show that

$$ \begin{align*}f_*:\mathrm{Shv}_{\acute{\mathrm{e}}\mathrm{t}}^{\wedge}(\acute{\mathrm{E}}\mathrm{t}_Y;\Lambda) \to \mathrm{Shv}_{\acute{\mathrm{e}}\mathrm{t}}^{\wedge}(\acute{\mathrm{E}}\mathrm{t}_X;\Lambda)\end{align*} $$

is colimit-preserving. Replacing $\Lambda $ with $\tau _{\geq 0}\Lambda $ , we may assume that $\Lambda $ is connective. Since $\Lambda $ is eventually coconnective, we can find an integer $N\geq 0$ such that $\pi _i\Lambda =0$ for $i\geq N+1$ . If $N\geq 1$ and the result holds for $\tau _{\leq N-1}\Lambda $ , we may use the exact triangle

$$ \begin{align*}f_*(-\otimes_{\Lambda}\tau_{\geq N}\Lambda) \to f_*(-) \to f_*(-\otimes_{\Lambda}\tau_{\leq N-1}\Lambda)\end{align*} $$

and the fact that $\tau _{\geq N}\Lambda $ is a $\tau _{\leq N-1}\Lambda $ -module to conclude. Thus, by induction on N, we can reduce to the case where $\Lambda $ is an ordinary commutative ring. This case was treated in [Reference AyoubAyo14a, Lemme 4.2]. Note that the assumption in loc. cit. is indeed satisfied for any noetherian scheme S by [Reference Illusie, Laszlo and OrgogozoILO14, Exposé XVIII ${}_{\mathrm{A}}$ , Corollary 1.2].

Notation 1.2.3. Let W be a topological space and $\Lambda \in \mathrm {CAlg}$ a commutative ring spectrum. We denote by $\mathrm {Sh}(W;\Lambda )^{\otimes }$ the symmetric monoidal $\infty $ -category of sheaves on W with coefficients in $\Lambda $ . In other words, we set

$$ \begin{align*}\mathrm{Sh}(W;\Lambda)=\mathrm{Shv}^{\wedge}_{\mathrm{cl}}(\mathrm{Op}(W);\Lambda)\end{align*} $$

with $(\mathrm {Op}(W),\mathrm {cl})$ the site of opens in W with the open cover topology.

Proof. The first equivalence is the content of [Reference AyoubAyo10, Théorèmes 1.8]. The second equivalence follows from [Reference AyoubAyo10, Lemme 1.10] showing that $\mathrm {T}_V$ is already $\otimes $ -invertible in $\mathrm {AnSh}^{\mathrm {eff}}(V;\Lambda )$ .

Proof. One argues as in [Reference RobaloRob15, §9.1] for the first assertion. The second assertion is the content of [Reference AyoubAyo10, Proposition 2.5].

Proof. The implication (ii) $\Rightarrow $ (i) is obvious and the implication (iii) $\Rightarrow $ (ii) follows from the classical fact that a complex analytic variety is locally contractible. (See [Reference HardtHar75, Corollary 3.4] for a much more general result.) The implication (ii) $\Rightarrow $ (iii) is standard, but we include an argument for the reader’s convenience. Assume that (ii) is satisfied. Replacing W by U, we may assume that W is contractible. In particular, it is connected and all the fibres of F are equivalent to a fixed object $F_0\in \mathrm {Mod}_{\Lambda }$ . Let $Q=\mathrm {Eqv}_{\mathrm {Mod}_{\Lambda }}(F_0,F)$ be the hypersheaf on W sending a connected open subset $V\subset W$ to the space of equivalences from $F_0$ to $F(V)$ . Then, Q admits an action of the group space $G=\mathrm {Auteq}_{\mathrm {Mod}_{\Lambda }}(F_0)$ . In fact, condition (ii) implies that Q is a G-torsor over W. By [Reference ŁojasiewiczŁoj64], W is a CW-complex, and this ensures that the G-torsor Q is classified by a map $W\to \mathrm {B}(G)$ , with $\mathrm {B}(G)$ the classifying space of G. Since W is contractible, every such map is null-homotopic, which implies that the G-torsor Q is necessarily trivial. A global section of Q yields an equivalence between F and the constant sheaf with value $F_0$ on W. This proves (iii).

It remains to show the implication (i) $\Rightarrow $ (ii). Assume that F is dualizable. We fix a point $x\in W$ , and we show that F is constant in the neighbourhood of x. The functor $A\mapsto A_x$ is symmetric monoidal. This implies that the $\Lambda $ -module $F_x$ is dualizable and hence perfect and compact. Writing $F_x = \mathrm {colim}_{x\in U} F(U)$ , where the colimit is over the open neighbourhoods of x, we deduce that the identity of $F_x$ lifts to a morphism of $\Lambda $ -modules $F_x\to F(U)$ , for U small enough. Replacing W by U, we obtain a morphism $(F_x)_{\mathrm {cst}} \to F$ from the constant sheaf $(F_x)_{\mathrm {cst}}$ with value $F_x$ , inducing the identity on the stalks at x. The cofibre G of this morphism is still dualizable, and it is enough to show that it is zero in the neighbourhood of x. Let $G^{\vee }$ be the dual of G, and consider the unit morphism $\eta :\Lambda _W \to G\otimes _{\Lambda } G^{\vee }$ . Restricting to an open neighbourhood U of x and passing to global sections, we obtain a morphism of $\Lambda $ -modules

(1.10) $$ \begin{align} \Lambda\to \Gamma(U;G\otimes_{\Lambda} G^{\vee}). \end{align} $$

Taking the colimit over U, we obtain the map $\Lambda \to G_x\otimes _{\Lambda }G^{\vee }_x \simeq 0$ . Since $\Lambda $ is compact, we deduce that the morphism (1.10) is zero for U small enough. Then, the unit map $\Lambda _U\to G|_U\otimes _{\Lambda } G^{\vee }|_U$ is zero, which implies that $G|_U$ is zero as needed.

Proof. To show that $\mathrm {LS}(X;\Lambda )^{\omega }$ is preserved by the canonical truncation functors, we use the second characterisation of local systems given in Lemma 1.2.9 and the fact that perfect $\Lambda $ -modules are preserved by the canonical truncation functors when $\Lambda $ is regular. To show the same property for $\mathrm {Sh}_{\mathrm {ct}}(X;\Lambda )^{\omega }$ , we use the fact that the pullback functors for $\mathrm {Sh}((-)^{\mathrm {an}};\Lambda )$ are t-exact. The remaining assertions are clear.

Proof. Let $\mathcal {T}\subset \mathrm {Sh}(X^{\mathrm {an}};\Lambda )$ be the full sub- $\infty $ -category generated under colimits by $\mathrm {LS}(X^{\mathrm {an}};\Lambda )^{\omega }$ . The functor (1.11) lands in $\mathcal {T}$ and restricts to the obvious inclusion on $\mathrm {LS}(X;\Lambda )^{\omega }= \mathrm {LS}(X^{\mathrm {an}};\Lambda )^{\omega }$ . Thus, it is enough to show that the objects of $\mathrm {LS}(X^{\mathrm {an}};\Lambda )^{\omega }$ are compact objects of $\mathcal {T}$ .

Since local systems are dualizable, we are reduced to showing that $\Lambda _{\mathrm {cst}}$ is a compact object of $\mathcal {T}$ or, equivalently, that the functor $\Gamma (X^{\mathrm {an}};-):\mathcal {T} \to \mathrm {Mod}_{\Lambda }$ commutes with direct sums. Let $(M_{\alpha })_{\alpha }$ be a family of objects in $\mathcal {T}$ and $M=\bigoplus _{\alpha } M_{\alpha }$ . We need to show that the natural map

(1.12) $$ \begin{align} \bigoplus_{\alpha}\Gamma(X^{\mathrm{an}};M_{\alpha})\to \Gamma(X^{\mathrm{an}};M) \end{align} $$

is an equivalence. Both sides of the map (1.12) satisfy cdh excision on X. Using resolution of singularities in characteristic zero [Reference HironakaHir64a, Reference HironakaHir64b], we reduce to the case where X is smooth, admitting an open immersion $j:X\hookrightarrow \overline {X}$ with $\overline {X}$ smooth and proper, and $\overline {X}\smallsetminus j(X)$ a strict normal crossings divisor. By [Reference LurieLur09, Corollary 7.3.4.12], the functor $\Gamma (\overline {X}{}^{\mathrm {an}};-)$ commutes with direct sums. Thus, we are left to show that the morphism

(1.13) $$ \begin{align} \bigoplus_{\alpha} j_*M_{\alpha} \to j_*M \end{align} $$

is an equivalence in $\mathrm {Sh}(\overline {X}{}^{\mathrm {an}};\Lambda )$ . We check this on stalks, and fix a point $x\in \overline {X}{}^{\mathrm {an}}$ . For U in a cofinal system of neighbourhoods of x in $\overline {X}{}^{\mathrm {an}}$ , we can find isomorphisms

$$ \begin{align*}U\simeq \mathbb{D}^{m+n} \qquad \text{and} \qquad j^{-1}(U)\simeq \mathbb{D}^m\times (\mathbb{D}^1\smallsetminus \{0\})^n,\end{align*} $$

for some integers $m,n\geq 0$ . In particular, $j^{-1}(U)$ is then equivalent to the classifying space $\mathrm {B}(\mathbb {Z}^n)$ of $\mathbb {Z}^n$ . It is enough to show that

(1.14) $$ \begin{align} \bigoplus_{\alpha} \Gamma(j^{-1}(U),M_{\alpha}|_{j^{-1}(U)}) \to \Gamma(j^{-1}(U),M|_{j^{-1}(U)}) \end{align} $$

is an equivalence for such U’s. By Lemma 1.2.9, the sheaves $M_{\alpha }|_{j^{-1}(U)}$ and their direct sum $M|_{j^{-1}(U)}$ are constant over any contractible open subvariety of $j^{-1}(U)$ . The $\infty $ -category of such sheaves is equivalent to the $\infty $ -category of $\Lambda $ -modules endowed with an action of $\mathbb {Z}^n$ ; such an equivalence is given by taking the stalk at a base point $o\in j^{-1}(U)$ and endowing it with the natural action of $\mathbb {Z}^n\simeq \pi _1(j^{-1}(U),o)$ . Denote by $N_{\alpha }$ the stalk of $M_{\alpha }|_{j^{-1}(U)}$ and set $N=\bigoplus _{\alpha } N_{\alpha }$ . Then, the morphism (1.14) is equivalent to

$$ \begin{align*}\bigoplus_{\alpha}\Gamma(\mathbb{Z}^n;N_{\alpha}) \to \Gamma(\mathbb{Z}^n;N),\end{align*} $$

where $\Gamma (\mathbb {Z}^n;-)=(-)^{\mathbb {Z}^n}$ is the derived functor of invariants under the action of $\mathbb {Z}^n$ . Thus, we are reduced to showing that $\Gamma (\mathbb {Z}^n;-)$ commutes with direct sums. By induction, we may assume that $n=1$ . The result follows then from the fact that $\Gamma (\mathbb {Z};-)$ can be computed as the equaliser of the identity map and the action of $1\in \mathbb {Z}$ .

Proof. This is a well-known fact, at least when $\Lambda $ is an ordinary ring. For the reader’s convenience, we include a proof. To simplify notation, we shall write $f^*$ , $f_*$ , etc., instead of $f^{\mathrm {an},\,*}$ , $f^{\mathrm {an}}_*$ , etc. We split the proof into several steps.

Step 1. The result is obvious for the ordinary pullback and the tensor product. In this step, we assume that the result is also known for the ordinary direct image functors, and explain how to derive the result for the remaining operations.

Given a morphism f of k-varieties, to show that $f_!$ preserves constructible sheaves, we may assume that f is an open immersion or a proper morphism. The case of open immersions is clear and, in the case of proper morphisms, we use that $f_! \simeq f_*$ to conclude. Similarly, to prove that $f^!$ preserves constructible sheaves, we may assume that f is smooth or a closed immersion. If f is smooth, we conclude using that $f^!$ is equivalent to $f^*$ up to a twist and shift. If f is a closed immersion, we use the localisation triangle and the assumption that ordinary direct images along open immersions preserve constructible sheaves. Finally, to prove that $\underline {\mathrm {Hom}}(F,G)$ is constructible if $F,G\in \mathrm {Sh}(X^{\mathrm {an}};\Lambda )$ are constructible, we may assume that F is of the form $u_!L$ , where $u:Z \to X$ is a locally closed immersion and L is a local system on $Z^{\mathrm {an}}$ . In this case, we have equivalences

$$ \begin{align*}\underline{\mathrm{Hom}}(u_!L,G)\simeq u_* \underline{\mathrm{Hom}}(L,u^!G) \simeq u_*(L^{\vee}\otimes_{\Lambda} u^!G)\end{align*} $$

and the result follows from the previous considerations.

Step 2. It remains to prove that $f_*$ preserves constructible sheaves for every morphism $f:Y \to X$ of k-varieties.

We argue by induction on the dimension of Y. Let F be a constructible sheaf on Y. We may find a dense open subvariety $V\subset Y$ , which is smooth up to nil-immersion, and such that $F|_V$ is a local system on $V^{\mathrm {an}}$ . Let $v:V\hookrightarrow Y$ be the obvious inclusion and $s:Z\to Y$ the complementary closed immersion. We have an exact triangle

$$ \begin{align*}f_*F \to f_*v_*v^*F\to f_*s_*s^*C \to\end{align*} $$

with $C=\mathrm {cofib}(F \to v_*v^*F)$ . Applying the induction hypothesis to $f\circ s:Z \to X$ , we see that $f_*F$ is constructible if $(f\circ v)_*F|_V$ and $v_*F|_V$ are constructible. This proves that it is enough to treat the case where Y is smooth and F is a local system on $Y^{\mathrm {an}}$ .

Step 3. Using resolution of singularities in characteristic zero [Reference HironakaHir64a, Reference HironakaHir64b], we may find an open immersion $j:Y \to \overline {Y}$ over X, with $\overline {Y}$ smooth and $D=\overline {Y}\smallsetminus Y$ a strict normal crossings divisor. In this step, we prove that $j_*F$ is constructible. (Recall that F is a local system on Y, by Step 2.)

Let $D_1,\ldots , D_m$ be the irreducible components of D and, for $I\subset \{1,\ldots ,m\}$ nonempty, let

$$ \begin{align*}D_I=\bigcap_{a\in I} D_a \qquad \text{and}\qquad D_I^{\circ}=D_I\smallsetminus \bigcup_{b\not\in I}D_b.\end{align*} $$

We claim that the restriction of $j_*F$ to $D^{\circ }_I$ is a local system. Indeed, every point of $(D^{\circ }_I)^{\mathrm {an}}$ admits an open neighbourhood V in $\overline {Y}{}^{\mathrm {an}}$ such that

$$ \begin{align*}V\simeq \mathbb{D}^n\qquad \text{and} \qquad j^{-1}(V)\simeq \mathbb{D}^{n-r}\times (\mathbb{D}^1\smallsetminus \{0\})^r.\end{align*} $$

(Of course, n is the local dimension of Y and r is the cardinality of I.) We form the commutative diagram with cartesian squares

By homotopy invariance, the local system $F|_{j^{-1}(V)}$ is the pullback of a local system $F_0$ on $(\mathbb {D}^1\smallsetminus \{0\})^r$ . Using the (analytic) smooth base change theorem, it follows that the restriction of $j_*F$ to $V\cap (D^{\circ }_I)^{\mathrm {an}}$ is isomorphic to $p^*i_0^*j_{0,\,*}F_0$ . The $\Lambda $ -module $i_0^*j_{0,\,*}F_0$ is perfect since it is equivalent to $\Gamma (\mathbb {Z}^r;(F_0)_x)$ , where $x\in (\mathbb {D}^1\smallsetminus \{0\})^r$ is a base point. (See the proof of Lemma 1.2.12.) This proves our claim.

Step 4. Let $g:\overline {Y} \to X$ be the structural morphism of the X-scheme $\overline {Y}$ . Recall that our aim is to prove that $f_*F\simeq g_*j_*F$ is constructible.

Replacing $\overline {Y}$ with a connected component and X by the image of this component, we may assume that $\overline {Y}$ is integral and g dominant. Using the proper base change theorem, the constructibility of $j_*F$ and induction on the dimension, we can replace X with any dense open subvariety of X. Thus, we may assume that g is smooth and that $D=\overline {Y}\smallsetminus Y$ is a strict relative normal crossings divisor, that is, all the $D_I$ ’s from Step 3 are smooth over X. We claim that, in this situation, $g_*j_*F$ is a local system on X. This is typically proven using Ehresmann’s theorem. We will give below a different proof using the six-functor formalism.

Step 5. In this last step, we finish the proof of the proposition. We work in the situation we reached in Step 4. To show that $f_*F$ is a local system, it is enough to show that the obvious morphism

(1.15) $$ \begin{align} \underline{\mathrm{Hom}}(A,\Lambda)\otimes_{\Lambda} g_*j_*F \to \underline{\mathrm{Hom}}(A,g_*j_*F) \end{align} $$

is an equivalence for every $A\in \mathrm {Sh}(X^{\mathrm {an}};\Lambda )$ . We have canonical equivalences

$$ \begin{align*}\begin{array}{rcl} \underline{\mathrm{Hom}}(A,\Lambda)\otimes_{\Lambda} g_*j_*F & \simeq & g_*(g^*\underline{\mathrm{Hom}}(A,\Lambda)\otimes_{\Lambda} j_*F)\\ & \simeq & g_*(\underline{\mathrm{Hom}}(g^*A,\Lambda)\otimes_{\Lambda} j_*F),\\ & &\\ \text{and} \quad \underline{\mathrm{Hom}}(A,g_*j_*F) & \simeq & g_*\underline{\mathrm{Hom}}(g^*A,j_*F). \end{array}\end{align*} $$

The third equivalence is obvious, while the first two rely on the fact that g is proper and smooth. Therefore, it is enough to show that the natural morphism

(1.16) $$ \begin{align} \underline{\mathrm{Hom}}(g^*A,\Lambda)\otimes_{\Lambda} j_*F \to \underline{\mathrm{Hom}}(g^*A,j_*F) \end{align} $$

is an equivalence. This is a local question over $\overline {Y}{}^{\mathrm {an}}$ , and can be checked on stalks. We fix a point $y\in \overline {Y}{}^{\mathrm {an}}$ . For V varying in a fundamental system of neighbourhoods of y, and U the image of V in $X^{\mathrm {an}}$ , we have:

1. 1. $U\simeq \mathbb {D}^m$ , $V\simeq \mathbb {D}^n$ and, modulo these isomorphisms, $V\to U$ is given by the projection to the first m coordinates.

2. 2. $j^{-1}(V)\simeq \mathbb {D}^{n-r}\times (\mathbb {D}^1\smallsetminus \{0\})^r$ with $r\leq n-m$ .

It is enough to show that, for these V’s, the following morphism of $\Lambda $ -modules

(1.17) $$ \begin{align} \Gamma(V;\underline{\mathrm{Hom}}(g^*A,\Lambda))\otimes_{\Lambda} \Gamma(V;j_*F) \to \Gamma(V;\underline{\mathrm{Hom}}(g^*A,j_*F)) \end{align} $$

is an equivalence. We have a commutative diagram with cartesian squares

and a local system $F_0$ on $\mathbb {D}^{n-m-r}\times (\mathbb {D}^1\smallsetminus \{0\})^r$ such that $F|_{j^{-1}(V)}\simeq p^{\prime \prime *}F_0$ . Up to canonical equivalences, the morphism (1.17) can be rewritten as

(1.18) $$ \begin{align} \Gamma(U;\underline{\mathrm{Hom}}(A|_U,\Lambda)) \otimes_{\Lambda} \Gamma(\mathbb{D}^{n-m};j_{0,\,*}F_0) \to \Gamma(V;\underline{\mathrm{Hom}}(g^{\prime*}A|_U,p^{\prime*}j_{0,\,*}F_0)). \end{align} $$

We have a chain of natural equivalences

$$ \begin{align*}\begin{array}{rcl} \Gamma(V;\underline{\mathrm{Hom}}(g^{\prime*}A|_U,p^{\prime*}j_{0,\,*}F_0)) & \overset{(1)}{\simeq} & \Gamma(U;\underline{\mathrm{Hom}}(A|_U,g^{\prime}_*p^{\prime*}j_{0,\,*}F_0))\\ & \overset{(2)}{\simeq} & \Gamma(U;\underline{\mathrm{Hom}}(A|_U,p^* g_{0,\,*}j_{0,\,*}F_0))\\ & \overset{(3)}{\simeq} & \Gamma(U;\underline{\mathrm{Hom}}(A|_U,\Lambda)\otimes_{\Lambda} p^* g_{0,\,*}j_{0,\,*}F_0)\\ & \overset{(4)}{\simeq} & \Gamma(U;\underline{\mathrm{Hom}}(A|_U,\Lambda)) \otimes_{\Lambda} \Gamma(\mathbb{D}^{n-m};j_{0,\,*}F_0), \end{array}\end{align*} $$

where:

1. 1. is induced by the adjunction $(g^{\prime *},g^{\prime }_*)$ ;

2. 2. follows from the (analytic) smooth base change theorem applied to the second square in the above commutative diagram;

3. 3, 4. follow from the fact that $p^*g_{0,\,*}j_{0,\,*}F_0$ is a (constant) local system with value $\Gamma (\mathbb {D}^{n-m};j_{0,\,*}F_0)$ which is a perfect $\Lambda $ -module.

This clearly proves that the morphism (1.18) is an equivalence, and finishes the proof of the proposition.

Proof. We may replace k with its algebraic closure in $\mathbb {C}$ in order to reduce to the case where k has finite $\Lambda $ -cohomological dimension. Indeed, by [Reference AyoubAyo15b, Corollaire 1.A.4], the inverse image functor along a field extension $k'/k$ defines a morphism of six-functor formalisms

$$ \begin{align*}\mathrm{MSh}(-;\Lambda)^{\otimes} \to \mathrm{MSh}(-\otimes_kk';\Lambda)^{\otimes}.\end{align*} $$

Now that k is assumed to have finite $\Lambda $ -cohomological dimension, Proposition 1.1.12 implies that the $\infty $ -categories $\mathrm {MSh}(X;\Lambda )$ are compactly generated for all $X\in \mathrm {Sch}_k$ , that is, we have equivalences

$$ \begin{align*}\mathrm{Ind}(\mathrm{MSh}(X;\Lambda)^{\omega})\simeq \mathrm{MSh}(X;\Lambda).\end{align*} $$

By [Reference AyoubAyo07a, Théorème 2.2.37 & Corollaire 2.3.65], the six operations deduced from $\mathrm {MSh}(-;\Lambda )^{\otimes }$ respect compact objects. It is worth noting here that compactness for motivic sheaves (under our hypothesis on k) coincides with the notion of constructibility introduced in [Reference AyoubAyo07a, Définition 2.2.3] relative to the set $\{\Lambda (n); n\in \mathbb {Z}\}$ . (This is just Proposition 1.1.12.) Proposition 1.2.13 then implies that the functors $\mathrm {B}^*_X$ in (1.6) take compact motivic sheaves to constructible sheaves, and thus induce a morphism of $\mathrm {CAlg}(\mathrm {Cat}_{\infty })$ -valued presheaves

$$ \begin{align*}\mathrm{B}_{\mathrm{ct}}:\mathrm{MSh}(-;\Lambda)^{\omega,\,\otimes} \to \mathrm{Sh}_{\mathrm{ct}}(-;\Lambda)^{\omega,\,\otimes}.\end{align*} $$

By [Reference AyoubAyo10, Théorème 3.19], $\mathrm {B}_{\mathrm {ct}}$ is a morphism of six-functor formalisms. Applying indization, we obtain a morphism of presentable six-functor formalisms

$$ \begin{align*}\mathrm{B}^*_{\mathrm{ct}}:\mathrm{MSh}(-;\Lambda)^{\otimes} \to \mathrm{Sh}_{\mathrm{ct}}(-;\Lambda)^{\otimes}.\end{align*} $$

All the remaining assertions are clear.

Proof. Since $\mathrm {B}_{\mathrm {ct},\,*}$ is an exact functor between stable $\infty $ -categories, it is enough to show that it preserves filtered colimits. If the ground field k has finite virtual $\Lambda $ -cohomological dimension, the functor $\mathrm {B}_{\mathrm {ct}}^*$ belongs to $\mathrm {Pr}^{\mathrm {L}}_{\omega }$ and the claim follows from [Reference LurieLur09, Proposition 5.5.7.2(2)]. In general, we argue as follows. Let $\mathrm {MSh}_{\mathrm {nis}}(X;\Lambda )$ be the $\infty $ -category of motivic sheaves on X in the Nisnevich topology. (See Remark 1.1.22.) There is an obvious functor

$$ \begin{align*}\mathrm{a}_{\acute{\mathrm{e}}\mathrm{t}}:\mathrm{MSh}_{\mathrm{nis}}(X;\Lambda) \to \mathrm{MSh}(X;\Lambda)\end{align*} $$

which is a localisation functor, that is, its right adjoint functor $\mathrm {o}_{\acute {\mathrm {e}}\mathrm {t}}$ is fully faithful. By [Reference Ayoub, Gallauer and VezzaniAGV22, Proposition 3.2.3 or Remark 2.4.23], the $\infty $ -category $\mathrm {MSh}_{\mathrm {nis}}(X;\Lambda )$ is compactly generated and the composite functor $\mathrm {B}^*_{\mathrm {ct}}\circ \mathrm {a}_{\acute {\mathrm {e}}\mathrm {t}}$ preserves compact objects. Using [Reference LurieLur09, Proposition 5.5.7.2(2)] as above, we conclude that $\mathrm {o}_{\acute {\mathrm {e}}\mathrm {t}}\circ \mathrm {B}_{\mathrm {ct},\,*}$ is colimit-preserving. The result follows since $\mathrm {B}_{\mathrm {ct},\,*}\simeq \mathrm {a}_{\acute {\mathrm {e}}\mathrm {t}}\circ \mathrm {o}_{\acute {\mathrm {e}}\mathrm {t}}\circ \mathrm {B}_{\mathrm {ct},\,*}$ and $\mathrm {a}_{\acute {\mathrm {e}}\mathrm {t}}$ is colimit-preserving.

Proof. By Proposition 1.2.17, the domain and codomain of the morphism (1.20) are colimit-preserving in the variable M. Thus, we may assume M dualizable and use [Reference AyoubAyo14b, Lemme 2.8] to conclude. The last assertion follows from Proposition 1.1.13.

Proof. We split the proof into two steps. In the first step, we gather some preliminary results which are used in the second step.

Step 1. Refining $\mathcal {P}$ if necessary, we may assume that there is a morphism of k-varieties $f:X'\to X$ and a stratification $\mathcal {P}'$ of $X'$ with the following properties:

• ○ $X'$ is smooth and $\mathcal {P}'$ is regular in the sense of Definition 3.1.6(i);

• ○ f is proper, surjective, and compatible with the stratifications $\mathcal {P}$ and $\mathcal {P}'$ (as formulated in Remark 3.1.2(ii));

• ○ for every $\mathcal {P}$ -stratum Z, there is a unique open $\mathcal {P}'$ -stratum $Z'$ such that f induces an isomorphism $Z'\simeq Z$ ;

• ○ for every $\mathcal {P}'$ -stratum $Z'$ whose image in X is contained in the $\mathcal {P}$ -stratum Z, the morphism

  $$ \begin{align*}f_{Z'}:\overline{Z}{}^{\prime}_Z=\overline{Z}{}'\times_XZ \to Z\end{align*} $$
  is smooth and the complement of $Z'$ in $\overline {Z}{}^{\prime }_Z$ is a strict normal crossings divisor relative to Z.

The existence of f and $\mathcal {P}'$ as above can be obtained by noetherian induction using resolution of singularities in characteristic zero [Reference HironakaHir64a, Reference HironakaHir64b].

Given a locally closed subvariety $Z\subset X$ , we denote by $j_Z:Z \hookrightarrow X$ the obvious immersion. If Z is $\mathcal {P}$ -constructible (i.e., a union of $\mathcal {P}$ -strata), we write $\mathcal {P}_Z$ for the stratification of Z obtained by restricting $\mathcal {P}$ . In particular, we may consider the $\infty $ -category $\mathrm {Sh}_{\mathcal {P}_Z}(Z;\Lambda )$ defined as in the statement. Notice that, if Z is a $\mathcal {P}$ -stratum, then $\mathrm {Sh}_{\mathcal {P}_Z}(Z;\Lambda )=\mathrm {LS}(Z;\Lambda )$ . We use similar notations for locally closed subvarieties of $X'$ .

Let $Z\subset X$ be a $\mathcal {P}$ -constructible locally closed subvariety. We claim that $j_{Z,\,*}$ takes $\mathrm {Sh}_{\mathcal {P}_Z}(Z;\Lambda )^{\omega }$ to $\mathrm {Sh}_{\mathcal {P}}(X;\Lambda )^{\omega }$ . By induction on the dimension of Z and using localisation triangles, we easily reduce to the case where Z is a $\mathcal {P}$ -stratum. Let $Z'$ be the open $\mathcal {P}'$ -stratum mapping isomorphically to Z. By the third step of the proof of Proposition 1.2.13, we see that $j_{Z',\,*}$ takes $\mathrm {LS}(Z';\Lambda )^{\omega }$ to $\mathrm {Sh}_{\mathcal {P}'}(X';\Lambda )^{\omega }$ . (Indeed, $Z'$ is the complement of a strict normal crossings divisor in a connected component of $X'$ .) It remains to see that $f_*$ takes $\mathrm {Sh}_{\mathcal {P}'}(X';\Lambda )^{\omega }$ to $\mathrm {Sh}_{\mathcal {P}}(X;\Lambda )^{\omega }$ . We may check this on generators of the form $j_{T',\,!}M$ with $T'\subset X'$ a $\mathcal {P}'$ -stratum and $M\in \mathrm {LS}(T';\Lambda )^{\omega }$ . Letting $u:T'\to \overline {T}{}^{\prime }_T$ and $g:\overline {T}{}^{\prime }_T\to T$ be the obvious morphisms, it is sufficient to show that $g_*u_!M$ is a local system. By duality, we may as well prove that $g_*u_*M^{\vee }$ is a local system. This was proven in the fifth step of Proposition 1.2.13. (Recall that g is smooth and proper, and $T'$ is the complement of a strict relative normal crossings divisor in $\overline {T}{}^{\prime }_T$ .)

Similarly, under the assumption that $Z\subset X$ is $\mathcal {P}$ -constructible, the functor $j_Z^!$ takes $\mathrm {Sh}_{\mathcal {P}'}(X';\Lambda )$ to $\mathrm {Sh}_{\mathcal {P}_Z}(Z;\Lambda )^{\omega }$ . This can be deduced using a localisation triangle from the analogous property for $j_{T,\,*}$ , with $T=X\smallsetminus \overline {Z}$ .

Step 2. Let $\mathcal {T}_X\subset \mathrm {Sh}(X^{\mathrm {an}};\Lambda )$ be the full sub- $\infty $ -category generated under colimits by $\mathrm {Sh}_{\mathcal {P}}(X;\Lambda )^{\omega }$ . (We will use a similar notation when X is replaced by a locally closed $\mathcal {P}$ -constructible subvariety.) As in the proof of Lemma 1.2.12, we need to show that the objects of $\mathrm {Sh}_{\mathcal {P}}(X;\Lambda )^{\omega }$ are compact objects of $\mathcal {T}_X$ . It is enough to do so for objects of the form $j_{Z,\,!}M$ , where $Z\subset X$ is a $\mathcal {P}$ -stratum and M a local system on $Z^{\mathrm {an}}$ . Assume for a moment that $j_Z^!$ , restricted to $\mathcal {T}_X$ , is colimit-preserving. Then, by Step 1, it induces a functor

$$ \begin{align*}j_Z^!:\mathcal{T}_X \to \mathcal{T}_Z\end{align*} $$

which is right adjoint to $j_{Z,\,!}$ . By adjunction, we have

(1.22) $$ \begin{align} \mathrm{Map}_{\mathcal{T}_X}(j_{Z,\,!}M,-)\simeq \mathrm{Map}_{\mathcal{T}_Z}(M,j_Z^!(-)). \end{align} $$

By Lemma 1.2.12, we have an equivalence $\mathrm {LS}(Z;\Lambda )\simeq \mathcal {T}_Z$ showing that M is a compact object of $\mathcal {T}_Z$ . But we have also assumed that $j_Z^!:\mathcal {T}_X \to \mathcal {T}_Z$ was colimit-preserving. This shows that the right-hand side of the equivalence (1.22) preserves filtered colimits as needed.

To conclude, it remains to see that $j_Z^!:\mathcal {T}_X \to \mathrm {Sh}(Z^{\mathrm {an}};\Lambda )$ is colimit-preserving. We will show more generally that

(1.23) $$ \begin{align} j_{Y,\,Z}^!:\mathcal{T}_{Y}\to \mathrm{Sh}(Z^{\mathrm{an}};\Lambda) \end{align} $$

is colimit-preserving for any $\mathcal {P}$ -constructible closed subvariety $Y\subset X$ containing Z. (Of course, $j_{Y,\,Z}:Z\hookrightarrow Y$ is the obvious inclusion.) We argue by induction on the codimension of Z in Y. If Z is open in Y, there is nothing to prove. Thus, we may assume that Z has codimension at least one in Y. Let $Y^{\circ }\subset Y$ be the union of the $\mathcal {P}$ -strata that are open in Y, and let $u:Y^{\circ } \subset Y$ be the obvious inclusion. We claim that

(1.24) $$ \begin{align} u_*:\mathcal{T}_{Y^{\circ}} \to \mathrm{Sh}(Y^{\mathrm{an}};\Lambda) \end{align} $$

is colimit-preserving. Assuming this claim, we can conclude as follows. By Step 1, the functor $u_*$ induces a colimit-preserving functor $u_*:\mathcal {T}_{Y^{\circ }}\to \mathcal {T}_Y$ . Given an inductive system $(M_{\alpha })_{\alpha }$ in $\mathcal {T}_Y$ , consider the localisation triangles

(1.25) $$ \begin{align} v_*N_{\alpha} \to M_{\alpha} \to u_*u^*M_{\alpha}, \end{align} $$

where $v:Y\smallsetminus Y^{\circ } \to Y$ is the closed immersion complementary to u and $N_{\alpha }=v^!M_{\alpha }$ . Since v is a closed immersion, the functor $v_*$ is colimit-preserving. Thus, setting $M=\mathrm {colim}_{\alpha }\, M_{\alpha }$ and $N=\mathrm {colim}_{\alpha }\, N_{\alpha }$ , we have a localisation triangle

(1.26) $$ \begin{align} v_*N \to M \to u_*u^*M. \end{align} $$

Now, recall that our goal is to show that

$$ \begin{align*}\underset{\alpha}{\mathrm{colim}}\,j^!_{Y,\,Z}M_{\alpha} \to j^!_{Y,\,Z}M\end{align*} $$

is an equivalence. Using the exact triangles (1.25) and (1.26), and the fact that $j^!_{Y,\,Z}u_*=0$ , we are reduced to showing that the morphism

$$ \begin{align*}\underset{\alpha}{\mathrm{colim}}\, j^!_{Y,\,Z}v_*N_{\alpha}\to j^!_{Y,\,Z}v_*N\end{align*} $$

is an equivalence. Since $j^!_{Y,\,Z}v_*\simeq j^!_{Y\smallsetminus Y^{\circ },\,Z}$ , this follows from the induction hypothesis on the codimension of Z in Y.

We now finish the proof by establishing the claim that the functor (1.24) is colimit-preserving. Without loss of generality, we may assume that $Y=X$ . Since $X^{\circ }$ is the disjoint union of the open $\mathcal {P}$ -strata, it is enough to show that $j_{U,\,*}:\mathcal {T}_U \to \mathrm {Sh}(X^{\mathrm {an}};\Lambda )$ is colimit-preserving, with $U\subset X$ an open $\mathcal {P}$ -stratum. We appeal to the morphism $f:X'\to X$ and the stratification $\mathcal {P}'$ from Step 1. Let $U'\subset X'$ be an open stratum of $X'$ mapping isomorphically to U. Since f is proper, the functor $f_*$ is colimit-preserving. (For instance, this follows from the proper base change theorem for the Voevodsky pullback formalism $\mathrm {Sh}((-)^{\mathrm {an}};\Lambda )$ and [Reference LurieLur09, Corollary 7.3.4.12].) Thus, it is enough to show that $j_{U',\,*}:\mathrm {LS}(U';\Lambda ) \to \mathrm {Sh}(X^{\prime \mathrm {an}};\Lambda )$ is colimit-preserving. This was done in the proof of Lemma 1.2.12. (Recall that $U'$ is the complement of a strict normal crossings divisor in a connected component of $X'$ .)