$i^{*}\omega^{0}j_{*}(M)=\omega^{0}i^{*}j_{*}(M).$

Using Proposition 3.2.1, the Artin motive $\omega^{0}i^{*}j_{*}(M)$ is in
degree at least $\delta(X)$ with respect to the ordinary homotopy t-structure.
Therefore, we get

$\omega^{0}i^{*}j_{*}(M)\geqslant_{hp}-1.$

This ensures that the Artin motive $j_{!}(M)$ is perverse homotopy t-non-
negative. ∎

######  Proposition 5.3.5.

(Affine Lefschetz Theorem for Artin motives) Let $S$ be an excellent scheme,
let $f\colon X\rightarrow S$ be a quasi-finite affine morphism and let $K$ be
a number field. Assume that $X$ is nil-regular and that we are in one of the
following cases

  1. (a)

We have $\dim(S)\leqslant 2.$

  2. (b)

There is a prime number $p$ such that the scheme $S$ is of finite type over
$\mathbb{F}_{p}$.

Then, the functor

$f_{!}\colon\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(X,K)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,K)$

is $t_{hp}$-exact.

###### Proof.

We can assume that the scheme $S$ is connected. As in the proof of Proposition
5.3.4, it suffices to show that if $j\colon U\rightarrow S$ is an affine open
immersion such that the scheme $U$ is nil-regular and if $M$ is a
$t_{hp}$-non-negative smooth Artin motive over $U$, the Artin motive $j_{!}M$
is $t_{hp}$-non-negative.

Suppose that Assumption (b) holds, i.e. there is a prime number $p$ such that
the scheme $S$ is of finite type over $\mathbb{F}_{p}$. Let $v$ be a non-
archimedian valuation on $K$ which does not extend the $p$-adic valuation. We
have a commutative diagram

${\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(U,K)}$${\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,K)}$${\mathcal{D}^{smA}(U,K_{v})}$${\mathcal{D}^{A}(S,K_{v})}$$\scriptstyle{j_{!}}$$\scriptstyle{\rho_{v}}$$\scriptstyle{\rho_{v}}$$\scriptstyle{j_{!}}$

such that the vertical arrows are conservative and t-exact when the categories
which appear in the bottom row are endowed with the perverse homotopy
t-structure. Using Lemma 1.1.8, it suffices to show that the functor $j_{!}$
is t-exact which follows from [Rui22, 4.8.12].

We now suppose that Assumption (a) holds i.e. that $\dim(S)\leqslant 2$. By
Proposition 5.3.4, we can assume that $\dim(S)=2$. Take the convention that
$\delta(S)=2$. Let $i\colon F\rightarrow S$ be the reduced complementary
closed immersion of $j$. As in the proof of Proposition 5.3.4, it suffices to
show that the functor $\omega^{0}i^{*}j_{*}$ has cohomological amplitude
bounded below by $-1$.

Step 1: In this step, we show that we can assume that the scheme $S$ is
normal.

Let $\nu\colon\hat{S}\rightarrow S$ be the normalization of $S$. Write

${U_{\mathrm{red}}}$${\hat{S}}$${\hat{F}}$${U}$${S}$${F}$$\scriptstyle{\nu_{U}}$$\scriptstyle{\gamma}$$\scriptstyle{\nu}$$\scriptstyle{\nu_{F}}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\scriptstyle{i}$

the commutative diagram such that both squares are cartesian. Notice that
$U_{\mathrm{red}}$ is the reduced scheme associated to $U$ and therefore, the
functor $(\nu_{U})_{*}$ is a t-exact equivalence. Hence, it suffices to show
that the functor $\omega^{0}i^{*}j_{*}(\nu_{U})_{*}$ has cohomological
amplitude bounded below by $-1$. This functor is equivalent to
$\omega^{0}(\nu_{F})_{*}\iota^{*}\gamma_{*}$. Since $\nu_{F}$ is a finite
morphism, the functor $(\nu_{F})_{*}$ is $t_{hp}$-exact by Corollary 5.1.3 and
commutes with $\omega^{0}$ by Proposition 1.4.4. Therefore, suffices to show
that the functor $\omega^{0}\iota^{*}\gamma_{*}$ has cohomological amplitude
bounded below by $-1$. Therefore, we can assume that the scheme $S$ is normal.

Step 2: In this step, we show that it suffices to show that
$\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}\geqslant_{hp}1$.

Using Proposition 5.1.6, the objects of
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(U,K)$ are
bounded with respect to the perverse homotopy t-structure. Therefore, it
suffices to show that the objects of the heart of the t-category
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(U,R)$ are
send to objects of degree at least $-1$ with respect to the perverse homotopy
t-structure. Letting $\xi$ be a geometric point of $U$, the heart of
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(U,R)$ is
equivalent to
$\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(U,\xi),R)$ by
Theorem 2.1.1 and Proposition 1.5.9. Since $K$ is a $\mathbb{Q}$-algebra, the
latter a semi-simple category by Maschke’s Theorem. Thus, every object of the
heart is a retract of an object of the form $f_{*}\mathbbm{1}_{V}[2]$ with
$f\colon V\rightarrow U$ finite and étale. Therefore, it suffice to show that
$\omega^{0}i^{*}j_{*}f_{*}\mathbbm{1}_{V}\geqslant_{hp}1$.

Let $\overline{f}\colon S_{V}\rightarrow S$ be the normalization of $S$ in
$V$. Write

${V}$${S_{V}}$${F_{V}}$${U}$${S}$${F}$$\scriptstyle{f}$$\scriptstyle{\gamma}$$\scriptstyle{\overline{f}}$$\scriptstyle{p}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\scriptstyle{i}$

the commutative diagram such that both squares are cartesian. By proper base
change, we get

$\omega^{0}i^{*}j_{*}f_{*}\mathbbm{1}_{V}=\omega^{0}p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{V}.$

Since the map $p$ is finite, the functor $p_{*}$ is $t_{hp}$-exact by
Corollary 5.1.3 and commutes with the functor $\omega^{0}$ by Proposition
1.4.4. Therefore, it suffices to show that

$\omega^{0}\iota^{*}\gamma_{*}\mathbbm{1}_{V}\geqslant_{hp}1.$

Hence, we can assume that $V=U$.

Step 3: Since the scheme $S$ is normal its singular point are in codimension
$2$. Let $x$ be a point of $S$. Using Proposition 5.2.1, we can replace $S$
with a neighborhood of $x$. We can therefore assume that the singular locus of
$S$ is either empty or the single point $x$. Let
$f\colon\widetilde{S}\rightarrow S$ be a resolution of singularities of $S$
such that $E$ is a simple normal crossing divisor. Since the scheme $S$ is
excellent, such a resolution exists by Lipman’s Theorem on embedded resolution
of singularities (see [Sta23, 0BGP,0BIC,0ADX]. Write

${U}$${\widetilde{S}}$${E}$${U}$${S}$${F}$$\scriptstyle{\gamma}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\scriptstyle{i}$

the commutative diagram such that both squares are cartesian.

Then, by proper base change, we get

$\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}=\omega^{0}p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U}.$

The latter is by Proposition 1.4.4 equivalent to
$\omega^{0}p_{*}\iota^{*}\omega^{0}\gamma_{*}\mathbbm{1}_{U}$ which by Lemma
3.2.2 is equivalent to $\omega^{0}p_{*}\mathbbm{1}_{E}$. Therefore, it
suffices to show that $\omega^{0}p_{*}\mathbbm{1}_{E}\geqslant_{hp}1$.

Write $E=\bigcup\limits_{i\in J}E_{i}$ with $J$ finite, where the $E_{i}$ are
regular and of codimension $1$, the $E_{ij}=E_{i}\cap E_{j}$ are of
codimension $2$ and regular if $i\neq j$ and the intersections of $3$ distinct
$E_{i}$ are empty. By cdh descent, we have an exact triangle

$\omega^{0}p_{*}\mathbbm{1}_{E}\rightarrow\bigoplus\limits_{i\in
J}\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}}\rightarrow\bigoplus\limits_{\\{i,j\\}\subseteq
J}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}.$

Since we have $\delta(E_{i})=1$ for any $i$ and since $\delta(E_{ij})=0$ if
$i\neq j$, Propositions 5.2.4 and 5.1.2 imply that the Artin motives
$\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}}$ and
$(p_{ij})_{*}\mathbbm{1}_{E_{ij}}$ are $t_{hp}$-non-negative. Hence, we get
$\omega^{0}p_{*}\mathbbm{1}_{E}\geqslant_{hp}0$ and that
${}^{hp}H^{0}(\omega^{0}p_{*}\mathbbm{1}_{E})$ is the kernel of the map

$\bigoplus\limits_{i\in
J}{}^{hp}H^{0}(\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}})\rightarrow\bigoplus\limits_{\\{i,j\\}\subseteq
J}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}.$

Hence, to finish the proof, it suffices to show that the kernel of the above
map vanishes.

Let

$F_{0}=\\{y\in F\mid\dim(f^{-1}(y))>0\\}.$

The scheme $F_{0}$ is $0$-dimensional. Since we can work locally around $x$,
we may assume that $F_{0}$ is either empty or the single point $x$ an that the
image of any $E_{ij}$ through the map $p$ is $\\{x\\}$.

Assume that the scheme $F_{0}$ is empty, then, the morphism $p$ is finite.
Furthermore, for any index $i$ in $J$, the Artin motive $\mathbbm{1}_{E_{i}}$
is in degree $1$ with respect to the perverse homotopy t-structure by
Proposition 5.2.4, therefore, the Artin motive
${}^{hp}H^{0}(\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}})$ vanishes. Thus, the
motive ${}^{hp}H^{0}(\omega^{0}p_{*}\mathbbm{1}_{E})$ vanishes.

Assume now that $F_{0}=\\{x\\}$. Let

$I=\\{i\in J\mid p(E_{i})=\\{x\\}\\}$

and let $k\colon\\{x\\}\rightarrow F$ be the inclusion. If an index $i$
belongs to $J\setminus I$, the morphism $p_{i}$ is finite and therefore, the
motive ${}^{hp}H^{0}(\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}})$ vanishes.

If an index $i$ belongs to $I$, let

$E_{i}\rightarrow G_{i}\overset{g_{i}}{\rightarrow}\\{x\\}$

be the Stein factorization of $p_{i}$. Corollary 3.1.8 yields

${}^{hp}H^{0}(\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}})=k_{*}(g_{i})_{*}\mathbbm{1}_{G_{i}}$

and therefore

$\bigoplus\limits_{i\in
I}{}^{hp}H^{0}(\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}})=k_{*}\bigoplus\limits_{i\in
I}(g_{i})_{*}\mathbbm{1}_{G_{i}}.$

Now, if $i\neq j$, the morphism $p_{ij}$ factors through $\\{x\\}$, so that we
have a morphism $g_{ij}\colon E_{ij}\rightarrow\\{x\\}$ with $p_{ij}=k\circ
g_{ij}$.

We have

$\bigoplus\limits_{\\{i,j\\}\subseteq
J}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}=k_{*}\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}(g_{ij})_{*}\mathbbm{1}_{E_{ij}}.$

Hence, the Artin motive ${}^{hp}H^{0}(\omega^{0}p_{*}\mathbbm{1}_{E})$ is the
kernel of the map

$k_{*}\bigoplus\limits_{i\in I}(g_{i})_{*}\mathbbm{1}_{G_{i}}\rightarrow
k_{*}\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}(g_{ij})_{*}\mathbbm{1}_{E_{ij}}.$

Since the functor $k_{*}$ is t-exact, we get
${}^{hp}H^{0}(\omega^{0}p_{*}\mathbbm{1}_{E})=k_{*}P$ with $P$ the kernel of
the map

$\bigoplus\limits_{i\in
I}(g_{i})_{*}\mathbbm{1}_{G_{i}}\rightarrow\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}(g_{ij})_{*}\mathbbm{1}_{E_{ij}}.$

Let $\kappa=k(x)$ and let $\Gamma=G_{\kappa}$. Recall that if $g\colon
X\rightarrow\operatorname{Spec}(\kappa)$ is a finite morphism, the Artin
motive $g_{*}\mathbbm{1}_{X}$ corresponds to the Artin representation
$K[X_{\overline{\kappa}}]$ of $\Gamma$ through the equivalence $\alpha_{!}$
(see 3.1.7. Hence, we have

$P=\alpha_{!}P^{\prime}$

where $P^{\prime}$ is the kernel of the morphism

$\bigoplus\limits_{i\in
I}K[(G_{i})_{\overline{\kappa}}]\rightarrow\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}K[(E_{ij})_{\overline{\kappa}}]$

of Artin representations of $\Gamma$ with coefficients in $K$.

Now, since $K$ is flat over $\mathbb{Z}$, we have

$P^{\prime}=P_{0}\otimes_{\mathbb{Z}}K$

where $P_{0}$ is the kernel of the map

$\bigoplus\limits_{i\in
I}\mathbb{Z}[(G_{i})_{\overline{\kappa}}]\rightarrow\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}\mathbb{Z}[(E_{ij})_{\overline{\kappa}}].$

Since the underlying $\mathbb{Z}$-module of $P_{0}$ is of finite type, and
since $K$ is a $\mathbb{Q}$-algebra, it suffices to show that $P_{0}$ is of
rank $0$.

Denote by $N$ the $\mathbb{Z}[\Gamma]$-module $\bigoplus\limits_{i\in
I}\mathbb{Z}[(G_{i})_{\overline{\kappa}}]$, by $Q$ the
$\mathbb{Z}[\Gamma]$-module
$\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}\mathbb{Z}[(E_{ij})_{\overline{\kappa}}]$
and by $R$ the image of $N$ through the map $N\rightarrow Q$.

We have an exact sequence

$0\rightarrow P_{0}\rightarrow N\rightarrow R\rightarrow 0.$

Let $\ell$ be a prime number, let $n$ be a positive integer and denote by
$\Lambda$ the ring $\mathbb{Z}/\ell^{n}\mathbb{Z}$. We get an exact sequence

$\mathrm{Tor}^{1}_{\mathbb{Z}}(R,\Lambda)\rightarrow
P_{0}\otimes_{\mathbb{Z}}\Lambda\rightarrow
N\otimes_{\mathbb{Z}}\Lambda\rightarrow
R\otimes_{\mathbb{Z}}\Lambda\rightarrow 0.$

Assume that the map $N\otimes_{\mathbb{Z}}\Lambda\rightarrow
Q\otimes_{\mathbb{Z}}\Lambda$ is injective. Then, the induced map
$N\otimes_{\mathbb{Z}}\Lambda\rightarrow R\otimes_{\mathbb{Z}}\Lambda$ is also
injective and we get a surjection

$\mathrm{Tor}^{1}_{\mathbb{Z}}(R,\Lambda)\rightarrow
P_{0}\otimes_{\mathbb{Z}}\Lambda.$

Since $\mathrm{Tor}^{1}_{\mathbb{Z}}(R,\Lambda)$ is the $\ell^{n}$-torsion
subgroup of $R$ and since the latter is of finite type, there is an integer
$n_{0}$ such that if $n\geqslant n_{0}$, the group
$\mathrm{Tor}^{1}_{\mathbb{Z}}(R,\Lambda)$ is of $\ell^{n_{0}}$-torsion. If
$P_{0}$ is not of rank $0$, the group $P_{0}\otimes_{\mathbb{Z}}\Lambda$ is
not of $\ell^{n_{0}}$-torsion for $n>n_{0}$.

Hence, to show that $P_{0}$ is of rank $0$ and to finish the proof, it
suffices to show that there is a prime number $\ell$ such that for any
positive integer $n$, letting $\Lambda=\mathbb{Z}/\ell^{n}\mathbb{Z}$, the map

$\bigoplus\limits_{i\in
I}\Lambda[(G_{i})_{\overline{\kappa}}]\rightarrow\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}\Lambda[(E_{ij})_{\overline{\kappa}}]$

has a trivial kernel.

Step 4: Let $\ell$ be a prime number which is invertible in $\kappa$.
Replacing $S$ with a neighborhood of $x$ if needed, we can assume that $\ell$
is invertible on $S$. Let $n$ be a positive integer. Denote by $\Lambda$ the
ring $\mathbb{Z}/\ell^{n}\mathbb{Z}$. In this step, we show that the map

$\bigoplus\limits_{i\in
I}\Lambda[(G_{i})_{\overline{\kappa}}]\rightarrow\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}\Lambda[(E_{ij})_{\overline{\kappa}}]$

has trivial kernel.

The classical Affine Lefschetz Theorem [LYF14, XV.1.1.2] on
$\mathcal{D}^{b}_{c}(S,\Lambda)$ endowed with its perverse t-structure ensures
that

${}^{p}H^{0}(i^{*}j_{*}\underline{\Lambda})=0.$

As before, the sheaf $i^{*}j_{*}\underline{\Lambda}$ is equivalent to the
sheaf $p_{*}\iota^{*}\gamma_{*}\underline{\Lambda}$. Now, the localization
triangle in $\mathcal{D}^{b}_{c}(S,\Lambda)$ yields an exact triangle

$\iota_{*}\iota^{!}\underline{\Lambda}\rightarrow\underline{\Lambda}\rightarrow\gamma_{*}\underline{\Lambda}.$

Applying $p_{*}\iota^{*}$, we get an exact triangle

$p_{*}\iota^{!}\underline{\Lambda}\rightarrow
p_{*}\underline{\Lambda}\rightarrow
p_{*}\iota^{*}\gamma_{*}\underline{\Lambda}.$

Furthermore, by cdh descent, we have an exact triangle

$\bigoplus\limits_{\\{i,j\\}\subseteq
J}(p_{ij})_{*}\underline{\Lambda}(-2)[-4]\rightarrow\bigoplus\limits_{i\in
J}(p_{i})_{*}\underline{\Lambda}(-1)[-2]\rightarrow
p_{*}\iota^{!}\underline{\Lambda}$

Thus, the perverse sheaf ${}^{p}H^{k}(p_{*}\iota^{!}\underline{\Lambda})$
vanishes if $k<2$. Therefore, the map

${}^{p}H^{0}(p_{*}\underline{\Lambda})\rightarrow{}^{p}H^{0}(p_{*}\iota^{*}\gamma_{*}\underline{\Lambda})$

is an equivalence and therefore, the perverse sheaf
${}^{p}H^{0}(p_{*}\underline{\Lambda})$ vanishes.

Finally, the same method as in the third step of our lemma shows that

${}^{p}H^{0}(p_{*}\underline{\Lambda})=k_{*}P_{\Lambda}$

where $P_{\Lambda}$ is the étale sheaf of $\kappa_{\operatorname{\acute{e}t}}$
which corresponds through Galois-Grothendieck theory to the kernel
$P_{\Lambda}^{\prime}$ of the map

$\bigoplus\limits_{i\in
I}\Lambda[(G_{i})_{\overline{\kappa}}]\rightarrow\bigoplus\limits_{\\{i,j\\}\in\mathcal{I}}\Lambda[(E_{ij})_{\overline{\kappa}}]$

of $\Gamma$-modules. Therefore, the $\Gamma$-module $P_{\Lambda}^{\prime}$
vanishes which finishes the proof. ∎

######  Remark 5.3.6.

Let $k$ be a sub-field of $\mathbb{C}$, let $S$ be a $k$-scheme of finite
type, let $f\colon X\rightarrow S$ be a quasi-finite affine morphism and let
$K$ be a number field. By using the Betti realization instead of the $v$-adic
realization and by adapting the proof of [Rui22, 4.8.2] to the setting of
Mixed Hodge Modules, it should be possible to prove that the functor

$f_{!}\colon\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(X,K)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,K)$

is perverse homotopy t-exact.

We can now prove the main result of this section.

######  Theorem 5.3.7.

Let $S$ be an excellent scheme, let $f\colon X\rightarrow S$ be a quasi-finite
morphism of schemes and let $R$ be a regular good ring. Assume that $X$ is
nil-regular and that we are in one of the following cases

  1. (a)

We have $\dim(S)\leqslant 2.$

  2. (b)

There is a prime number $p$ such that the scheme $S$ is of finite type over
$\mathbb{F}_{p}$.

Then, the functor

$f_{!}\colon\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},\mathbb{Q}-c}(X,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},\mathbb{Q}-c}(X,R)$

is $t_{hp}$-exact.

###### Proof.

Proposition 4.1.2 implies that the functor $f_{!}$ is right $t_{hp}$-exact.
Hence, it suffices to show that it is left $t_{hp}$-exact.

Let $M$ be a $\mathbb{Q}$-constructible motive over $X$ which is $t_{hp}$-non-
negative. We have an exact triangle

$M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1]\rightarrow M\rightarrow
M\otimes_{\mathbb{Z}}\mathbb{Q}.$

By Proposition 4.1.6, the motive $M\otimes_{\mathbb{Z}}\mathbb{Q}$ is
$t_{hp}$-non-negative. Since the subcategory of $t_{hp}$-non-negative is
closed under limits, the motive
$M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1]$ is also $t_{hp}$-non-
negative.

Furthermore, we have an exact triangle

$f_{!}(M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1])\rightarrow
f_{!}(M)\rightarrow f_{!}(M\otimes_{\mathbb{Z}}\mathbb{Q}).$

By Proposition 5.3.5, the motive $f_{!}(M\otimes_{\mathbb{Z}}\mathbb{Q})$ is
$t_{hp}$-non-negative and by Proposition 1.3.18, the motive
$f_{!}(M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1])$ is $t_{hp}$-non-
negative. Since the subcategory of $t_{hp}$-non-negative Artin motives is
closed under extensions, the motive $f_{!}(M)$ is also $t_{hp}$-non-negative.
∎

We also have a $t_{hp}$-exactness result about the functor $\omega^{0}f_{*}$
when $f$ is a proper morphism.

######  Proposition 5.3.8.

Let $S$ be a scheme allowing resolution of singularities by alterations, let
$f\colon X\rightarrow S$ be a proper morphism and let $K$ be a number field.
Then, the functor

$\omega^{0}f_{*}\colon\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(X,K)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,K)$

is right $t_{hp}$-exact.

###### Proof.

Recall that since $f$ is proper, we have

$\omega^{0}f_{*}=\omega^{0}f_{!}.$

Since the functor $f_{!}$ is a left adjoint functor, it respects small
colimits. Furthermore, with rational coefficients, the functor $\omega^{0}$
also respects small colimits by [Leh19b, 3.5] and the proof applies in the
setting of motives with coefficients in $K$. Hence, it suffices to show that
the Artin motives of the form $\omega^{0}f_{*}h_{X}(Y)[\delta(Y)]$ with $Y$
finite over $S$ are perverse homotopy t-non-positive. Hence, replacing $X$
with $Y$, it suffices to show that the Artin motive
$\omega^{0}f_{*}\mathbbm{1}_{X}[\delta(X)]$ is perverse homotopy t-non-
positive.

This follows from Lemma 5.3.9. ∎

######  Lemma 5.3.9.

Let $K$ be a number field and let $f\colon X\rightarrow S$ be a morphism of
finite type. Then, the Artin motive
$\omega^{0}f_{*}\mathbbm{1}_{X}[\delta(X)]$ belongs to the smallest
subcategory of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,K)$ which is
closed under finite colimits, extensions and retracts and contains the motives
of the form $g_{*}\mathbbm{1}_{Y}[\delta(Y)]$ with $g\colon Y\rightarrow S$
finite.

###### Proof.

We now proceed by noetherian induction on $X$. If $\dim(X)=0$, the morphism
$f$ is finite and the result follows from Corollary 5.1.3.

Step 1: Assume first that the map $f$ is proper.

Using the induction hypothesis and cdh-descent, we can assume the scheme $X$
to be normal. Let $q\colon\widetilde{X}\rightarrow X$ be a resolution of
singularities by alterations and let $q^{\prime}\colon X^{\prime}\rightarrow
X$ be the relative normalization of $X$ in $\widetilde{X}\setminus B$ where
$B$ is the inverse image of the singular locus of $X$. Using [Ayo07, 2.1.165],
the motive $\omega^{0}f_{*}q^{\prime}_{*}\mathbbm{1}_{X^{\prime}}$ is a
retract of the motive $\omega^{0}f_{*}q_{*}\mathbbm{1}_{\widetilde{X}}$.
Hence, using the induction hypothesis and cdh-descent, we can assume the
scheme $X$ to be regular and connected.

Let $\eta=\operatorname{Spec}(\mathbb{K})$ be a generic point of $S$. Let
$f_{\mathbb{K}}\colon X_{\mathbb{K}}\rightarrow\mathbb{K}$ be the pullback of
$f$ to $\mathbb{K}$. De Jong’s resolution of singularities yield a proper
alteration $X^{\prime\prime}_{\mathbb{K}}\rightarrow X_{\mathbb{K}}\rightarrow
X_{\mathbb{K}}$ such that the map $X_{\mathbb{K}}\rightarrow X_{\mathbb{K}}$
is proper and birational, the map $X^{\prime\prime}_{\mathbb{K}}\rightarrow
X_{\mathbb{K}}$ is a finite morphism which is generically the composition of
an étale cover and of a purely inseparable morphism and such that the
structural map $X^{\prime\prime}_{\mathbb{K}}\rightarrow\mathbb{K}$ is the
composition of a purely inseparable morphism and of a smooth morphism.
Therefore by [Gro66, 8], we have a dense open subset $U$ and a commutative
diagram

${X^{\prime\prime}_{U}}$${X^{\prime}_{U}}$${X_{U}}$${U}$$\scriptstyle{q}$$\scriptstyle{f^{\prime\prime}_{U}}$$\scriptstyle{p}$$\scriptstyle{f^{\prime}_{U}}$$\scriptstyle{f_{U}}$

such that the map $f_{U}$ is the pullback of the map $f$ to $U$, the map $q$
is proper and birational, the map $q$ is finite and is generically the
composition of an étale Galois cover and of a purely inseparable morphism and
the map $f^{\prime\prime}_{U}$ is the composition of a purely inseparable
morphism and of a smooth morphism.

Shrinking $U$ if needed, we can assume that it regular and that it is the
complement of a simple normal crossing divisor. Denote by $j\colon
U\rightarrow S$ the immersion, by $i\colon F\rightarrow S$ its reduced closed
complement and by $f_{F}\colon X_{F}\rightarrow F$ the pullback of $f$ along
the map $i$.

By localization, we have an exact triangle

$i_{*}i^{!}f_{*}\mathbbm{1}_{X}\rightarrow f_{*}\mathbbm{1}_{X}\rightarrow
j_{*}(f_{U})_{*}\mathbbm{1}_{X_{U}}.$

By Corollary 3.1.12, the motive $\omega^{0}i^{!}f_{*}\mathbbm{1}_{X}$ vanishes
since the scheme $X$ is regular and since $X_{F}$ is a simple normal crossing
divisor in $X$. This yields an equivalence

$\omega^{0}f_{*}\mathbbm{1}_{X}=\omega^{0}j_{*}(f_{U})_{*}\mathbbm{1}_{X_{U}}$

By Proposition 1.4.4, we therefore get that

$\omega^{0}f_{*}\mathbbm{1}_{X}=\omega^{0}j_{*}\omega^{0}(f_{U})_{*}\mathbbm{1}_{X_{U}}.$

Let $Z$ be the complement of the open subset of $X_{U}$ over which the map $p$
is an isomorphism and let $Z^{\prime}$ be its pullback along the map $p$. By
cdh-descent, and by using the induction hypothesis on the structural maps
$Z\rightarrow U\rightarrow S$ and $Z^{\prime}\rightarrow U\rightarrow S$, it
suffices to show that

$\omega^{0}j_{*}\omega^{0}(f_{U}^{\prime})_{*}\mathbbm{1}_{X_{U}^{\prime}}[\delta(X)]\leqslant_{hp}0.$

Furthermore, by [Ayo07, 2.1.165], the motive
$\omega^{0}(f_{U}^{\prime})_{*}\mathbbm{1}_{X^{\prime}_{U}}$ is a retract of
the motive
$\omega^{0}(f_{U}^{\prime\prime})_{*}\mathbbm{1}_{X^{\prime\prime}_{U}}$ and
therefore, it suffices to show that

$\omega^{0}j_{*}\omega^{0}(f_{U}^{\prime\prime})_{*}\mathbbm{1}_{X_{U}^{\prime\prime}}[\delta(X)]\leqslant_{hp}0.$

Since the map $f_{U}^{\prime\prime}$ is the composition of a smooth and proper
morphism and of a purely inseparable morphism, letting

$X_{U}^{\prime\prime}\rightarrow Y\overset{g}{\rightarrow}U$

be the Stein factorization of $f_{U}^{\prime\prime}$ yields by Proposition
1.4.4(5) that

$\omega^{0}(f_{U}^{\prime\prime})_{*}\mathbbm{1}_{X_{U}^{\prime\prime}}=g_{*}\mathbbm{1}_{Y}.$

On the other hand, the localization property also gives an exact triangle

$j_{!}g_{*}\mathbbm{1}_{Y}\rightarrow\omega^{0}j_{*}g_{*}\mathbbm{1}_{Y}\rightarrow
i_{*}\omega^{0}i^{*}j_{*}g_{*}\mathbbm{1}_{Y}$

and therefore, using Proposition 5.1.2, it suffices to show that

$\omega^{0}i^{*}j_{*}g_{*}\mathbbm{1}_{Y}[\delta(X)-1]\leqslant_{hp}0.$

Let $Y_{1}$ be a relative normalization of $S$ in $Y$ and let $\overline{Y}$
be a resolution of singularities by alterations of $Y_{1}$. Let
$k\colon\overline{Y}\rightarrow S$ be the structural map. The scheme
$\overline{Y}$ is regular and there is a map $e\colon Y^{\prime}\rightarrow Y$
which the composition of a finite étale cover and of a purely inseparable
morphism such that we have a comutative diagam

${Y^{\prime}}$${\overline{Y}}$${Y_{F}}$${U}$${S}$${F}$$\scriptstyle{g\circ
e}$$\scriptstyle{\gamma}$$\scriptstyle{k}$$\scriptstyle{g_{F}}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\scriptstyle{i}$

which is made of cartesian square.

The Artin motive $\omega^{0}i^{*}j_{*}g_{*}\mathbbm{1}_{Y}$ is by [Ayo07,
2.1.165] a retract of the motive $\omega^{0}i^{*}j_{*}(g\circ
e)_{*}\mathbbm{1}_{Y^{\prime}}$ and by proper base change, we get

$\omega^{0}i^{*}j_{*}(g\circ
e)_{*}\mathbbm{1}_{Y^{\prime}}=\omega^{0}(g_{F})_{*}\omega^{0}\iota^{*}\gamma_{*}\mathbbm{1}_{Y}.$

But using Propositions 1.4.4 and 3.2.2, we have

$\omega^{0}\iota^{*}\gamma_{*}\mathbbm{1}_{Y}=\iota^{*}\omega^{0}\gamma_{*}\mathbbm{1}_{Y}=\mathbbm{1}_{Y_{F}}.$

Therefore, it suffices to show that

$\omega^{0}(g_{F})_{*}\mathbbm{1}_{Y_{F}}[\delta(X)-1]\leqslant_{hp}0.$

Since

$\dim(Y_{F})\leqslant\dim(Y)-1\leqslant\dim(X)-1,$

the result follows by induction.

Step 2: Assume now that the map $f$ is a general morphism.

Using Nagata’s Theorem [Sta23, 0F41], we can write

$f=j\circ\overline{f}$

where $\overline{f}$ is proper and $j$ is an open immersion. Therefore, we can
assume that $f=j\colon U\rightarrow S$ is an open immersion.

We want to show that

$\omega^{0}j_{*}\mathbbm{1}_{U}[\delta(U)]\geqslant_{hp}0.$

Let now $p\colon Y\rightarrow S$ be a resolution of singularities by
alterations. We can proceed as before to show that the motive
$\omega^{0}j_{*}\mathbbm{1}_{U}$ is a retract of the motive
$\omega^{0}p_{*}\mathbbm{1}_{Y}$. The result then follows from the case of
proper morphisms. ∎

######  Corollary 5.3.10.

Let $S$ be a scheme allowing resolution of singularities by alteration, let
$f\colon X\rightarrow S$ be a quasi-finite morphism of schemes and let $K$ be
a number field. Assume that $X$ is nil-regular.

Then, the functor

$\omega^{0}f_{*}\colon\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(X,K)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(X,K)$

is t-exact when both sides are endowed with the perverse homotopy t-structure.

###### Proof.

Using Proposition 5.1.6, the objects of
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(X,K)$ are
bounded with respect to the perverse homotopy t-structure. Therefore, it
suffices to show that the objects of the heart of the t-category
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(X,R)$ are
send to objects of degree $0$ with respect to the perverse homotopy
t-structure. Since the latter a semi-simple category by Maschke’s Theorem,
every object of the heart is a retract of an object of the form
$g_{*}\mathbbm{1}_{V}[\delta(X)]$ with $g\colon V\rightarrow X$ finite and
étale. Therefore, it suffice to show that

$\omega^{0}f_{*}g_{*}\mathbbm{1}_{V}[\delta(V)]\geqslant_{hp}0$

which follows from Lemma 5.3.9. ∎

###  5.4. Induced t-structure on constructible motives over a base scheme of
dimension at most $2$

######  Theorem 5.4.1.

Let $S$ be an excellent scheme of dimension $2$ or less and let $R$ be a
regular good ring, then, the perverse homotopy t-structure on the stable
category $\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)$
induces a t-structure on the stable subcategory
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(S,R)$.

###### Proof.

We can assume that $S$ is reduced and connected. We proceed by noetherian
induction on $S$. If $\dim(S)=0$, the result follows from Proposition 5.1.8.

Assume now that $\dim(S)>0$. If $\mathfrak{p}$ be a prime ideal of
$\mathbb{Z}$, if $X$ is a scheme and if $M$ is an Artin motive over $X$, we
denote by $M_{\mathfrak{p}}$ the image of $M$ in
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
(with the notations of Section 1.1.5) and we say that an object $N$ of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
is constructible when it belongs to
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(X,R)_{\mathfrak{p}}$.
Recall that an Artin motive $M$ is _$\mathfrak{p}$ -constructible_ when
$M_{\mathfrak{p}}$ is constructible. By Proposition 1.1.12, it suffices to
show that for any constructible Artin motive $M$, the Artin motive
${}^{hp}\tau_{\leqslant 0}(M)$ is $\mathfrak{p}$-constructible for any maximal
ideal $\mathfrak{p}$ of $\mathbb{Z}$.

Let $\mathfrak{p}$ be a maximal ideal of $\mathbb{Z}$. By Proposition 1.1.10,
there is a unique t-structure on the stable category
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
such that the canonical functor

$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$

is t-exact. We still call this t-structure the _perverse homotopy
t-structure_. If $M$ is an Artin motive, we get by t-exactness of the functor
$M\mapsto M_{\mathfrak{p}}$ that

${}^{hp}\tau_{\leqslant 0}(M)_{\mathfrak{p}}={}^{hp}\tau_{\leqslant
0}(M_{\mathfrak{p}}).$

Hence, if the perverse homotopy t-structure on the stable category
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
induces a t-structure on the subcategory
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$,
then the Artin motive ${}^{hp}\tau_{\leqslant 0}(M)$ is
$\mathfrak{p}$-constructible for any Artin motive $M$.

Therefore, it suffices to show that the perverse homotopy t-structure induces
a t-structure on the subcategory
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$
for any maximal ideal $\mathfrak{p}$ of $\mathbb{Z}$.

The properties of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},(c)}(-,R)$
transfer to
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},(c)}(-,R)_{\mathfrak{p}}$:
we get functors of the form $f^{*}$ for any morphism $f$, functors of the form
$\omega^{0}f_{*}$ for any morphism of finite type $f$, functors of the form
$f_{!}$ and $\omega^{0}f^{!}$ for any quasi-finite morphism $f$ and
localization triangles. The t-exactness properties of Propositions 5.1.2 and
5.1.3 and the Affine Lefschetz Theorem (Theorem 5.3.7) remain true in this
setting.

Let $p$ be a generator of $\mathfrak{p}$. If $p$ is invertible on a regular
connected scheme $X$, the functor $\rho_{!}$ of Proposition 1.4.3 induces a
t-exact fully faithful functor

$\rho_{!}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(X,R)_{\mathfrak{p}}\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(X,R)_{\mathfrak{p}}$

when the left hand side is endowed with the ordinary t-structure shifted by
$\delta(X)$ and the right hand side is endowed with the perverse homotopy
t-structure. Indeed, the full faithfulness follows from Lemma 4.2.8 while the
t-exactness follows from Proposition 5.2.1.

On the other hand, if a regular scheme $X$ is of characteristic $p$, the
functor $\rho_{!}$ induces a t-exact fully faithful functor

$\rho_{!}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(X,R[1/p])_{\mathfrak{p}}\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(X,R)_{\mathfrak{p}}$

when the left hand side is endowed with the ordinary t-structure shifted by
$\delta(X)$ and the right hand side is endowed with the perverse homotopy
t-structure. Indeed, the full faithfulness follows from Corollary 2.1.6 while
the t-exactness follows from Proposition 5.2.1.

We say that a connected scheme $X$ has the property $\mathscr{P}$ if one of
the two following properties hold:

  * •

The scheme $X$ is of characteristic $p$.

  * •

The prime number $p$ is invertible on $X$.

We say that a scheme $X$ has the property $\mathscr{P}$ if all its connected
components have the property $\mathscr{P}$. The above discussion implies that
in that case, the perverse homotopy t-structure induces a t-structure on
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(X,R)_{\mathfrak{p}}$
and we let $\mathrm{M}_{\mathrm{perv}}^{smA}(X,R)_{\mathfrak{p}}$ be the heart
of this t-structure.

Notice furthermore that if $F$ is any closed subscheme of $S$ which is of
positive codimension, the induction hypothesis and Proposition 1.1.10 imply
that the perverse homotopy t-structure induces a t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$.
We let $\mathrm{M}_{\mathrm{perv}}^{A}(F,R)_{\mathfrak{p}}$ be the heart of
this induced t-structure.

Using Lemma 5.4.2 below, it suffices to show that if $j\colon U\rightarrow S$
is a dense affine open immersion with $U$ regular and having the property
$\mathscr{P}$, letting $i\colon F\rightarrow S$ be the reduced closed
complementary immersion, for any object $M$ of
$\mathrm{M}_{\mathrm{perv}}^{A}(F,R)_{\mathfrak{p}}$, any object $N$ of
$\mathrm{M}_{\mathrm{perv}}^{smA}(U,R)_{\mathfrak{p}}$ and any map

$f\colon M\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N),$

the kernel $K$ of $f$ is a constructible Artin motive.

If $\dim(S)=1$, the scheme $F$ is $0$-dimensional. For any profinite group $G$
and any noetherian ring $\Lambda$, the subcategory
$\operatorname{Rep}^{A}(G,\Lambda)$ of $\mathrm{Mod}(G,\Lambda)$ is Serre.
Therefore, Propositions 5.1.8 and 2.1.7 imply that the subcategory
$\mathrm{M}_{\mathrm{perv}}^{A}(F,R)_{\mathfrak{p}}$ of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)_{\mathfrak{p}}^{\heartsuit}$
is Serre. Since $K$ is a subobject of $M$, it is an object of
$\mathrm{M}_{\mathrm{perv}}^{A}(F,R)_{\mathfrak{p}}$ and thus it
constructible.

If $\dim(S)=2$, the scheme $F$ is of dimension at most $1$.

By Lemma 5.4.3 below, we have an exact sequence

$0\overset{u}{\rightarrow}M_{1}\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)\overset{v}{\rightarrow}M_{2}$

in
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)_{\mathfrak{p}}^{\heartsuit}$
such that $M_{1}$ and $M_{2}$ are a perverse Artin motives. Let $K_{1}$ be the
kernel of the map

$M\overset{f}{\rightarrow}{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)\rightarrow
M_{2},$

it is a perverse Artin motive. Furthermore, the map $f_{0}\colon
K_{1}\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)$ induced by $f$ factors
through $M_{1}$ since

$u\circ f_{0}=0$

and the kernel of the induced map

$K_{1}\rightarrow M_{1}$

is $K$ which is therefore a perverse Artin motive. ∎

######  Lemma 5.4.2.

Let $S$ be a reduced excellent scheme, let $R$ be a regular good ring, let
$\mathfrak{p}$ be a maximal ideal of $\mathbb{Z}$ and let $p$ be a generator
of $\mathfrak{p}$.

Assume that:

  1. (i)

The perverse homotopy t-structure on the stable category
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
induces a t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(F,R)_{\mathfrak{p}}$,
where $F$ is any closed subscheme of $S$ which is of positive codimension.

  2. (ii)

If $j\colon U\rightarrow S$ is a dense affine open immersion with $U$ regular
and having the property $\mathscr{P}$, letting $i\colon F\rightarrow S$ be the
reduced closed complementary immersion, for any object $M$ of
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$, any object $N$ of
$\mathrm{M}_{\mathrm{perv}}^{smA}(U,R)_{\mathfrak{p}}$ and any map

$f\colon M\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N),$

the kernel of $f$ is a perverse Artin motive on $F$.

Then, the perverse homotopy t-structure induces a t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$.

###### Proof.

Step 1: Let $j\colon U\rightarrow S$ be a dense affine open immersion with $U$
regular and having the property $\mathscr{P}$. Let $i\colon F\rightarrow S$ be
the reduced closed complementary immersion. Let $M$ be an object of
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$, let $N$ be an object of
$\mathrm{M}_{\mathrm{perv}}^{smA}(U,R)_{\mathfrak{p}}$. Recall that the object
$i_{*}M$ is constructible and belongs to
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}^{\heartsuit}$.
Furthermore, because of the Affine Lefschetz Theorem, the object $j_{!}N$ also
belongs to
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}^{\heartsuit}$.
Let $f\colon i_{*}M\rightarrow j_{!}N$ be an arbitrary morphism in
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}^{\heartsuit}$.
In this step we prove that the objects $\ker(f)$ and $\operatorname{coker}(f)$
are constructible.

The localization property yields an exact triangle

$j_{!}N\rightarrow\omega^{0}j_{*}N\rightarrow i_{*}\omega^{0}i^{*}j_{*}N$

which, since $j$ is affine yields an exact sequence

$0\rightarrow i_{*}{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)\rightarrow
j_{!}N\rightarrow{}^{hp}H^{0}(\omega^{0}j_{*}N).$

Notice that, by Proposition 5.1.2, we have

${}^{hp}H^{0}(\omega^{0}j_{*}N)={}^{hp}\tau_{\leqslant 0}(\omega^{0}j_{*}N).$

and since $j^{*}i_{*}=0$, we get

$\operatorname{Hom}(i_{*}M,\tau_{\leqslant
0}(\omega^{0}j_{*}N))=\operatorname{Hom}(i_{*}M,\omega^{0}j_{*}N)=0,$

thus, the map $f$ factors through the object
$i_{*}{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)$. Since the functor $i_{*}$ is
$t_{hp}$-exact, we get

$\ker(f)=i_{*}\ker\left(M\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)\right).$

Therefore, our second assumption implies that the object $\ker(f)$ is
constructible.

To finish the first step, let $C$ be the kernel of the map
$j_{!}N\rightarrow\operatorname{coker}(f)$. We have exact triangles

$\ker(f)\rightarrow i_{*}M\rightarrow C$

and

$C\rightarrow j_{!}N\rightarrow\operatorname{coker}(f).$

The first exact triangle shows that $C$ is a constructible. Thus, using the
second exact triangle, the object $\operatorname{coker}(f)$ is also
constructible.

Step 2: We now prove the lemma. By Proposition 5.1.6, the objects of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$
are bounded with respect to the perverse homotopy t-structure. Thus, by
dévissage, it suffices to show that if $P$ is an object of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$
and if $n$ is an integer, then, the object ${}^{hp}H^{n}(P)$ is constructible.

Using Proposition 2.3.1 we have $j\colon U\rightarrow S$ a dense open
immersion such that $j^{*}P$ is an object of
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$.
Shrinking $U$ if needed, we can assume that $U$ is regular since $S$ is
excellent. Shrinking $U$ further, we can assume that it has the property
$\mathscr{P}$. Finally, using [Rui22, 3.3.7], we may assume that the morphism
$j$ is affine.

Let $i\colon F\rightarrow S$ be the reduced complementary closed immersion of
$j$. The localization triangle (0.0.1) yields an exact sequence

$i_{*}{}^{hp}H^{n-1}(i^{*}P)\overset{\alpha}{\rightarrow}j_{!}{}^{hp}H^{n}(j^{*}P)\rightarrow{}^{hp}H^{n}(P)\rightarrow
i_{*}{}^{hp}H^{n}(i^{*}P)\overset{\beta}{\rightarrow}j_{!}{}^{hp}H^{n+1}(j^{*}P)$

which yields an exact triangle:

$\operatorname{coker}(\alpha)\rightarrow{}^{hp}H^{n}(P)\rightarrow\ker(\beta).$

For any integer $m$, the object ${}^{hp}H^{m}(i^{*}P)$ lies in
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$ by our first assumption
and the object ${}^{hp}H^{m}(j^{*}P)$ lies in
$\mathrm{M}_{\mathrm{perv}}^{smA}(U,R)_{\mathfrak{p}}$. Thus, using our first
step, the objects $\operatorname{coker}(\alpha)$ and $\ker(\beta)$ are
constructible and therefore, so is the object ${}^{hp}H^{n}(P)$, which
finishes the proof. ∎

######  Lemma 5.4.3.

Let $S$ be a reduced excellent connected scheme of dimension $2$, let $R$ be a
regular good ring, let $j\colon U\rightarrow S$ be a dense affine open
immersion with $U$ regular and having property $\mathscr{P}$, let $i\colon
F\rightarrow S$ be the reduced closed complementary immersion and let $N$ be
an object of $\mathrm{M}_{\mathrm{perv}}^{smA}(U,R)_{\mathfrak{p}}$.

Then, the object ${}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)$ fits in
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)_{\mathfrak{p}}^{\heartsuit}$
into an exact sequence of the form

$0\rightarrow M_{1}\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)\rightarrow
M_{2}$

where $M_{1}$ and $M_{2}$ belong to
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$.

###### Proof.

We use the convention that $\delta(S)=2$. Replacing $S$ with the closure of a
connected component of $U$, we may assume that $U$ is connected. Thus, since
$U$ is regular and has property $\mathscr{P}$, we have

$\mathrm{M}_{\mathrm{perv}}^{smA}(U,R)_{\mathfrak{p}}=\rho_{!}\operatorname{Loc}_{U}(\Lambda_{0})_{\mathfrak{p}}[2]$

where $\Lambda_{0}=R$ if $p$ is invertible on $U$ and $\Lambda_{0}=R[1/p]$ if
$U$ is of characteristic $p$.

Notice that the result holds when the ring $R$ is torsion by Corollary 1.4.10
and therefore, we can assume that the ring $R$ is the localization of the ring
of integers of a number field $K$.

Denote by $\Lambda$ the ring
$\Lambda_{0}\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}}$. By Propositions
1.5.17 and 1.1.10, the canonical functor

$\operatorname{Loc}_{U}(\Lambda_{0})_{\mathfrak{p}}\rightarrow\operatorname{Loc}_{U}(\Lambda)$

is an equivalence.

Step 1: Let $\xi$ be a geometric point of $U$ and let

$\varepsilon_{!}\colon\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(U,\xi),\Lambda)\rightarrow\operatorname{Loc}_{U}(\Lambda)$

be the inverse of the equivalence of Proposition 1.5.9, let
$\alpha_{!}=\varepsilon_{!}\rho_{!}$ and let $\alpha^{!}$ be the inverse of
$\alpha_{!}$.

Let $P=\alpha^{!}N[-2]$ and let $P_{\mathrm{tors}}$ be the sub-representation
of $P$ made of the torsion elements. There is a positive integer $n$ such that
$p^{n}P_{\mathrm{tors}}$ vanishes; moreover, the
$\Lambda[\pi_{1}^{\operatorname{\acute{e}t}}(U,\xi)]$-module
$P_{\mathrm{tors}}$ is an Artin representation. Therefore, the
$\Lambda[\pi_{1}^{\operatorname{\acute{e}t}}(U,\xi)]$-module
$P/P_{\mathrm{tors}}$ is also an Artin representation and it is torsion free.

Using the Affine Lefschetz Theorem (Theorem 5.3.7), the functor
$\omega^{0}i^{*}j_{*}$ is of perverse homotopy cohomological amplitude bounded
below by $-1$. Therefore, the exact triangle

$\omega^{0}i^{*}j_{*}\alpha_{!}P_{\mathrm{tors}}[2]\rightarrow\omega^{0}i^{*}j_{*}N\rightarrow\omega^{0}i^{*}j_{*}\alpha_{!}(P/P_{\mathrm{tors}})[2]$

induces an exact sequence

$0\rightarrow{}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\alpha_{!}P_{\mathrm{tors}})\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}N)\rightarrow{}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\alpha_{!}(P/P_{\mathrm{tors}})).$

Using Corollary 1.4.10, the object
${}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\alpha_{!}P_{\mathrm{tors}})$ lies in
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$ and therefore, it
suffices to prove that the object
${}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\alpha_{!}(P/P_{\mathrm{tors}}))$ of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)_{\mathfrak{p}}^{\heartsuit}$
is a subobject of an object of
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$.

Since the Artin representation $P/P_{\mathrm{tors}}$ is torsion free, the
canonical map

$P/P_{\mathrm{tors}}\rightarrow P/P_{\mathrm{tors}}\otimes_{\Lambda}K$

is injective. The category
$\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(U,\xi),K)$ is
semi-simple by Maschke’s lemma. Therefore, the representation
$P/P_{\mathrm{tors}}\otimes_{\Lambda}K$ is a direct factor of a representation

$Q_{K}:=K[G_{1}]\bigoplus\cdots\bigoplus K[G_{n}]$

where $G_{1},\ldots,G_{n}$ are finite quotients of
$\pi_{1}^{\operatorname{\acute{e}t}}(U,\xi)$.

Denote by $\lambda_{K}$ the map
$P/P_{\mathrm{tors}}\otimes_{\Lambda}K\rightarrow Q_{K}$. There is an integer
$m$ such that the image of $P/P_{\mathrm{tors}}$ in $Q_{K}$ is contained in
the lattice

$Q_{0}:=\frac{1}{p^{m}}\left(\Lambda[G_{1}]\bigoplus\cdots\bigoplus\Lambda[G_{n}]\right)$

in $Q_{K}$. The representation $Q_{0}$ is isomorphic to
$\Lambda[G_{1}]\bigoplus\cdots\bigoplus\Lambda[G_{n}]$. Thus, we get an
embedding

$P/P_{\mathrm{tors}}\rightarrow\Lambda[G_{1}]\bigoplus\cdots\bigoplus\Lambda[G_{n}].$

Since the functor $\omega^{0}i^{*}j_{*}$ is of perverse homotopy cohomological
amplitude bounded below by $-1$, we get an embedding

${}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\alpha_{!}(P/P_{\mathrm{tors}}))\rightarrow\bigoplus_{i=1}^{n}{}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\alpha_{!}\Lambda[G_{i}]).$

Furthermore, if $1\leqslant i\leqslant n$, the motive
$\alpha_{!}\Lambda[G_{i}]$ is of the form $f_{*}\mathbbm{1}_{V}$ where
$f\colon V\rightarrow U$ is a finite étale map.

Therefore, it suffices to show that the following claim: "If $f\colon
V\rightarrow U$ is a finite étale map, and if $N=f_{*}\mathbbm{1}_{V}$, the
object ${}^{hp}H^{1}(\omega^{0}i^{*}j_{*}N)$ belongs to
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$."

Step 2: In this step we show that to prove the above claim, we can assume $S$
to be normal and that $N=\mathbbm{1}_{U}$.

Let $\overline{V}$ be the relative normalization of $S$ in $V$. Since $S$ is
excellent and noetherian, it is Nagata by [Sta23, 033Z]. Thus, [Sta23, 0AVK]
ensures that the structural map $\nu:\overline{V}\rightarrow S$ is finite.
Furthermore, the scheme $\overline{V}$ is normal by [Sta23, 035L].

Consider the following commutative diagram

${V}$${\overline{V}}$${F^{\prime}}$${U}$${S}$${F}$$\scriptstyle{f}$$\scriptstyle{\gamma}$$\scriptstyle{\nu}$$\scriptstyle{\iota}$$\scriptstyle{\nu_{F}}$$\scriptstyle{j}$$\scriptstyle{i}$

made of cartesian squares.

Now, we have

$\omega^{0}i^{*}j_{*}N=\omega^{0}i^{*}j_{*}f_{*}\mathbbm{1}_{V}=\omega^{0}i^{*}\nu_{*}\gamma_{*}\mathbbm{1}_{V}=(\nu_{F})_{*}\omega^{0}\iota^{*}\gamma_{*}\mathbbm{1}_{V}.$

Since $\nu_{F}$ is finite, it is perverse homotopy t-exact. Thus, we have

${}^{hp}H^{1}(\omega^{0}i^{*}j_{*}N)=(\nu_{F})_{*}{}^{hp}H^{1}(\omega^{0}\psi^{*}\gamma_{*}\mathbbm{1}_{V}).$

Therefore, we can replace $S$ with $\overline{V}$ and $U$ with $V$ in order to
assume that $S$ is normal and that $N=\mathbbm{1}_{U}$.

Step 3: In this step, we study the singularities of $S$.

First, notice that if $Y\rightarrow S$ is a closed immersion and if $F\cap
Y=\varnothing$, we can factor $i$ into a closed immersion $u\colon
F\rightarrow S\setminus Y$ and an open immersion $\xi\colon S\setminus
Y\rightarrow S$. Letting $j^{\prime}\colon U\setminus Y\rightarrow S\setminus
Y$ be the canonical immersion, smooth base change implies that

$\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}=\omega^{0}u^{*}\xi^{*}j_{*}\mathbbm{1}_{U}=\omega^{0}u^{*}j^{\prime}_{*}\mathbbm{1}_{U\setminus
Y}.$

We can therefore remove any closed subset which does not intersect $F$ from
$S$ without changing ${}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}).$
Since $S$ is normal, its singular locus lies in codimension $2$ and we can
therefore assume that it is contained in $F$.

Since $S$ is excellent, Lipman’s Theorem on embedded resolution of
singularities applies (see [Sta23, 0BGP,0BIC,0ADX] for precise statements). We
get a cdh-distinguished square

${E}$${\widetilde{S}}$${F}$${S}$$\scriptstyle{i_{E}}$$\scriptstyle{p}$$\scriptstyle{f}$$\scriptstyle{i}$

such that the map $f$ induces an isomorphism over $U$, the scheme
$\widetilde{S}$ is regular and the subscheme $E$ of $\widetilde{S}$ is a
simple normal crossing divisor.

Let $\gamma\colon U\rightarrow\widetilde{S}$ be the complementary open
immersion of $i_{E}$. We have $j=f\circ\gamma$. Therefore, we get

$\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}=\omega^{0}i^{*}f_{*}\gamma_{*}\mathbbm{1}_{U}=\omega^{0}p_{*}i_{E}^{*}\gamma_{*}\mathbbm{1}_{U}.$

We have a localization exact triangle

$(i_{E})_{*}i_{E}^{!}\mathbbm{1}_{\widetilde{S}}\rightarrow\mathbbm{1}_{\widetilde{S}}\rightarrow\gamma_{*}\mathbbm{1}_{U},$

applying $\omega^{0}p_{*}i_{E}^{*}$, we get an exact triangle

$\omega^{0}p_{*}i_{E}^{!}\mathbbm{1}_{\widetilde{S}}\rightarrow\omega^{0}p_{*}\mathbbm{1}_{E}\rightarrow\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}$

and therefore, we get an exact sequence

${}^{hp}H^{1}\left(\omega^{0}p_{*}i_{E}^{!}\mathbbm{1}_{\widetilde{S}}\right)\rightarrow{}^{hp}H^{1}(\omega^{0}p_{*}\mathbbm{1}_{E})\rightarrow{}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\mathbbm{1}_{U})\rightarrow{}^{hp}H^{2}\left(\omega^{0}p_{*}i_{E}^{!}\mathbbm{1}_{\widetilde{S}}\right)$

and an equivalence

${}^{hp}H^{0}\left(\omega^{0}p_{*}i_{E}^{!}\mathbbm{1}_{\widetilde{S}}\right)={}^{hp}H^{0}(\omega^{0}p_{*}\mathbbm{1}_{E})$

(to get this last equivalence, we use the Affine Lefschetz Theorem).

Since $E$ is a simple normal crossing divisor, there is a finite set $I$ and
regular 1-dimensional closed subschemes $E_{i}$ of $\widetilde{X}$ for $i\in
I$, such that $E=\bigcup\limits_{i\in I}E_{i}$. Chose a total order on the
finite set $I$. If $i<j$, write $E_{ij}=E_{i}\cap E_{j}$. Consider for
$J\subseteq I$ of cardinality at most $2$ the obvious diagram:

${E_{J}}$${E}$${F}$$\scriptstyle{u_{E_{J}}}$$\scriptstyle{p_{J}}$$\scriptstyle{p}$

By cdh-descent and absolute purity, we have an exact triangle

$\bigoplus_{i<j}(u_{E_{ij}})_{*}\mathbbm{1}_{E_{ij}}(-2)[-4]\rightarrow\bigoplus_{i\in
I}(u_{E_{i}})_{*}\mathbbm{1}_{E_{i}}(-1)[-2]\rightarrow
i_{E}^{!}\mathbbm{1}_{\widetilde{S}}$

and applying $\omega^{0}p_{*}$ yields an exact triangle

$\bigoplus_{i<j}\omega^{0}\left((p_{ij})_{*}\mathbbm{1}_{E_{ij}}(-2)\right)[-4]\rightarrow\bigoplus_{i\in
I}\omega^{0}\left((p_{i})_{*}\mathbbm{1}_{E_{i}}(-1)\right)[-2]\rightarrow\omega^{0}p_{*}i_{E}^{!}\mathbbm{1}_{\widetilde{S}}.$

Let $J\subseteq I$ be of cardinality at most $2$ and let $c$ be the
cardinality of $J$. We have an exact triangle

$\omega^{0}\left((p_{J})_{*}(\mathbbm{1}_{E_{J}}\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z})(-c)\right)[-1]\rightarrow\omega^{0}\left((p_{J})_{*}\mathbbm{1}_{E_{J}}(-c)\right)\rightarrow\omega^{0}\left((p_{J})_{*}(\mathbbm{1}_{E_{J}}\otimes_{\mathbb{Z}}\mathbb{Q})(-c)\right)$

[Leh19b, 3.9] ensures that the motive
$\omega^{0}\left((p_{J})_{*}(\mathbbm{1}_{E_{J}}\otimes_{\mathbb{Z}}\mathbb{Q})(-c)\right)$
vanishes. Therefore, using Corollary 1.4.10, we get an equivalence

$\omega^{0}\left((p_{J})_{*}\mathbbm{1}_{E_{J}}(-c)\right)=(p_{J})_{*}(\mathbbm{1}_{E_{J}}\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z})(-c)[-1]$

But then, using Proposition 5.1.9 and the usual properties of the perverse
t-structure described in [BBDG18, 4.2.4], we get

$(p_{J})_{*}(\mathbbm{1}_{E_{J}}\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z})(-c)\geqslant_{hp}0.$

Therefore, we have

$\omega^{0}\left((p_{J})_{*}\mathbbm{1}_{E_{J}}(-c)\right)\geqslant_{hp}1$

and thus, we have

$\omega^{0}\left((p_{J})_{*}\mathbbm{1}_{E_{J}}(-c)\right)[-2c]\geqslant_{hp}1+2c$

which yields

$\omega^{0}p_{*}i_{E}^{!}\mathbbm{1}_{\widetilde{S}}\geqslant_{hp}3.$

Hence, we get an equivalence

${}^{hp}H^{1}(\omega^{0}p_{*}\mathbbm{1}_{E})={}^{hp}H^{1}(\omega^{0}i^{*}j_{*}\mathbbm{1}_{U})$

and the motive ${}^{hp}H^{0}(\omega^{0}p_{*}\mathbbm{1}_{E})$ vanishes.

Step 4: In this step we compute the motive
${}^{hp}H^{1}(\omega^{0}p_{*}\mathbbm{1}_{E})$ and we finish the proof. By
cdh-descent, we have an exact triangle

$\mathbbm{1}_{E}\rightarrow\bigoplus_{i\in
I}(u_{E_{i}})_{*}(u_{E_{i}})^{*}\mathbbm{1}_{E}\rightarrow\bigoplus\limits_{i<j}(u_{E_{ij}})_{*}(u_{E_{ij}})^{*}\mathbbm{1}_{E}.$

and applying $\omega^{0}p_{*}$, we get an exact triangle

$\omega^{0}p_{*}\mathbbm{1}_{E}\rightarrow\bigoplus_{i\in
I}\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}}\rightarrow\bigoplus\limits_{i<j}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}.$

Let

$I_{0}=\\{i\in I\mid\delta(p(E_{i}))=0\\}.$

If $i\in I\setminus I_{0}$, we have $\delta(E_{i})=1$ and the morphism $p_{i}$
is finite. If $i\in I_{0}$, the map $p_{i}$ factors through the singular locus
$Z$ of $S$ which is $0$-dimensional and we can write $p_{i}=\iota\circ\pi_{i}$
with $\iota\colon Z\rightarrow S$ the closed immersion. Finally, notice that
$\delta(Z)=0$ and thus, for any $i<j$, we have $\delta(E_{ij})=0$. Hence,
since the motive ${}^{hp}H^{0}(\omega^{0}p_{*}\mathbbm{1}_{E})$ vanishes, we
get an exact sequence

$0\rightarrow
P\rightarrow\bigoplus\limits_{i<j}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}\rightarrow{}^{hp}H^{1}(\omega^{0}p_{*}\mathbbm{1}_{E})\rightarrow\bigoplus_{i\in
I\setminus I_{0}}(p_{i})_{*}\mathbbm{1}_{E_{i}}\bigoplus Q\rightarrow 0$

where $P=\iota_{*}\bigoplus_{i\in
I_{0}}{}^{hp}H^{0}(\omega^{0}(\pi_{i})_{*}\mathbbm{1}_{E_{i}})$ and
$Q=\iota_{*}\bigoplus_{i\in
I_{0}}{}^{hp}H^{1}(\omega^{0}(\pi_{i})_{*}\mathbbm{1}_{E_{i}})$.

If $i\in I_{0}$, let

$E_{i}\rightarrow F_{i}\overset{q_{i}}{\rightarrow}Z$

be the Stein factorization of $\pi_{i}$. Corollary 3.1.8 yields

$P=\iota_{*}\bigoplus_{i\in I_{0}}(q_{i})_{*}\mathbbm{1}_{F_{i}}$

and

$Q=0.$

Hence, the objects $P$ and $Q$ belong to
$\mathrm{M}^{A}_{\mathrm{perv}}(F,R)_{\mathfrak{p}}$ and therefore, so does
the object ${}^{hp}H^{1}(\omega^{0}p_{*}\mathbbm{1}_{E})$ which finishes the
proof. ∎

######  Proposition 5.4.4.

Let $S$ be an excellent scheme of dimension $2$ or less, let $f\colon
X\rightarrow S$ be a quasi-finite morphism of schemes and let $R$ be a regular
good ring. Assume that $X$ is nil-regular.

Then, the functor

$f_{!}\colon\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(X,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(X,R)$

is t-exact when both sides are endowed with the perverse homotopy t-structure.

###### Proof.

This follows from Theorem 5.3.7. ∎

######  Example 5.4.5.

Let $X$ be a normal scheme of dimension $4$ with a single singular point $x$
of codimension $4$, let $k$ be the residue field at $x$, let $p$ be the
characteristic exponent of $k$ and let $f\colon\widetilde{X}\rightarrow X$ be
a resolution of singularities of $X$.

Assume that the exceptional divisor above $x$ is isomorphic to
$\mathbb{P}^{3}_{k}$ and that the ring of coefficients is $\mathbb{Z}$. Take
the convention that $\delta(X)=4$.

We have a cdh-distinguished square

${\mathbb{P}^{3}_{k}}$${\widetilde{X}}$${\operatorname{Spec}(k)}$${X}$$\scriptstyle{i_{E}}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\scriptstyle{i}$

which yields an exact triangle

$\mathbbm{1}_{X}\rightarrow\omega^{0}f_{*}\mathbbm{1}_{\widetilde{X}}\oplus
i_{*}\mathbbm{1}_{k}\rightarrow
i_{*}\omega^{0}\pi_{*}\mathbbm{1}_{\mathbb{P}^{3}_{k}}.$

Furthermore, letting
$U=X\setminus\\{x\\}=\widetilde{X}\setminus\mathbb{P}^{3}_{k}$ and letting
$j\colon X\setminus\\{x\\}\rightarrow X$ be the open immersion, the
localization property (0.0.2) yields an exact triangle

$i_{*}\omega^{0}\pi_{*}(i_{E})^{!}\mathbbm{1}_{\widetilde{X}}\rightarrow\omega^{0}f_{*}\mathbbm{1}_{\widetilde{X}}\rightarrow\omega^{0}j_{*}\mathbbm{1}_{U}.$

Since by Propositions 5.1.2 and 5.2.4, the Artin motive
$\omega^{0}j_{*}\mathbbm{1}_{U}$ lies in degree at least $4$, we get that

${}^{hp}H^{3}(\omega^{0}f_{*}\mathbbm{1}_{\widetilde{X}})=i_{*}{}^{hp}H^{3}(\omega^{0}\pi_{*}(i_{E})^{!}\mathbbm{1}_{\widetilde{X}}).$

On the other hand, the Artin motives
${}^{hp}H^{2}(\omega^{0}\pi_{*}\mathbbm{1}_{\mathbb{P}^{3}_{k}})$ and
${}^{hp}H^{3}(\omega^{0}\pi_{*}\mathbbm{1}_{\mathbb{P}^{3}_{k}})$ vanish by
Corollary 3.1.8. Moreover, the Artin motive $i_{*}\mathbbm{1}_{k}$ is in the
heart of the perverse homotopy t-structure. Hence, we get

${}^{hp}H^{3}(\mathbbm{1}_{X})=i_{*}{}^{hp}H^{3}(\omega^{0}\pi_{*}(i_{E})^{!}\mathbbm{1}_{\widetilde{X}}).$

Using the absolute purity property, we have

$\omega^{0}\pi_{*}(i_{E})^{!}\mathbbm{1}_{\widetilde{X}}=\omega^{0}\left(\pi_{*}\mathbbm{1}_{\mathbb{P}^{3}_{k}}(-1)\right)[-2]$

and therefore, we get

${}^{hp}H^{3}(\mathbbm{1}_{X})=i_{*}{}^{hp}H^{1}\left(\omega^{0}\left(\pi_{*}\mathbbm{1}_{\mathbb{P}^{3}_{k}}(-1)\right)\right).$

Finally, Corollary 3.1.8 yields

${}^{hp}H^{3}(\mathbbm{1}_{X})=i_{*}\left(\alpha_{!}\mathbb{Q}/\mathbb{Z}[1/p](-1)\right)$

using 3.1.7. Hence, the Artin motive ${}^{hp}H^{3}(\mathbbm{1}_{X})$ is not
constructible.

######  Example 5.4.6.

Let $X$ be the affine cone over $\mathbb{P}^{3}_{k}$ embedded in
$\mathbb{P}^{N}$ through a Veronese embedding of degree at least $2$. The
scheme $X$ has a single singular point $x$ and the blow up of $X$ at $x$ is a
resolution of singularities of $X$ with exceptional divisor
$\mathbb{P}^{3}_{k}$.

By Noether’s normalization lemma, there is a finite map
$X\rightarrow\mathbb{A}^{4}_{k}$. Thus, the perverse homotopy t-structure does
not induce a t-structure on
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(\mathbb{A}^{4}_{k},\mathbb{Z})$.
As $\mathbb{A}^{4}_{k}$ is a closed subscheme of $\mathbb{A}^{n}_{k}$ for
$n\geqslant 4$, the perverse t-structure does not induce a t-structure on
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(\mathbb{A}^{n}_{k},\mathbb{Z})$
if $n\geqslant 4$.

######  Remark 5.4.7.

Loosely speaking, if the scheme $X$ becomes more singular, the perverse
homotopy cohomology sheaves of $\mathbbm{1}_{X}$ should become more
complicated. Here, the simplest possible singularity on a scheme of dimension
$4$ over a finite field already renders the cohomology sheaves not
constructible. Therefore, Theorem 5.4.1 should not hold for schemes of
dimension $4$ or more. The case of $3$-folds, however, remains open.

##  6\. The abelian category of perverse Artin étale motives

###  6.1. Definition and first properties

######  Definition 6.1.1.

Let $S$ be an excellent scheme and let $R$ be a good ring. Assume that the
perverse homotopy t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)$ induces a
t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$. We
define the _abelian category of perverse Artin étale motives over $S$ with
coefficients in $R$_, as the heart of the perverse homotopy t-structure on
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(S,R)$. We denote by
$\mathrm{M}^{A}_{\mathrm{perv}}(S,R)$ this category.

######  Proposition 6.1.2.

Let $R$ be a good ring. Assume that the perverse homotopy t-structure induces
a t-structure on the category of constructible Artin étale motives with $R$
coefficients over all schemes appearing below.

  1. (1)

Let $f\colon T\rightarrow S$ be a quasi-finite morphism of schemes. Then, the
functor $f_{!}$ induces a right exact functor

${}^{hp}H^{0}f_{!}\colon\mathrm{M}^{A}_{\mathrm{perv}}(T,R)\rightarrow\mathrm{M}^{A}_{\mathrm{perv}}(S,R).$

Furthermore, if the functor $f$ is finite, we have ${}^{hp}H^{0}f_{!}=f_{!}$
and this functor is exact.

  2. (2)

Let $g\colon T\rightarrow S$ be a morphism of schemes. Assume that
$\dim(g)\leqslant d$. Then, the functor $g^{*}$ induces a right exact functor

${}^{hp}H^{d}g^{*}\colon\mathrm{M}^{A}_{\mathrm{perv}}(S,R)\rightarrow\mathrm{M}^{A}_{\mathrm{perv}}(T,R).$

Furthermore, if the functor $f$ is étale, we have $H^{0}f^{*}=f^{*}$ and this
functor is exact.

  3. (3)

Consider a cartesian square of schemes

${Y}$${X}$${T}$${S}$$\scriptstyle{g}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{f}$

such that $p$ is quasi-finite and $\dim(f)\leqslant d$. Then we have a
canonical equivalence

${}^{hp}H^{d}f^{*}{}^{hp}H^{0}p_{!}\rightarrow{}^{hp}H^{0}q_{!}{}^{hp}H^{d}g^{*}.$

  4. (4)

Let $S$ be a scheme. Assume that $\dim(S)\leqslant 2$. Let $f:T\rightarrow S$
be a quasi-finite and affine morphism. Then, the functor

${}^{hp}H^{0}f_{!}\colon\mathrm{M}^{A}_{\mathrm{perv}}(T,R)\rightarrow\mathrm{M}^{A}_{\mathrm{perv}}(S,R).$

is exact and ${}^{hp}H^{0}f_{!}=f_{!}$.

  5. (5)

Let $i:F\rightarrow S$ be a closed immersion and $j:U\rightarrow S$ be the
open complement. Assume that $\dim(S)\leqslant 2$ and that $j$ is affine.
Then, we have an exact sequence:

$0\rightarrow i_{*}{}^{hp}H^{-1}i^{*}M\rightarrow j_{!}j^{*}M\rightarrow
M\rightarrow i_{*}{}^{hp}H^{0}i^{*}M\rightarrow 0.$

  6. (6)

Let $p$ be a prime number. Assume that the ring $R=K$ is a number field and
that the scheme $S$ is of finite type over $\mathbb{F}_{p}$. Let $v$ be a non-
archimedian valuation on $K$ which does not extend the $p$-adic valuation.
Then, the $v$-adic realization functor $\rho_{v}$ induces an exact and
conservative functor

$\mathrm{M}^{A}_{\mathrm{perv}}(S,K)\rightarrow\mathrm{Perv}^{A}(S,K_{v})^{\\#}.$

  7. (7)

Let $R$ be a $\mathbb{Z}/n\mathbb{Z}$-algebra where $n$ is invertible on $S$.
Then, the functor $\rho_{!}$ induces an equivalence of categories

$\rho_{!}\colon\mathrm{Perv}(S,R)\rightarrow\mathrm{M}^{A}_{\mathrm{perv}}(S,R).$

###### Proof.

This follows from Propositions 5.1.2, 5.4.4, 5.3.1 and 5.1.9. ∎

###  6.2. Intermediate extension, simple objects and the Ayoub-Zucker
weightless intersection complex (rational coefficients)

We now show that the abelian category of perverse Artin motives with
coefficients in a number field $K$ has similar features as the category of
perverse sheaves.

First, we recall the definition of the intermediate extension functor given by
the gluing formalism (see [BBDG18, 1.4.22]):

######  Definition 6.2.1.

Let $S$ be a scheme, let $K$ be a number field and let $j\colon U\rightarrow
S$ be an open immersion. The _intermediate extension functor_ $j_{!*}^{A}$ is
the functor which to an object $M$ in $\mathrm{M}^{A}_{\mathrm{perv}}(S,K)$
associates the image of the morphism

${}^{hp}H^{0}j_{!}M\rightarrow{}^{hp}H^{0}\omega^{0}j_{*}M.$

Using [BBDG18, 1.4.23], we have the following description of this functor
(compare with [BBDG18, 2.1.9]):

######  Proposition 6.2.2.

Let $S$ be a scheme, let $i\colon F\rightarrow S$ be a closed immersion, let
$j\colon U\rightarrow S$ be the open complementary immersion and let $M$ be in
$\mathrm{M}^{A}_{\mathrm{perv}}(U,K)$. Then, the Artin motive $j_{!*}^{A}(M)$
is the only extension $P$ of $M$ to $S$ such that $i^{*}P<_{hp}0$ and
$\omega^{0}i^{!}P>_{hp}0$.

The following result is similar to [BBDG18, 4.3.1].

######  Proposition 6.2.3.

Let $S$ be an excellent scheme.

  1. (1)

The abelian category of perverse Artin motives with rational coefficients on
$S$ is artinian and noetherian: every object is of finite length.

  2. (2)

If $j\colon V\hookrightarrow S$ is the inclusion of a regular connected
subscheme and if $L$ is a simple object of $\operatorname{Loc}_{V}(K)$, then
the perverse Artin motive $j_{!*}^{A}(\rho_{!}L[\delta(V)])$ is simple. Every
simple perverse Artin motive is obtained this way.

###### Proof.

The proof is the same as in [BBDG18, 4.3.1] replacing [BBDG18, 4.3.2, 4.3.3]
with the following lemmas. ∎

######  Lemma 6.2.4.

Let $V$ be a regular connected scheme. Let $L$ be a simple object of
$\operatorname{Loc}_{V}(K)$. Then, letting $F=\rho_{!}L[\delta(V)]$, for any
open immersion $j\colon U\hookrightarrow V$, we have

$F=j_{!*}^{A}j^{*}F.$

###### Proof.

Let $i\colon F\rightarrow V$ be the reduced complementary closed immersion to
$j$. Then, the motive $i^{*}F$ belongs to
$\rho_{!}\operatorname{Loc}_{F}(R)[\delta(X)]$ at is therefore in degree at
most $\delta(F)-\delta(X)<0$ with respect to the perverse homotopy
t-structure.

Now, if $x$ is a point of $F$, we have

$\omega^{0}i_{x}^{!}F=0$

by absolute purity and using Corollary 3.1.8. Therefore, by Proposition 5.2.1,
we get

$\omega^{0}i^{!}F>_{hp}0.$

Therefore, the motive $F$ is an extension of $j^{*}F$ such that
$i^{*}F<_{hp}0$ and $\omega^{0}i^{!}F>_{hp}0$ and the proposition follows from
Proposition 6.2.2. ∎

######  Lemma 6.2.5.

Let $V$ be a regular connected scheme. If $L$ is a simple object of
$\operatorname{Loc}_{V}(K)$, then $\rho_{!}L[\delta(V)]$ is simple in
$\mathrm{M}^{A}_{\mathrm{perv}}(V,K)$.

###### Proof.

The proof is exactly the same as [BBDG18, 4.3.3]. ∎

######  Proposition 6.2.6.

(compare with [Rui22, 4.9.7]) Let $X$ be a scheme allowing resolution of
singularities by alterations, let $j\colon U\rightarrow X$ be an open
immersion with $U$ nil-regular and let $d=\delta(X)$. Ayoub and Zucker defined
(see [AZ12, 2.20]) a motive

$\mathbb{E}_{X}=\omega^{0}j_{*}\mathbbm{1}_{U}.$

Then,

  1. (1)

$\mathbb{E}_{X}[d]=j_{!*}^{A}(\mathbbm{1}_{U}[d])$.

  2. (2)

$\mathbb{E}_{X}[d]$ is a simple perverse Artin motive over $X$. In particular,
$\omega^{0}j_{*}\mathbbm{1}_{U}$ is concentrated in degree $d$ with respect to
the perverse homotopy t-structure.

###### Proof.

First, notice that Assertion (2) follows from Assertion (1) and Proposition
6.2.3. Corollary 5.3.10 ensures that $\omega^{0}j_{*}\mathbbm{1}_{U}$ is in
degree $d$ with respect to the perverse homotopy t-structure. Moreover, we
have a triangle:

$j_{!}\mathbbm{1}_{U}\rightarrow\omega^{0}j_{*}\mathbbm{1}_{U}\rightarrow
i_{*}\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}.$

thus, it suffices to show that
${}^{hp}H^{d}(\omega^{0}i^{*}j_{*}\mathbbm{1}_{U})=0$. To prove this, note
that we can assume $X$ to be normal and connected as in the first step of the
proof of Proposition 5.3.5. By assumption, there is a proper and surjective
map $p\colon Y\rightarrow X$ such that $Y$ is regular and integral and $p$ is
generically the composition of an étale Galois cover and of a finite
surjective purely inseparable morphism.

Therefore, there is a closed subscheme $S$ of $X$ of positive codimension such
that $p:Y\setminus p^{-1}(S)\rightarrow X\setminus S$ is the composition of an
étale Galois cover of group $G$ and of a finite surjective purely inseparable
morphism.

Let $X^{\prime}$ be the relative normalization of $X$ in $Y\setminus
p^{-1}(S)$. Then, the scheme $X^{\prime}$ is finite over $X$ and endowed with
a $G$-action and the morphism $X^{\prime}/G\rightarrow X$ is purely
inseparable since $X$ is normal. We have a commutative diagram

${U^{\prime}}$${X^{\prime}}$${F^{\prime}}$${U}$${X}$${F}$$\scriptstyle{p_{U}}$$\scriptstyle{j^{\prime}}$$\scriptstyle{p}$$\scriptstyle{i^{\prime}}$$\scriptstyle{p_{F}}$$\scriptstyle{j}$$\scriptstyle{i}$

made of cartesian squares. Using [Ayo07, 2.1.165], the motive
$\mathbbm{1}_{U}$ is a retract of the motive
$(p_{U})_{*}\mathbbm{1}_{U^{\prime}}$. Moreover, we have

$\omega^{0}i^{*}j_{*}(p_{U})_{*}\mathbbm{1}_{U^{\prime}}=(p_{F})_{*}\omega^{0}(i^{\prime})^{*}j^{\prime}_{*}\mathbbm{1}_{U^{\prime}}.$

Thus, we can replace $X$ with $X^{\prime}$ and assume that the group $G$ is
trivial. In this case, the morphism $p$ is birational. Let $\pi\colon
E\rightarrow F$ be the pullback of $p$ along $i$. Using the same argument as
in the first step of the proof of Proposition 5.3.5, we get that

$\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}=\omega^{0}\pi_{*}\mathbbm{1}_{E}.$

The proof then follows from Proposition 5.3.8. ∎

###  6.3. Description of perverse Artin motives and t-exactness of the
realization

The following description of Artin perverse motives is a variation on
Beilinson’s gluing method developped in [Bei87]. Let $M$ be an Artin perverse
motive over an excellent base scheme $S$ of dimension at most $2$. Proposition
2.3.1 gives a stratification of $S$ with regular strata and such that for any
stratum $T$, the motive $M|_{T}$ is a constructible smooth Artin motive. We
can recover the motive $M$ from the motives $M|_{T}$ and additional data.

We take the convention that $\delta(S)=2$ and assume for simplicity that the
scheme $S$ is reduced. Let $U$ be the (disjoint) union of the strata
containing the generic points of $S$. Without loss of generality, we can
assume that $U$ is a single stratum over which the motive $M$ a constructible
smooth Artin motive. Shrinking the open subset $U$, we can also assume that
the immersion $j\colon U\rightarrow S$ is affine. Write $i\colon F\rightarrow
S$ the reduced complementary closed immersion. The motive $M|_{U}$ is a
perverse Artin motive and the constructible Artin motive $M|_{F}$ is in
perverse degree $[-1,0]$.

By localization, we have an exact triangle

$i_{*}M|_{F}[-1]\rightarrow j_{!}M|_{U}\rightarrow M.$

Furthermore, by Proposition 6.1.2, we have an exact sequence

$0\rightarrow i_{*}{}^{hp}H^{-1}M|_{F}\rightarrow j_{!}M|_{U}\rightarrow
M\rightarrow i_{*}{}^{hp}H^{0}M|_{F}\rightarrow O$

Hence, we can recover $M$ as the cofiber of a map $i_{*}M|_{F}[-1]\rightarrow
j_{!}M|_{U}$ such that the induced map $i_{*}{}^{hp}H^{-1}M|_{F}\rightarrow
j_{!}M|_{U}$ is a monomorphism.

Conversely, a smooth Artin perverse sheaf $M_{U}$ over $U$, a constructible
Artin motive $M_{F}$ in perverse degrees $[-1,0]$ over $F$ and a connecting
map $i_{*}M_{F}[-1]\rightarrow j_{!}M_{U}$ such that the induced map
$i_{*}{}^{hp}H^{-1}M_{F}\rightarrow j_{*}M_{U}$ is a monomorphism, give rise
to a unique perverse Artin motive $M$ over $S$.

Now, recall that we have an exact triangle

$j_{!}\rightarrow\omega^{0}j_{*}\rightarrow i_{*}\omega^{0}i^{*}j_{*}$

and that $j^{*}i_{*}=0$. Therefore, we have an exact sequence

$0\rightarrow i_{*}{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}M_{U})\rightarrow
j_{!}M_{U}\rightarrow{}^{hp}H^{0}(\omega^{0}j_{*}M_{U})\rightarrow
i_{*}{}^{hp}H^{0}(\omega^{0}i^{*}j_{*}M_{U})\rightarrow 0.$

and we have

$\operatorname{Hom}(i_{*}M_{F}[-1],j_{!}M_{U})=\operatorname{Hom}(M_{F},\omega^{0}i^{*}j_{*}M_{U}).$

Therefore, if $\phi\colon i_{*}M_{F}[-1]\rightarrow j_{!}M_{U}$ is any map,
the map ${}^{hp}H^{0}(\phi)$ is a monomorphism if and only if the induced map

${}^{hp}H^{-1}(M_{F})\rightarrow{}^{hp}H^{-1}(\omega^{0}i^{*}j_{*}M_{U})$

is a monomorphism.

Similarly, since the stratification of $S$ induces a stratification of $F$,
the Artin motive $M_{F}$ is uniquely encoded by the data of a smooth
constructible Artin motive $M_{1}$ concentrated in perverse degrees $[-1,0]$
over an affine open subset $U_{1}$ of $F$, a constructible Artin motive
$N_{1}$ in perverse degrees $[-2,0]$ on $F_{1}=F\setminus U_{1}$ and a
connecting map with a similar condition.

This discussion provides the following proposition:

######  Proposition 6.3.1.

Let $S$ be an excellent scheme of dimension $2$ or less and let $R$ be a
regular good ring. Then, perverse Artin étale motive over $S$ are uniquely
encoded by the following data.

  * •

A triple $(U_{0},U_{1},U_{2})$ of locally closed subschemes of $S$ such that
each $U_{k}$ is nil-regular, purely of codimension $k$, is open in the scheme
$F_{k-1}:=S\setminus(U_{0}\cup\cdots\cup U_{k-1})$ and such that the open
immersion

$j_{k}\colon U_{k}\rightarrow F_{k-1}$

is affine (take the conventions that $F_{-1}=S$). Denote by

$i_{k}\colon F_{k}\rightarrow F_{k-1}$

the closed immersion.

  * •

A triple $(M_{0},M_{1},M_{2})$ of constructible smooth Artin motives on
$U_{i}$ such that each $M_{i}$ is placed in perverse degree $[-i,0]$.

  * •

Two connecting maps

$\begin{cases}\phi_{2}\colon
M_{2}\rightarrow\omega^{0}i_{1}^{*}(j_{1})_{*}M_{1}\\\ \phi_{1}\colon
M_{12}\rightarrow\omega^{0}i_{0}^{*}(j_{0})_{*}M_{0}\end{cases}$

such that ${}^{hp}H^{-k}(\phi_{k})$ is a monomorphism and where $M_{12}$ is
the cofiber of the map $(i_{1})_{*}M_{2}[-1]\rightarrow(j_{1})_{!}M_{1}$
induced by $\phi_{2}$.

######  Remark 6.3.2.

If $\dim(S)\leqslant 1$, $U_{2}=\emptyset$, $M_{2}=0$ and the data of
$\phi_{1}$ is trivial so that $M_{12}=M_{1}$.

This description yields the t-exactness if the $\ell$-adic realization
functor.

######  Theorem 6.3.3.

Let $S$ be an excellent scheme of dimension $2$ or less, let $R$ be a
localization of the ring of integers of a number field $K$, let $v$ be a non-
archimedian valuation on $K$ and let $\ell$ be the prime number such that $v$
extends the $\ell$-adic valuation. Then, the reduced $v$-adic realization
functor

$\overline{\rho}_{v}\colon\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(S,R)\rightarrow\mathcal{D}^{b}_{c}(S[1/\ell],R_{v})$

of Definition 2.2.1 is t-exact when the left hand side is endowed with the
perverse homotopy t-structure and the right hand side is endowed with the
perverse t-structure.

###### Proof.

First, notice that $\rho_{v}$ is right t-exact because the generators of the
perverse homotopy t-structure are send to perverse t-non-positive objects.
Using same trick as in the proof of Theorem 4.2.6, we can assume that $R=K$.
We may also assume that $\ell$ is invertible on $S$. Now, it suffices to show
that any perverse Artin étale motive on $S$ is sent to a complex which is non-
negative for the perverse t-structure. Let $M$ be such a motive on $S$.

By Proposition 6.3.1, the motive $M$ is given by

  * •

A triple $(U_{0},U_{1},U_{2})$ of locally closed subschemes of $S$ such that
each $U_{k}$ is nil-regular, purely of codimension $k$, is open in the scheme
$F_{k-1}:=S\setminus(U_{0}\cup\cdots\cup U_{k-1})$ and such that the open
immersion

$j_{k}\colon U_{k}\rightarrow F_{k-1}$

is affine (take the conventions that $F_{-1}=S$). Denote by

$i_{k}\colon F_{k}\rightarrow F_{k-1}$

the closed immersion.

  * •

A triple $(M_{0},M_{1},M_{2})$ of constructible smooth Artin motives on
$U_{i}$ such that each $M_{i}$ is placed in perverse degree $[-i,0]$.

  * •

Two connecting maps

$\begin{cases}\phi_{2}\colon
M_{2}\rightarrow\omega^{0}i_{1}^{*}(j_{1})_{*}M_{1}\\\ \phi_{1}\colon
M_{12}\rightarrow\omega^{0}i_{0}^{*}(j_{0})_{*}M_{0}\end{cases}$

such that ${}^{hp}H^{-k}(\phi_{k})$ is a monomorphism and where $M_{12}$ is
the cofiber of the map $(i_{1})_{*}M_{2}[-1]\rightarrow(j_{1})_{!}M_{1}$
induced by $\phi_{2}$.

By Proposition 5.2.1, the $\ell$-adic complex
$\rho_{v}(M)|_{U_{i}}=\rho_{v}(M_{i})$ is in perverse degree $[-i,0]$. To show
that $\rho_{v}(M)$ is a perverse sheaf, it suffices to show that the natural
maps

$\begin{cases}\psi_{2}\colon\rho_{v}(M_{2})\rightarrow\rho_{v}(i_{1}^{*}(j_{1})_{*}M_{1})\\\
\psi_{1}\colon\rho_{v}(M_{12})\rightarrow\rho_{v}(i_{0}^{*}(j_{0})_{*}M_{0})\end{cases}$

are such that ${}^{p}H^{-k}(\psi_{k})$ is a monomorphism.

Notice now that if $k\in\\{1,2\\}$ and $M_{\geqslant
k}=\begin{cases}M_{12}\text{ if }k=1\\\ M_{2}\text{ if }k=2\end{cases}$, we
have a commutative diagram

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Working by induction on $\dim(S)$, we may assume that the maps

$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(F_{k-1},K)\rightarrow\mathcal{D}^{b}_{c}(F_{k-1},K_{v})$

are t-exact. Thus, the map ${}^{p}H^{-k}(\rho_{v}(\phi_{k}))$ is a
monomorphism. To prove the theorem, it therefore suffices to show that the map
${}^{p}H^{-k}\left[\rho_{v}(\delta(i^{*}_{k-1}(j_{k-1})_{*}M_{k-1}))\right]$
is a monomorphism.

Write

$M_{1}=A[1]\bigoplus B[0],$

which is possible since the category of smooth Artin motives with coefficients
in $K$ is semi-simple by Maschke’s theorem. Then, using Theorem 5.3.7 and the
induction hypothesis, the $\ell$-adic complex
$\rho_{v}(\omega^{0}i^{*}_{1}(j_{1})_{*}B)$ is in degree at least $-1$ with
respect to the perverse t-structure. By the usual affine Lefschetz theorem, so
is $\rho_{v}(i^{*}_{1}(j_{1})_{*}B)$, since by Proposition 5.2.1, the
$\ell$-adic complex $\rho_{v}(B)$ is a perverse sheaf. Hence, the map
${}^{p}H^{-2}\left[\rho_{v}(\delta(i^{*}_{1}(j_{1})_{*}M_{1}))\right]$ is a
monomorphism if and only if the map

${}^{p}H^{-1}\left[\rho_{v}(\delta)\right]\colon{}^{p}H^{-1}\left[\rho_{v}(\omega^{0}i_{1}^{*}(j_{1})_{*}A)\right]\longrightarrow{}^{p}H^{-1}\left[\rho_{v}(i_{1}^{*}(j_{1})_{*}A)\right]$

is a monomorphism. Hence, to prove the theorem, it suffices to show that the
maps

${}^{p}H^{-1}\left[\rho_{v}(\delta)\right]\colon{}^{p}H^{-1}\left[\rho_{v}(\omega^{0}i_{1}^{*}(j_{1})_{*}A)\right]\longrightarrow{}^{p}H^{-1}\left[\rho_{v}(i_{1}^{*}(j_{1})_{*}A)\right]$

and

${}^{p}H^{-1}\left[\rho_{v}(\delta)\right]\colon{}^{p}H^{-1}\left[\rho_{v}(\omega^{0}i_{0}^{*}(j_{0})_{*}M_{0})\right]\longrightarrow{}^{p}H^{-1}\left[\rho_{v}(i_{0}^{*}(j_{0})_{*}M_{0})\right]$

are monomorphisms which follows from lemma 6.3.4 below. ∎

######  Lemma 6.3.4.

Let $S$ be an excellent scheme of dimension $2$ or less, let $K$ be a number
field, let $v$ be a non-archimedian valuation on $K$, let $\ell$ be the prime
number such that $v$ extends the $\ell$-adic valuation, let $i\colon
F\rightarrow S$ be a closed immersion and let $j\colon U\rightarrow S$ be the
open complementary immersion. Assume that the prime number $\ell$ is
invertible on $S$, that the morphism $j$ is affine and that the scheme $U$ is
nil-regular. Let $M$ be an object of $\mathrm{M}^{smA}(U,K)$. Then, the map

${}^{p}H^{-1}\rho_{v}(\delta)\colon{}^{p}H^{-1}\left[\rho_{v}(\omega^{0}i^{*}j_{*}M)\right]\longrightarrow{}^{p}H^{-1}\left[\rho_{v}(i^{*}j_{*}M)\right]$

is a monomorphism.

###### Proof.

Let $d=\dim(S)$. Take the convention that $\delta(S)=d$. As in the proof of
Proposition 5.3.5, we can assume that $M=\mathbbm{1}_{U}[d]$, that the scheme
$S$ is normal and that the scheme $U$ is regular.

Case 1: Assume that $\dim(S)=1$. Since the closed subscheme $F$ is
$0$-dimensional, we can assume it to be reduced to a point $\\{x\\}$. Recall
that the $2$-morphism

$\Theta\colon\rho_{!}i^{*}j_{*}\rightarrow\omega^{0}i^{*}j_{*}\rho_{!}$

of Proposition 3.2.1 is defined so that we have a commutative diagram

${\rho_{!}i^{*}j_{*}}$${\omega^{0}i^{*}j_{*}}$${i^{*}j_{*}\rho_{!}}$$\scriptstyle{\Theta}$$\scriptstyle{Ex}$$\scriptstyle{\delta}$

where $Ex$ is the exchange map.

Moreover, the map $\Theta(\underline{K})$ is an equivalence by Proposition
3.2.1. Therefore, it suffices to show that the map

${}^{p}H^{0}\rho_{v}(Ex)\colon{}^{p}H^{0}\rho_{v}(\rho_{!}i^{*}j_{*}\underline{K})\longrightarrow{}^{p}H^{0}(i^{*}j_{*}\underline{K_{v}})$

is a monomorphism. Now, this map can be by [AGV73, VIII.5] identified with the
canonical map

$\underline{K_{v}}\rightarrow
H^{0}_{\operatorname{\acute{e}t}}(\mathcal{O}_{S,x}^{sh}\times_{S}U,K_{v})$

where $\mathcal{O}_{S,x}^{sh}$ is the strict henselization of the local ring
of $S$ at $x$. Since $S$ is normal, this map is an isomorphism by Proposition
1.7.2.

Case 2: Assume that $\dim(S)=2$. Since $S$ is normal its singular point are of
codimension $2$. Let $f\colon\widetilde{S}\rightarrow S$ be a resolution of
singularities of $S$ such that the inverse image of $F$ is a simple normal
crossing divisor (which exists by Lipman’s theorem). We have a commutative
diagram

${U}$${\widetilde{S}}$${E}$${U}$${S}$${F}$$\scriptstyle{\gamma}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\scriptstyle{i}$

made of cartesian squares which yields a commutative diagram

${\rho_{v}(\omega^{0}i^{*}j_{*}\mathbbm{1}_{U})}$${\rho_{v}(\omega^{0}p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})}$${\rho_{v}(i^{*}j_{*}\mathbbm{1}_{U})}$${\rho_{v}(p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})}$$\scriptstyle{\rho_{v}(\delta)}$$\scriptstyle{\rho_{v}(\delta)}$

where the horizontal maps are equivalences. Thus, it suffices to show that the
map

${}^{p}H^{1}\rho_{v}(\delta)\colon{}^{p}H^{1}\rho_{v}(\omega^{0}p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})\longrightarrow{}^{p}H^{1}\rho_{v}(p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})$

is a monomorphism.

The localization triangle (0.0.2) induces a morphism of exact triangles:

${{\rho_{v}(\omega^{0}p_{*}\iota^{!}\mathbbm{1}_{\widetilde{S}})}}$${{\rho_{v}(\omega^{0}p_{*}\mathbbm{1}_{E})}}$${{\rho_{v}(\omega^{0}p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})}}$${{\rho_{v}(p_{*}\iota^{!}\mathbbm{1}_{\widetilde{S}})}}$${{\rho_{v}(p_{*}\mathbbm{1}_{E})}}$${{\rho_{v}(p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})}}$$\scriptstyle{\rho_{v}(\delta)}$$\scriptstyle{\rho_{v}(\delta)}$$\scriptstyle{\rho_{v}(\delta)}$

By Lemma 3.2.2, we have

$\omega^{0}p_{*}\iota^{!}\mathbbm{1}_{\widetilde{S}}=0.$

Moreover, using the cdh-descent argument which appears the last step of the
proof of Proposition 5.3.5 we can show that

${}^{p}H^{1}(\rho_{v}(p_{*}\iota^{!}\mathbbm{1}_{\widetilde{S}}))=0.$

Hence, we get a commutative diagram with exact rows

${{0}}$${{{}^{p}H^{1}\rho_{v}(\omega^{0}p_{*}\mathbbm{1}_{E})}}$${{{}^{p}H^{1}\rho_{v}(\omega^{0}p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})}}$${0}$${{0}}$${{{}^{p}H^{1}\rho_{v}(p_{*}\mathbbm{1}_{E})}}$${{{}^{p}H^{1}\rho_{v}(p_{*}\iota^{*}\gamma_{*}\mathbbm{1}_{U})}}$$\scriptstyle{{}^{p}H^{1}\rho_{v}(\delta)}$$\scriptstyle{{}^{p}H^{1}\rho_{v}(\delta)}$

Therefore, it suffices to show that the map

${}^{p}H^{1}\rho_{v}(\delta)\colon{}^{p}H^{1}\rho_{v}(\omega^{0}p_{*}\mathbbm{1}_{E})\rightarrow{}^{p}H^{1}\rho_{v}(p_{*}\mathbbm{1}_{E})$

is a monomorphism.

Write $E=\bigcup\limits_{i\in J}E_{i}$ with $J$ finite, where for any index
$i$, the scheme $E_{i}$ is regular and of codimension $1$, where for any
distinct indices $i$ and $j$ the scheme $E_{ij}=E_{i}\cap E_{j}$ is of
codimension $2$ and regular and where the intersections of $3$ distinct
$E_{i}$ are empty.

By cdh-descent, we have a morphism exact triangles

${\rho_{v}\omega^{0}p_{*}\mathbbm{1}_{E}}$${\bigoplus\limits_{i\in
J}\rho_{v}\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}}}$${\bigoplus\limits_{\\{i,j\\}\subseteq
J}\rho_{v}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}}$${\rho_{v}p_{*}\mathbbm{1}_{E}}$${\bigoplus\limits_{i\in
J}\rho_{v}(p_{i})_{*}\mathbbm{1}_{E_{i}}}$${\bigoplus\limits_{\\{i,j\\}\subseteq
J}\rho_{v}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}}$$\scriptstyle{\rho_{v}(\delta)}$$\scriptstyle{\rho_{v}(\delta)}$

which yields a commutative diagram with exact rows

${\bigoplus\limits_{i\in
J}{}^{p}H^{0}\rho_{v}\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}}}$${\bigoplus\limits_{\\{i,j\\}\subseteq
J}\rho_{v}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}}$${{}^{p}H^{1}\rho_{v}\omega^{0}p_{*}\mathbbm{1}_{E}}$${\bigoplus\limits_{i\in
J}{}^{p}H^{1}\rho_{v}\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}}}$${0}$${\bigoplus\limits_{i\in
J}{}^{p}H^{0}\rho_{v}(p_{i})_{*}\mathbbm{1}_{E_{i}}}$${\bigoplus\limits_{\\{i,j\\}\subseteq
J}\rho_{v}(p_{ij})_{*}\mathbbm{1}_{E_{ij}}}$${{}^{p}H^{1}\rho_{v}p_{*}\mathbbm{1}_{E}}$${\bigoplus\limits_{i\in
J}{}^{p}H^{1}\rho_{v}\omega^{0}(p_{i})_{*}\mathbbm{1}_{E_{i}}}$${0}$$\scriptstyle{{}^{p}H^{0}\rho_{v}(\delta)}$$\scriptstyle{{}^{p}H^{1}\rho_{v}(\delta)}$$\scriptstyle{{}^{p}H^{1}\rho_{v}(\delta)}$

The proof then follows from the Five Lemma and from Lemma 6.3.5 below. ∎

######  Lemma 6.3.5.

Let $S$ be an excellent $1$-dimensional scheme and let $f\colon X\rightarrow
S$ be a proper map, let $K$ be a number field, let $v$ be a non-archimedian
valuation on $K$, let $\ell$ be the prime number such that $v$ extends the
$\ell$-adic valuation. Assume that the scheme $X$ is regular and connected.
Then,

  1. (1)

The map

${}^{p}H^{0}(\rho_{v}\omega^{0}f_{*}\mathbbm{1}_{X})\rightarrow{}^{p}H^{0}(\rho_{v}f_{*}\mathbbm{1}_{X})$

is an isomorphism.

  2. (2)

The map

${}^{p}H^{1}(\rho_{v}\omega^{0}f_{*}\mathbbm{1}_{X})\rightarrow{}^{p}H^{1}(\rho_{v}f_{*}\mathbbm{1}_{X})$

is a monomorphism.

###### Proof.

If $f$ is finite, then, $\omega^{0}f_{*}\mathbbm{1}_{X}=f_{*}\mathbbm{1}_{X}$
and both statement hold. Otherwise, the image of $X$ is a point and the lemma
follows from Corollary 3.1.8. ∎

###  6.4. Explicit description of perverse Artin motives over a
$1$-dimensional scheme

This description can be made more explicit in the case of $1$-dimensional
excellent schemes. We begin with some notations:

######  Notations 6.4.1.

Let $C$ be a regular connected $1$-dimensional scheme and $K$ be its field of
functions. Then, if $U$ is an open subset of $C$ and if $x$ is a closed point
of $C$.

  * •

We let $K_{x}$ be the completion of $K$ with respect to the valuation defined
by $x$ on $K$.

  * •

We let
$\phi_{x}\colon\operatorname{Spec}(K_{x})\rightarrow\operatorname{Spec}(K)$ be
the induced map.

  * •

We let $G_{0}(x)$ be inertia subgroup of $G_{K_{x}}$.

  * •

We let $p(x)$ be the characteristic exponent of the residue field $k(x)$ of
$C$ at $x$.

The following lemma is classical (see for example [Rui22, 5.1.2]).

######  Lemma 6.4.2.

Let $C$ be a regular connected $1$-dimensional scheme of generic point $\eta$,
$U$ an open subset of $C$ and $x$ a closed point. Let $K=k(\eta)$ be the field
of regular functions on $C$. Then,

  1. (1)

We have an exact sequence of groups:

$1\rightarrow G_{0}(x)\rightarrow G_{K_{x}}\rightarrow G_{k(x)}\rightarrow 1$

  2. (2)

If $x\notin U$, the map

$G_{K_{x}}\rightarrow
G_{K}\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(U,\overline{\eta})$

is injective.

In this setting, we denote by

$\partial_{x}\colon\operatorname{Rep}^{A}(G_{K_{x}},R)\rightarrow\operatorname{Rep}^{A}(G_{k(x)},R[1/p(x)])$

the functor given by $M\mapsto M^{G_{0}(x)}\otimes_{R}R[1/p(x)]$.

######  Proposition 6.4.3.

Let $C$ be a regular connected $1$-dimensional scheme of generic point $\eta$
and field of regular functions $k(C)$, let $R$ be a localization of the ring
of integers of a number field, let $i\colon F\rightarrow C$ be a reduced
closed immersion and let $j\colon U\rightarrow C$ be the open complement. If
$x$ is a point of $F$, we denote by $i_{x}\colon\\{x\\}\rightarrow F$ be the
closed immersion. Assume that all the residue characteristics of $U$ are
invertible in $R$.

Then, the diagram

${{\mathrm{M}^{smA}(U,R)}}$${{\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)}}$${{\operatorname{Loc}_{U}(R)}}$${{\prod\limits_{x\in
F}\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(k(x),R)}}$${{\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(U,\overline{\eta}),R)}}$${{\prod\limits_{x\in
F}\operatorname{Loc}_{x}(R[1/p(x)])}}$${{\operatorname{Rep}^{A}(G_{K},R)}}$${{\operatorname{Rep}^{A}(G_{K_{x}},R)}}$${{\operatorname{Rep}^{A}(G_{k(x)},R[1/p(x)])}}$$\scriptstyle{\rho_{!}[+1]}$$\scriptstyle{\overline{\eta}^{*}}$$\scriptstyle{\prod\limits_{x\in
F}\phi_{x}^{*}}$$\scriptstyle{\prod\limits_{x\in
F}\partial_{x}}$$\scriptstyle{\prod\limits_{x\in
F}\overline{x}^{*}}$$\scriptstyle{\prod\limits_{x\in
F}\rho_{!}[+1]}$$\scriptstyle{\bigoplus\limits_{x\in
F}(i_{x})_{*}}$$\scriptstyle{{}^{hp}\tau_{\leqslant 0}\omega^{0}i^{*}j_{*}}$

is commutative.

######  Remark 6.4.4.

Less formally, this diagram shows that, under the identification of Theorem
2.1.1, the functor

${}^{hp}\tau_{\leqslant
0}\omega^{0}i^{*}j_{*}\colon\mathrm{M}^{smA}(U,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)$

can be identified with the functor

$\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(U,\overline{\eta}),R)[+1]\rightarrow\prod\limits_{x\in
F}\mathcal{D}^{b}(\operatorname{Rep}^{A}(G_{k(x)},R[1/p(x)]))$

that sends the object $M[1]$ where $M$ is an Artin representation of
$\pi_{1}^{\operatorname{\acute{e}t}}(U,\overline{\eta})$ to the complex

$\bigoplus\left(\left[\phi_{x}^{*}(M)\right]^{G_{0}(x)}\otimes_{R}R[1/p(x)][+1]\right)_{x\in
F}.$

###### Proof.

We can assume that the closed subscheme $F$ is reduced to a point $x$. By
Proposition 3.2.1, it suffices to show that the diagram

${{\operatorname{Loc}_{U}(R)}}$${{\operatorname{\mathcalboondox{Sh}}(k(x),R[1/p(x)])}}$${{\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(U,\overline{\eta}),R)}}$${{\operatorname{Loc}_{x}(R[1/p(x)])}}$${{\operatorname{Rep}^{A}(G_{K},R)}}$${{\operatorname{Rep}^{A}(G_{K_{x}},R)}}$${{\operatorname{Rep}^{A}(G_{k(x)},R[1/p(x)])}}$$\scriptstyle{\overline{\eta}^{*}}$$\scriptstyle{\phi_{x}^{*}}$$\scriptstyle{\partial_{x}}$$\scriptstyle{\overline{x}^{*}}$$\scriptstyle{\tau_{\leqslant
1}i^{*}j_{*}}$

is commutative.

Let $M$ be a locally constant étale sheaf over $U$ with finitely presented
fibers. Identify $M$ with the corresponding Artin representation of
$\pi_{1}^{\operatorname{\acute{e}t}}(U,\overline{\eta})$. By [AGV73, VIII.5],
if $n$ is an integer, the étale sheaf $H^{n}(i^{*}j_{*}M)$ identifies with the
$G_{k(x)}$-module

$H^{n}_{\operatorname{\acute{e}t}}(\operatorname{Spec}(L),\phi_{x}^{*}M)=H^{n}(G_{0}(x),\phi_{x}^{*}M)$

where $L$ is the maximal unramified extension of $K_{x}$. Therefore, the sheaf
$H^{1}(i^{*}j_{*}M)$ vanishes and we have

$H^{0}(i^{*}j_{*}M)=\left[\phi_{x}^{*}(M)\right]^{G_{0}(x)}\otimes_{R}R[1/p(x)]$

which yields the result. ∎

We now give an analog to this statement in the general case.

######  Notations 6.4.5.

Let $C$ be an excellent $1$-dimensional scheme, let
$\nu\colon\widetilde{C}\rightarrow C$ be the normalization of $C$, let
$\Gamma$ be the set of generic points of $\widetilde{C}$, let $U$ is an open
subset of $C$, let $y$ be a closed point of $\widetilde{C}$ and let
$\eta\in\Gamma$.

  * •

We let $U_{\eta}=\overline{\\{\eta\\}}\cap\nu^{-1}(U)\subseteq\widetilde{C}$.

  * •

We let $K_{\eta}$ be the field of regular functions on $C_{\eta}$.

  * •

We let $\eta(y)$ be the unique element of $\Gamma$ such that $y\in
C_{\eta(y)}$.

  * •

We let $K_{y}$ be the completion of $K_{\eta(y)}$ with respect to the
valuation defined by $y$ on $K_{\eta(y)}$.

  * •

We let
$\phi_{y}:\operatorname{Spec}(K_{y})\rightarrow\operatorname{Spec}(K_{\eta(y)})$
be the induced map.

  * •

We let $G_{0}(y)$ be the inertia subgroup of $G_{K_{y}}$.

######  Lemma 6.4.6.

Let $C$ be a $1$-dimensional excellent scheme, let
$\nu\colon\widetilde{C}\rightarrow C$ be the normalization of $C$, let
$\Gamma$ be the set of generic points of $\widetilde{C}$, let $U$ be a nil-
regular open subset of $C$, let $y$ be a closed point of $\widetilde{C}$ and
let $\eta\in\Gamma$. Then,

  1. (1)

We have an exact sequence of groups:

$1\rightarrow G_{0}(y)\rightarrow G_{K_{y}}\rightarrow G_{k(y)}\rightarrow 1$

  2. (2)

If $y\notin U_{\eta(y)}$, the map

$G_{K_{y}}\rightarrow
G_{K_{\eta(y)}}\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(U_{\eta(y)},\overline{\eta(y)})$

is injective.

In this setting, we denote by

$\partial_{y}\colon\operatorname{Rep}^{A}(G_{K_{y}},R)\rightarrow\operatorname{Rep}^{A}(G_{k(y)},R[1/p(y)])$

the functor given by $M\mapsto M^{G_{0}(y)}\otimes_{R}R[1/p(y)]$.

######  Corollary 6.4.7.

Let $C$ be a $1$-dimensional excellent scheme, let
$\nu\colon\widetilde{C}\rightarrow C$ be the normalization of $C$, let
$\Gamma$ be the set of generic points of $C$, let $i\colon F\rightarrow C$ be
a closed immersion, let $j\colon U\rightarrow C$ be the open complement and
let $R$ be the localization of a number field. Assume that the scheme $U$ is
nil-regular and that the residue characteristic exponents of $U$ are
invertible in $R$.

Then, the diagram

${{\mathrm{M}^{smA}_{\mathrm{perv}}(U,R)}}$${{\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)}}$${{\operatorname{Loc}_{U}(R)}}$${{\prod\limits_{x\in
F}\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(k(x),R)}}$${{\prod\limits_{\eta\in\Gamma}\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(U_{\eta},\overline{\eta}^{*}),R)}}$${{\prod\limits_{x\in
F}\operatorname{Loc}_{x}(R[1/p(x)])}}$${{\prod\limits_{\eta\in\Gamma}\operatorname{Rep}^{A}(G_{K_{\eta}},R)}}$${{\prod\limits_{x\in
F}\operatorname{Rep}^{A}\left(G_{k(x)},R[1/p(x)]\right)}}$${{\prod\limits_{\eta\in\Gamma}\left(\prod\limits_{y\in\overline{\\{\eta\\}}\cap\nu^{-1}(F)}\operatorname{Rep}^{A}(G_{K_{y}},R)\right)}}$${{\prod\limits_{y\in\nu^{-1}(F)}\operatorname{Rep}^{A}(G_{K_{y}},R)}}$${{\prod\limits_{y\in\nu^{-1}(F)}\operatorname{Rep}^{A}\left(G_{k(y)},R[1/p(y)]\right)}}$${{\prod\limits_{x\in
F}\prod\limits_{\nu(y)=x}\operatorname{Rep}^{A}\left(G_{k(y)},R[1/p(x)]\right)}}$$\scriptstyle{\rho_{!}[+1]}$$\scriptstyle{\prod\limits_{\eta\in\Gamma}\overline{\eta}^{*}}$$\scriptstyle{\prod\limits_{\eta\in\Gamma}\left(\prod\limits_{y\in\overline{\\{\eta\\}}\cap\nu^{-1}(F)}\phi_{y}^{*}\right)}$$\scriptstyle{\prod\limits_{y\in\nu^{-1}(F)}\partial_{y}}$$\scriptstyle{\prod\limits_{x\in
F}\bigoplus\limits_{\nu(y)=x}\operatorname{Ind}^{G_{k(y)}}_{G_{k(x)}}}$$\scriptstyle{\prod\limits_{x\in
F}\overline{x}^{*}}$$\scriptstyle{\rho_{!}[+1]}$$\scriptstyle{\bigoplus\limits_{x\in
F}(i_{x})_{*}}$$\scriptstyle{{}^{hp}\tau_{\leqslant 0}\omega^{0}i^{*}j_{*}}$
is commutative.

######  Remark 6.4.8.

Less formally, the commutativity of this diagram shows that, under the
identification of Theorem 2.1.1, the functor the functor

${}^{hp}\tau_{\leqslant
0}\omega^{0}i^{*}j_{*}\colon\mathrm{M}^{smA}(U,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)$

can be identified with the functor that sends the object
$(M_{\eta}[+1])_{\eta\in\Gamma}$ where each $M_{\eta}$ is an Artin
representation of $G_{K_{\eta}}$ to

$\left(\bigoplus\limits_{\nu(y)=x}\mathrm{Ind}_{G_{k(y)}}^{G_{k(x)}}\left(\left[\phi_{y}^{*}(M_{\eta(y)})\right]^{G_{0}(y)}\otimes_{R}R[1/p(x)]\right)[+1]\right)_{x\in
F}.$

###### Proof.

Let $\nu_{U}\colon U_{\mathrm{red}}\rightarrow U$ (resp.
$\nu_{F}\colon\widetilde{F}\rightarrow F$) be the pullback of the map $\nu$
along $j$ (resp. $i$).

Let $\widetilde{j}$ (resp.$\widetilde{i}$) be the inclusion of
$U_{\mathrm{red}}$ (resp. $\widetilde{F}$) in $\widetilde{C}$. Recall that the
functor $\nu_{U}^{*}$ is an equivalence. Thus, we have

$\omega^{0}i^{*}j_{*}=\omega^{0}i^{*}j_{*}(\nu_{U})_{*}\nu_{U}^{*}=\omega^{0}i^{*}\nu_{*}\widetilde{j}_{*}\nu_{U}^{*}=(\nu_{F})_{*}\omega^{0}\widetilde{i}^{*}\widetilde{j}_{*}\nu_{U}^{*}.$

The result then follows from Proposition 6.4.3 and from Proposition 1.5.10. ∎

We can now describe the category of Artin motives over a $1$-dimensional
excellent scheme.

######  Definition 6.4.9.

Let $C$ be a $1$-dimensional excellent scheme, let $\Gamma$ be the set of
generic points of $C$, let $\nu\colon\widetilde{C}\rightarrow C$ be a
normalization of $C$ and let $R$ be a localization of a number field. We
define an abelian category $\mathrm{N}(C,R)$ as follows.

  * •

An object of $\mathrm{N}(C,R)$ is a quadruple
$(U,(M_{\eta})_{\eta\in\Gamma},(M_{x})_{x\in C\setminus U},(f_{x})_{x\in
C\setminus U})$ where

    1. (1)

$U$ is a nil-regular open subset of $C$ such that the immersion $U\rightarrow
C$ is affine and such that the residue characteristic exponents of $U$ are
invertible in $U$,

    2. (2)

for all $\eta\in\Gamma$, $M_{\eta}$ is a representation of
$\pi_{1}^{\operatorname{\acute{e}t}}(U_{\eta},\overline{\eta})$,

    3. (3)

for all $x\in C\setminus U$, $M_{x}$ is a complex of representations of
$G_{k(x)}$ placed in degrees $[0,1]$

    4. (4)

for all $x\in C\setminus U$, $f_{x}$ is an element of

$\operatorname{Hom}_{\mathrm{D}(\operatorname{Rep}^{A}(G_{k(x)},R))}\left(M_{x},\bigoplus\limits_{\nu(y)=x}\mathrm{Ind}_{G_{k(y)}}^{G_{k(x)}}\left(\partial_{y}\phi_{y}^{*}(M_{\eta(y)})\right)\right)$

such that $H^{0}(f_{x})$ is a monomorphism.

  * •

An element of
$\operatorname{Hom}_{\mathrm{N}(C,R)}\left((U,(M_{\eta}),(M_{x}),(f_{x})),(V,(N_{\eta}),(N_{x}),(g_{x}))\right)$
is a couple $((\Phi_{\eta})_{\eta\in\Gamma},(\Phi_{x})_{x\in(C\setminus
U)\cap(C\setminus V)})$ where $\Phi_{\eta}\colon M_{\eta}\rightarrow N_{\eta}$
is a map of representations of $G_{K_{\eta}}$, where $\Phi_{x}\colon
M_{x}\rightarrow N_{x}$ is a map of representations of $G_{k(x)}$ and the
diagram

${M_{x}}$${\bigoplus\limits_{\nu(y)=x}\mathrm{Ind}_{G_{k(y)}}^{G_{k(x)}}\left(\partial_{y}\phi_{y}^{*}(M_{\eta(y)})\right)}$${N_{x}}$${\bigoplus\limits_{\nu(y)=x}\mathrm{Ind}_{G_{k(y)}}^{G_{k(x)}}\left(\partial_{y}\phi_{y}^{*}(N_{\eta(y)})\right)}$$\scriptstyle{f_{x}}$$\scriptstyle{\Phi_{x}}$$\scriptstyle{\bigoplus\limits_{\nu(y)=x}\mathrm{Ind}_{G_{k(y)}}^{G_{k(x)}}\left(\partial_{y}\phi_{y}^{*}(\Phi_{\eta(y)})\right)}$$\scriptstyle{g_{x}}$

is commutative.

Proposition 6.3.1 and Corollary 6.4.7 give us the following proposition:

######  Proposition 6.4.10.

Let $C$ be an excellent $1$-dimensional scheme, and let $R$ be a localization
of the ring of integers of a number field. Assume that the residue
characteristic exponents of a dense open subset of $C$ are invertible in $R$.
Then, the category $\mathrm{M}^{A}_{\mathrm{perv}}(C,R)$ is equivalent to
$\mathrm{N}(C,R)$.

######  Remark 6.4.11.

As in Corollary 2.3.2 we can also give a description in the case where $S$ has
finitely many residue characteristics.

######  Example 6.4.12.

Let $k$ be an algebraically closed field of characteristic $0$. A perverse
Artin étale motive over $\mathbb{P}^{1}_{k}$ is given by a quadruple
$(\mathbb{P}^{1}_{k}\setminus F,M,(M_{x})_{x\in F},(f_{x})_{x\in F})$ where
$F$ is a finite set of points of $\mathbb{P}^{1}_{k}$, where $M$ is an Artin
representation of
$\pi_{1}^{\operatorname{\acute{e}t}}(\mathbb{P}^{1}_{k}\setminus F)$, where
$M_{x}$ is a complex of $R$-modules of finite type placed in degrees $[0,1]$
and where

$f_{x}\colon M_{x}\rightarrow\left[\phi_{x}^{*}(M)\right]^{G_{0}(x)}$

is such that the morphism $H^{0}(f_{x})$ of $R$-modules is injective.

We can always assume that the point at infinity belongs to $F$. Write
$F=\\{\infty\\}\sqcup F^{\prime}$ and let $m=|F^{\prime}|$.

Let $x\in F^{\prime}$, then, the field $k((X-x))$ is the completion of the
field $k(X)$ of regular functions on $\mathbb{P}^{1}_{k}$ with respect to the
valuation defined by $x$.

On the other hand, the field $k((1/X))$ is the completion of $k(X)$ with
respect to the valuation given by the point at infinity.

Furthermore, recall that the absolute Galois group of the field $k((X))$ is
$\hat{\mathbb{Z}}$.

The étale fundamental group of the scheme $\mathbb{P}^{1}_{k}\setminus F$
where $F$ is a finite set of closed points is the profinite completion of the
free group over $F^{\prime}$; we denote by $g_{x}$ for $x\in F^{\prime}$ its
topological generators. The map

$G_{k((x-x))}\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(\mathbb{P}^{1}_{k}\setminus
F)$

is the only continuous map that sends the topological generator of
$\hat{\mathbb{Z}}$ to $g_{x}$.

The map
$G_{k((1/X))}\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(\mathbb{P}^{1}_{k}\setminus
F)$ is the only continuous map which that sends the topological generator to a
certain product $(g_{x_{1}}\cdot\cdots\cdot g_{x_{m}})^{-1}$ where
$F^{\prime}=\\{x_{1},\ldots,x_{m}\\}$.

Now, the action of
$\pi_{1}^{\operatorname{\acute{e}t}}(\mathbb{P}^{1}_{k}\setminus F)$ on $M$
factors through a finite quotient $G$ and is therefore the same as a
representation of a finite group with $m$ marked generators which are the
images of the $g_{x}$ for $x\in F^{\prime}$.

Thus, a perverse Artin étale motive over $\mathbb{P}^{1}_{k}$ is equivalent to
the following data:

  * •

A finite number of distinct points $x_{1},\ldots,x_{m}$ of
$\mathbb{A}^{1}_{k}$ (i.e of elements of $k$).

  * •

A finite group $G$ generated by $m$ elements $g_{1},\ldots,g_{m}$; let
$\phi_{i}:\hat{\mathbb{Z}}\rightarrow G$ be the map given by $g_{i}$ and
$\phi_{\infty}:\hat{\mathbb{Z}}\rightarrow G$ be the map given by
$g_{\infty}=(g_{1}\cdot\cdots\cdot g_{m})^{-1}$.

  * •

An $R$-linear representation $M$ of $G$.

  * •

Complexes $M_{1},\ldots,M_{m},M_{\infty}$ of $R$-modules of finite type placed
in degrees $[0,1]$.

  * •

Maps $f_{i}\colon M_{i}\rightarrow(\phi_{i}^{*}M)^{g_{i}}$ for
$i=1,\ldots,m,\infty$ such that

$H^{0}(f_{i})\colon H^{0}(M_{i})\rightarrow(\phi_{i}^{*}M)^{g_{i}}$

is injective.

######  Example 6.4.13.

Let $S=\operatorname{Spec}(\mathbb{Z}_{p})$. A perverse Artin étale motives
over $S$ is given by a triple $(M,N,f)$ where $M$ is an Artin representation
of $G_{\mathbb{Q}_{p}}$, where $N$ is a complex of Artin representation of
$\hat{\mathbb{Z}}$ with coefficients in $R[1/p]$ placed in degrees $[0,1]$ and
$f\colon N\rightarrow M^{G_{0}}\otimes_{R}R[1/p]$ (where $G_{0}$ is the
inertia group) is a map such that the morphism $H^{0}(f)$ of representations
of $\hat{\mathbb{Z}}$ with coefficients in $R[1/p]$ is injective.

######  Example 6.4.14.

Let $k$ be an algebraically closed field of characteristic $0$. Let $A$ be the
localization of the ring $k[X,Y]/(XY)$ at the prime ideal $(X,Y)$. It is the
local ring at the intersection point of two lines in $\mathbb{P}^{2}_{k}$.

Let $S=\operatorname{Spec}(A)$. The scheme $S$ has two generic points of
residue field $k(X)$ and the residue field at the closed point is $k$. The
normalization of $S$ is given by the scheme

$\operatorname{Spec}(k[X]_{(X)}\times k[Y]_{(Y)}).$

Let $\varepsilon$ be the topological generator of $\hat{\mathbb{Z}}$ and let

$\phi\colon\hat{\mathbb{Z}}=G_{k((X))}\rightarrow G_{k(X)}$

be the inclusion.

A perverse Artin étale motive on $S$ is given by a quadruple
$(M_{1},M_{2},N,f)$ where $M_{1}$ and $M_{2}$ are Artin representations of
$G_{k(X)}$, where $N$ is a complex of $R$-modules placed in degrees $[0,1]$
and where

$f\colon N\rightarrow(\phi^{*}(M_{1}\oplus M_{2}))^{\varepsilon}$

is such that $H^{0}(f)$ is injective.

######  Example 6.4.15.

Let $S=\operatorname{Spec}(\mathbb{Z}[\sqrt{5}]_{(2)})$. This scheme has two
points: its generic point has residue field $\mathbb{Q}(\sqrt{5})$ and its
closed point has residue field $\mathbb{F}_{2}$ (the field with two elements).
The normalization of $S$ is
$\operatorname{Spec}\left(\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]_{(2)}\right)$
which is a discrete valuation ring. The residue field of its closed point is
the field $\mathbb{F}_{4}$ with $4$ elements. The completion of the field
$\mathbb{Q}(\sqrt{5})$ with respect to the valuation defined by $2$ is the
field $\mathbb{Q}_{2}(\sqrt{5})$. Let $G_{0}$ be the inertia subgroup of
$G_{\mathbb{Q}(\sqrt{5})}$.

Let $\phi\colon\hat{\mathbb{Z}}\rightarrow\hat{\mathbb{Z}}$ be the
multiplication by $2$. The map $\phi^{*}$ has a right adjoint functor

$\operatorname{Ind}_{\phi}\colon\operatorname{Rep}(\hat{\mathbb{Z}},\mathbb{Z}_{\ell})\rightarrow\operatorname{Rep}(\hat{\mathbb{Z}},\mathbb{Z}_{\ell}).$

Thus a perverse Artin motive on $S$ is given by a triple $(M,N,f)$ where $M$
is an Artin representation of $G_{\mathbb{Q}(\sqrt{5})}$, where $N$ is a
complex Artin representations of $\hat{\mathbb{Z}}$ placed in degrees $[0,1]$
and where

$f\colon N\rightarrow\operatorname{Ind}(M^{G_{0}})$

is such that $H^{0}(f)$ is injective.

Recall that there is an analogous description in the $\ell$-adic setting (see
[Rui22, 5.1.7]). If $G$ is a profinite group and if $\Lambda$ is an
$\ell$-adic field or its ring of integers, we denote by
$BG_{\mathrm{pro\acute{e}t}}$ the site of profinite continuous $G$-sets with
covers given by continous surjections (see [BS15, 4.1.10]). If $k$ is a field,
the stable category $\mathcal{D}(k,\Lambda)$ identifies with the stable
category
$\operatorname{\mathcalboondox{Sh}}((BG_{k})_{\mathrm{pro\acute{e}t}},\Lambda)$
of hypersheaves of modules over the hypersheaf of rings on
$(BG_{k})_{\mathrm{pro\acute{e}t}}$ defined by the topological ring $\Lambda$
endowed with its trivial action using [BS15, 4.3.2]). The stable category
$\mathcal{D}^{b}_{c}(k,\Lambda)$ then identifies with the stable subcategory
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}((BG_{k})_{\mathrm{pro\acute{e}t}},\Lambda)$
of
$\operatorname{\mathcalboondox{Sh}}((BG_{k})_{\mathrm{pro\acute{e}t}},\Lambda)$
made of those objects which are dualizable. The heart of
$\mathcal{D}^{b}_{c}(k,\Lambda)$ then identifies with the category
$\operatorname{Rep}(G_{k},\Lambda)$ of continuous representations of $G_{k}$
with coefficients in $\Lambda$.

Let $C$ be an excellent scheme of dimension $1$, let
$\nu\colon\widetilde{C}\rightarrow C$ be a normalization of $C$, let $\Gamma$
be the set of generic points of $\widetilde{C}$, let $U$ be a nil-regular open
subset of $C$, let $y$ be closed point of $\widetilde{C}$, let $\eta$ be an
element of $\Gamma$, let $R$ be a localization of a number field $L$ and let
$v$ be a non-archemedian valuation on $L$. Assume that the prime number $\ell$
such that the valuation $v$ extends the $\ell$-adic valuation is invertible on
$C$. We denote by

$\partial_{y}^{v}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}((BG_{K_{y}})_{\mathrm{pro\acute{e}t}},R_{v})\rightarrow\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}((BG_{k(y)})_{\mathrm{pro\acute{e}t}},R_{v})$

the derived functor of the map given by $M\mapsto M^{G_{0}(y)}$.

We still denote by $\partial_{y}^{v}$ the induced map

$\partial_{y}^{v}\colon\operatorname{Rep}^{A}(G_{K_{y}},R_{v})\rightarrow\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}((BG_{k(y)})_{\mathrm{pro\acute{e}t}},R_{v})$

We have a $2$-morphism

$e_{y}\colon\partial_{y}\otimes_{R}R_{v}\rightarrow\partial_{y}^{v}.$

We can describe the $\ell$-adic valuation in terms of our description and of
the description of [Rui22, 5.1.7].

######  Proposition 6.4.16.

Let $C$ be an excellent $1$-dimensional scheme, let $R$ be a localization of
the ring of integers of a number field $L$, let $v$ be a non-archimedian
valuation on $L$ and let $\ell$ be the prime number such that $v$ extends the
$\ell$-adic valuation. We have a commutative diagram

${\mathrm{M}^{A}_{\mathrm{perv}}(C,R_{(v)})}$${\mathrm{Perv}^{A}(C[1/\ell],R_{v})}$${\mathrm{N}(C,R_{(v)})}$${\mathrm{P}(C[1/\ell],R_{v})}$$\scriptstyle{\overline{\rho}_{v}}$$\scriptstyle{\theta_{v}}$

where

$\displaystyle\theta_{v}(U,(M_{\eta})_{\eta\in\Gamma},(M_{x})_{x\in C\setminus
U},(f_{x})_{x\in C\setminus U})=$
$\displaystyle\left(U,(M_{\eta}\otimes_{R_{(v)}}R_{v})_{\eta\in\Gamma},(M_{x}\otimes_{R_{(v)}}R_{v})_{x\in
C\setminus U},(\psi_{x}^{v}(f_{x}))_{x\in C\setminus U}\right)$

setting

$\psi_{x}^{v}(f_{x})=\left(\bigoplus\limits_{\nu(y)=x}\mathrm{Ind}_{G_{k(y)}}^{G_{k(x)}}\left(e_{y}\left[\phi_{y}^{*}(M_{\eta(y)}\otimes_{R_{(v)}}R_{v})\right]\right)\right)\circ(f_{x}\otimes_{R_{(v)}}R_{v})$

for any closed point $x$ of $C$.

## Appendix: The six functors formalism for cohomological étale motives
without resolution of singularities (following Gabber)

In [Leh19b, 1.12], Pépin Lehalleur proved that (constructible) cohomological
motives are stable under the six functors. However, one has to assume that
$R=\mathbb{Q}$ and to have the functors $f^{!}$ and $f_{*}$ one also has to
assume to have resolutions of singularities by alterations. This is not
necessary: one can mimic the method of Gabber [CD16, 6.2] in the cohomological
case. Thus, we can prove that if $X$ (resp. $Y$) is a quasi-excellent scheme,
and $f:Y\rightarrow X$, $f_{*}$ (resp. $f^{!}$) preserves cohomological
constructible motives. We will outline how to do this.

  1. (1)

Let $R$ be a ring. The fibred subcategory
$\operatorname{\mathcal{DM}}^{\operatorname{coh}}_{\operatorname{\acute{e}t},c}(-,R)$
of $\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(-,R)$ is stable
under tensor product, negative Tate twists, $f^{*}$ if $f$ is any morphism and
$f_{!}$ if $f$ is separated and of finite type.

  2. (2)

From the absolute purity property and the stability of
$\operatorname{\mathcal{DM}}^{\operatorname{coh}}_{\operatorname{\acute{e}t},c}$
under negative Tate twists and using [Ayo07, 2.2.31], we deduce that if $i$ is
the immersion of a simple normal crossing divisor with regular target $X$,
$i^{!}\mathbbm{1}_{X}$ is a cohomological constructible étale motive.

  3. (3)

For any prime ideal $\mathfrak{p}$ of $\mathbb{Z}$, we define
$\mathfrak{p}$-constructible cohomological étale motives as in [CD16, 6.2.2]
and show as in [CD16, 6.2.3] that an étale motive is a cohomological
constructible étale motive if and only if it is $\mathfrak{p}$-cohomological
constructible for all prime ideal $\mathfrak{p}$. Moreover, if $p$ is a prime
number then as in [CD16, 6.2.4], an étale motive over a scheme of
characteristic $p$ is $(p)$-constructible cohomological if and only if it is
$(0)$-cohomological constructible.

  4. (4)

Let $X$ be a scheme and $\mathfrak{p}$ a prime ideal of $\mathbb{Z}$. Denote
by $\mathcal{C}_{0}(X)$ the category of $\mathfrak{p}$-cohomological
constructible étale motives. It has the properties [CD16, 6.2.9(b)(c)]. It
does not necessarily have property [CD16, 6.2.9(a)] however, in the proof of
[CD16, 6.2.7] we only need the following weaker version: $\mathcal{C}_{0}(X)$
is a thick subcategory of
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(X,R)$ which contains
the object $\mathbbm{1}_{X}$. From this discussion, we deduce Gabber’s lemma
for cohomological étale motives: if $X$ is quasi-excellent, and if either
$\mathfrak{p}=(0)$ or $\mathfrak{p}=(p)$ and for any point $x$ of $X$,
$\mathrm{char}(k(x))\neq p$, then for any dense immersion $U\rightarrow X$,
the étale motive $j_{*}(\mathbbm{1}_{X})$ is a $\mathfrak{p}$-cohomological
constructible étale motive.

  5. (5)

Follow the proofs of [CD16, 6.2.13, 6.2.14] to prove that
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{\operatorname{coh}}(-,R)$
is endowed with the six functors formalism over quasi-excellent schemes.

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