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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-102.57692pt}{2.09589pt}\pgfsys@lineto{-102.57692pt}{-15.38197pt}\pgfsys@stroke\pgfsys@invoke{
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}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
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}\hbox{$\scriptstyle{-\otimes_{K}K_{v}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
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}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-38.78847pt}{-25.91939pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-52.94858pt}{-23.56662pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $\alpha$ is the derived functor of the fully faithful exact functor

$\operatorname{Rep}(G_{k},K_{v})\rightarrow\operatorname{Sh}((BG_{k})_{\mathrm{pro\acute{e}t}},K_{v}),$

is commutative.

Since the functors $\alpha$ and $-\otimes_{K}K_{v}$ are conservative and since
the functor $\rho_{!}$ is an equivalence, the functor $\rho_{v}$ is
conservative.

With the assumptions of the second assertion, one of the prime numbers $\ell$
and $\ell^{\prime}$ is distinct from $p$ and the proof is the same.

We now prove the third assertion. Let $v$ be a valuation on $K$ such that
$v(p)=0$ (i.e. $v$ does not extend the $p$-adic valuation) Using Proposition
2.1.9, the diagram

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where $\alpha$ is the derived functor of the fully faithful exact functor

$\operatorname{Rep}(G_{k},R_{v})\rightarrow\operatorname{Sh}((BG_{k})_{\mathrm{pro\acute{e}t}},R_{v}),$

is commutative.

Now, if $M$ is a complex in
$\mathcal{D}^{b}(\operatorname{Rep}^{A}(G_{k},R[1/p])$ such that for any
valuation $v$ such that $v(p)=0$, we have

$M\otimes_{R}R_{v}=0,$

then, the complex $M$ vanishes. Hence, the family $(\overline{\rho}_{\ell})$
is conservative. ∎

###  2.3. Constructible Artin étale motives as stratified representations

Let $S$ be a scheme. Recall that a _stratification_ of $S$ is a finite
partition of $S$ into equidimensional locally closed connected subsets of $S$
(called _strata_) such that the topological closure of any stratum is a union
of strata.

######  Proposition 2.3.1.

Let $S$ be a scheme, let $M$ be a constructible Artin étale motive over $S$.
Then, there is a stratification of $S$ such that, for any stratum $T$, the
Artin motive $M|_{T}$ is smooth. Moreover if the scheme $S$ is excellent, we
can assume the strata to be regular.

###### Proof.

We can assume that $S$ is reduced and connected. We proceed by noetherian
induction on $S$. The local rings at the generic points of $S$ are fields. Let
$M$ be a constructible Artin étale motive over $S$. As Artin étale motives
over a field are smooth, by Proposition 1.2.8, there is an open dense
subscheme $U$ of $S$ such that $M|_{U}$ is smooth. If the scheme $S$ is
excellent, we can furthermore assume that $U$ is regular. Using the induction
hypothesis, there is a stratification of $S\setminus U$ such that for any
stratum $T$, the motive $M|_{T}$ is a constructible smooth Artin motive and we
are done. ∎

######  Corollary 2.3.2.

Let $S$ be an excellent scheme with finitely many residue characteristics and
let $R$ be a regular good ring. Write $S_{0}=S\times_{\mathbb{Z}}\mathbb{Q}$
and $S_{p}=S\times_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$ for any prime number
$p$. Then, we have prime numbers $p_{1},\ldots,p_{r}$ such that

$S=S_{0}\sqcup S_{p_{1}}\sqcup\cdots\sqcup S_{p_{r}}.$

Furthermore, the subscheme $S_{0}$ is open in $S$ and the subschemes
$S_{p_{1}},\ldots,S_{p_{r}}$ are closed in $S$.

Let $M$ be a constructible Artin étale motive over $S$. We have a
stratification $\mathcal{S}_{i}$ of every $S_{i}$ when $i$ runs trhough the
set $\\{0,p_{1},\ldots,p_{r}\\}$ (and therefore a stratification of $S$),
lisse étale sheaves
$M_{T}\in\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(T,R)$ for
$T$ in $\mathcal{S}_{0}$ and lisse étale sheaves
$M_{T}\in\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(T,R[1/p_{i}])$
for $T$ in $\mathcal{S}_{p_{i}}$, such that for any stratum $T$, we have

$M|_{T}=\rho_{!}(M_{T}).$

In particular, the motive $M$ can be constructed from the $M_{T}$ by gluing
using the localization triangle (0.0.1).

######  Corollary 2.3.3.

Let $S$ be an excellent scheme and let $R$ be a regular good ring. Assume that
all residue characteristics of $S$ are invertible in $R$.

Let $M$ be a constructible Artin étale motive over $S$. We have a
stratification of $S$ and lisse étale sheaves
$M_{T}\in\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(T,R)$ for
every stratum $T$, such that we have

$M|_{T}=\rho_{!}(M_{T}).$

In particular, the motive $M$ can be constructed from the $M_{T}$ by gluing
using the localization triangle (0.0.1).

######  Corollary 2.3.4.

Let $S$ be an excellent scheme, let $p$ be a prime number and let $R$ be a
$\mathbb{Z}_{(p)}$-algebra.

Write $S_{p}=S\times_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$. Then, we have

$S=S[1/p]\sqcup S_{p}.$

Furthermore the subscheme $S[1/p]$ is open in $S$ and the subscheme $S_{p}$ is
closed in $S$.

Let $M$ be a constructible Artin étale motive over $S$ with coefficients in
$R$. We have a stratification $\mathcal{S}[1/p]$ of $S[1/p]$ with regular
strata, a stratification $\mathcal{S}_{p}$ of $S_{p}$ with regular strata,
lisse étale sheaves
$M_{T}\in\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(T,R[1/p])$
for $T$ in $\mathcal{S}_{p}$ and lisse étale sheaves
$M_{T}\in\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(T,R)$ for
$T\in\mathcal{S}[1/p]$, such that for any stratum $T$, we have

$M|_{T}=\rho_{!}(M_{T}).$

In particular, the motive $M$ can be constructed from the $M_{T}$ by gluing
using the localization triangle (0.0.1).

##  3\. Computations around the Artin truncation functor

###  3.1. The Artin truncation over a field

With integral coefficients, Proposition 1.4.6 is (very) false, namely we have:

######  Proposition 3.1.1.

Let $R$ be a localization of the ring of integers of a number field $K$ and
let $f\colon X\rightarrow S$ be a separated morphism of finite type. If the
Artin motive $\omega^{0}f_{!}\mathbbm{1}_{X}$ is constructible, then we are in
one of the following cases:

  1. (1)

we have $R=K$.

  2. (2)

the map $f$ is quasi-finite.

  3. (3)

there is a prime number $p$ such that $R[1/p]$ is a $\mathbb{Q}$-algebra and
all the points $x$ of $S$ such that the set $f^{-1}(x)$ is infinite have the
same residue characteristic $p$.

Proposition 3.1.1 will follow from Theorem 3.1.4 below. To state this result,
we need to introduce the following notion:

######  Definition 3.1.2.

Let $R$ be a localization of the ring of integers of a number field and let
$k$ be a field of characteristic exponent $p$. If $n$ is a positive integer,
we denote by $\mu_{n}$ the sheaf of $n$-th roots of unity over the small étale
site of $k$.

Let $f\colon X\rightarrow\operatorname{Spec}(k)$ be a morphism of finite type.
Let $m$ be an integer, we define the _$m$ -th Tate twist_ of the étale sheaf
$f_{*}\left[\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}}\right]$
as the étale sheaf

$f_{*}\left[\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}}\right](m):=\operatorname{colim}_{(n,p)=1}f_{*}\left[\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}}\right]\otimes_{\mathbb{Z}}\mu_{n}^{\otimes
m}.$

######  Remark 3.1.3.

Notice that through the equivalence of Proposition 1.5.12, the $i$-th
cohomology group of
$f_{*}\left[\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}}\right](m)$
seen as a complex of discrete $G_{k}$-modules is the discrete $G_{k}$-module
$H^{i}_{\operatorname{\acute{e}t}}(X_{\overline{k}},R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z})(m)$
defined in Example 1.5.4.

We can now state the main result of this section:

######  Theorem 3.1.4.

Let $R$ be a localization of the ring of integers of a number field, let $k$
be a field of characteristic exponent $p$ and let $f\colon X\rightarrow S$ be
a morphism of finite type. Assume that the scheme $X$ is regular.

Then,

  1. (1)

The diagram

${\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(X,R)}$${\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{smA}(X,R)}$${\operatorname{\mathcalboondox{Sh}}(k_{\operatorname{\acute{e}t}},R)}$${\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(k,R)}$$\scriptstyle{\rho_{!}}$$\scriptstyle{f_{*}}$$\scriptstyle{\omega^{0}f_{*}}$$\scriptstyle{\rho_{!}}$

is commutative.

  2. (2)

We have

$\omega^{0}f_{*}\mathbbm{1}_{X}=\rho_{!}f_{*}\underline{R[1/p]}.$

  3. (3)

If $m$ is a positive integer, we have

$\omega^{0}(f_{*}\mathbbm{1}_{X}(-m))=\rho_{!}\left[f_{*}\left(\underline{R[1/p]\otimes\mathbb{Q}/\mathbb{Z}}\right)[-1](-m)\right].$

###### Proof.

We have pairs of adjoint functors

$f^{*}\colon\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(k,R)\leftrightarrows\mathcal{DM}^{smA}_{\operatorname{\acute{e}t}}(X,R)\colon\omega^{0}f_{*}$

and

$f^{*}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(k,R)\leftrightarrows\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(X,R)\colon
f_{*}.$

Since the functor $\rho_{!}$ is an equivalence by Theorem 2.1.1 and commutes
with the functor $f^{*}$ by Proposition 1.2.1, this yields the first
assertion.

The second assertion follows from the first one, since we have

$\mathbbm{1}_{X}=\rho_{!}\underline{R[1/p]}.$

We now prove the third assertion. Let
$\pi\colon\mathbb{P}^{m}_{X}\rightarrow\operatorname{Spec}(k)$ be the
canonical projection. We have

$\pi_{*}\mathbbm{1}_{\mathbb{P}^{n}_{X}}=\bigoplus\limits_{i=0}^{m}f_{*}\mathbbm{1}_{X}(-i)[-2i]$

by the projective bundle formula.

On the other hand, we have an exact triangle étale sheaves

$\pi_{*}\left(\underline{R[1/p]}\right)\rightarrow\pi_{*}\left(\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}}\right)\rightarrow\pi_{*}\left(\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}}\right).$

Proposition 1.7.2 yields

$\pi_{*}\left(\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}}\right)=f_{*}\left(\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}}\right)$

and the projective bundle formula in étale cohomology yields

$\pi_{*}\left(\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}}\right)=\bigoplus\limits_{i=0}^{m}f_{*}\left(\underline{R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}}\right)(-i)[-2i].$

The result follows by induction on $m$. ∎

######  Definition 3.1.5.

Let $f:X\rightarrow\operatorname{Spec}(k)$ be a morphism of finite type, let
$\overline{k}$ be the algebraic closure of $k$ and let $X_{\overline{k}}$ be
the pull-back of $X$ to $\overline{k}$. The set $\pi_{0}(X_{\overline{k}})$ of
connected components of $X_{\overline{k}}$ is a finite $G_{k}$-set and
therefore corresponds to a finite étale $k$-scheme through Galois-Grothendieck
correspondence [GR71, VI]. We let $\pi_{0}(X/k)$ be this scheme and call it
the _scheme of connected components of $X$_.

######  Remark 3.1.6.

When $f$ is proper, this scheme coincides with the scheme $\pi_{0}(X/k)$
defined in [AZ12] as the Stein factorization of $f$. In general, the scheme
$\pi_{0}(X/k)$ coincides with the Stein factorization of any compactification
of $X$.

######  Notations 3.1.7.

Let $k$ be a field of characteristic exponent $p$ and let
$\xi\colon\operatorname{Spec}(\overline{k})\rightarrow\operatorname{Spec}(k)$
be the algebraic closure of $k$.

  1. (1)

Let $R$ be a ring. We let

$\alpha_{!}\colon\mathcal{D}(\mathrm{Mod}(G_{k},R[1/p]))\rightarrow\mathcal{DM}^{A}(k,R)$

be the equivalence of Corollary 2.1.7 (which is the composition of $\rho_{!}$
and of the inverse of the fiber functor $\xi^{*}$ associated to $\xi$) and we
let $\alpha^{!}$ be its inverse.

  2. (2)

Let $R$ be a localization of the ring of integers of a number field, let
$f\colon X\rightarrow S$ be a morphism of finite type and let
$X_{\overline{k}}$ be the pull-back of $X$ to $\overline{k}$. We denote by
$\mu^{n}(X,R)$ the $G_{k}$-module
$H^{n}_{\operatorname{\acute{e}t}}(X_{\overline{k}},R[1/p]\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z})$.

######  Corollary 3.1.8.

Let $R$ be a localization of the ring of integers of a number field, let $k$
be a field of characteristic exponent $p$ and let $f\colon X\rightarrow S$ be
a morphism of finite type. Assume that the scheme $X$ is regular.

Then,

  1. (1)

We have

$H^{0}(\omega^{0}f_{*}\mathbbm{1}_{X})=h_{k}(\pi_{0}(X/k)).$

  2. (2)

We have

$H^{n}(\omega^{0}f_{*}\mathbbm{1}_{X})=\alpha_{!}\left[H^{n}_{\operatorname{\acute{e}t}}(X_{\overline{k}},R[1/p])\right]$

and

$H^{n}_{\operatorname{\acute{e}t}}(X_{\overline{k}},R[1/p])=\begin{cases}R[1/p][\pi_{0}(X_{\overline{k}})]&\text{
if }n=0\\\ \mu^{n-1}(X,R)&\text{ if }n\geqslant 2\\\ 0&\text{
otherwise}\end{cases}.$

  3. (3)

If $m$ is a positive integer and if $n$ is an integer, we have

$H^{n}(\omega^{0}f_{*}\mathbbm{1}_{X}(-m))=\alpha_{!}\left[\mu^{n-1}(X,R)(-m)\right].$

We can now prove Proposition 3.1.1. Notice that it follows form the following
lemma.

######  Lemma 3.1.9.

Let $R$ be a localization of the ring of integers of a number field, let $k$
be a field of characteristic exponent $p$ and let $f\colon
X\rightarrow\operatorname{Spec}(k)$ be a morphism of finite type. Assume that
the ring $R[1/p]$ is not a $\mathbb{Q}$-algebra. Then, the Artin motive
$\omega^{0}f_{!}\mathbbm{1}_{X}$ is not constructible.

Using Theorem 2.1.1, the lemma follows from the following more precise
statement.

######  Lemma 3.1.10.

Let $R$ be a localization of the ring of integers of a number field, let $k$
be a field of characteristic exponent $p$ and let $f\colon X\rightarrow S$ be
a separated morphism of finite type. Assume that the ring $R[1/p]$ is not a
$\mathbb{Q}$-algebra. Then, the motive
$H^{2d+1}(\omega^{0}f_{!}\mathbbm{1}_{X})$ is not the image of an Artin
representation through the map $\alpha_{!}$. Furthermore, if $n>2d+1$, the
motive $H^{n}(\omega^{0}f_{!}\mathbbm{1}_{X})$ vanishes.

###### Proof.

Since change of coefficients preserves constructible objects, we may assume
that there is a prime number $\ell\neq p$ such that the ring $R$ is a flat
$\mathbb{Z}_{(\ell)}$-algebra. We may also assume that the scheme $X$ is
connected.

We proceed by induction on the dimension of $X$. By compactifying $X$ using
Nagata’s theorem [Sta23, 0F41], and by using the localization triangle (0.0.1)
and the induction hypothesis, we may assume that the morphism $f$ is proper.
By normalizing $X$, by using cdh-descent and by using the induction hypothesis
again, we may also assume that the scheme $X$ is normal.

Using Gabber’s $\ell$-alterations [LYF14, IX.1.1], 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 of group of
order prime to $\ell$ and of a finite surjective purely inseparable morphism.

Therefore, there is a closed subscheme $F$ of $X$ of positive codimension such
that the induced map

$p\colon Y\setminus p^{-1}(F)\rightarrow X\setminus F$

is the composition of an étale Galois cover of group $G$, where the order of
$G$ is prime to $\ell$, and of a finite surjective purely inseparable
morphism.

Let $X^{\prime}$ be the relative normalization of $X$ in $Y\setminus
p^{-1}(F)$. Then, the map $X^{\prime}\rightarrow X$ is finite and the scheme
$X^{\prime}$ is endowed with a $G$-action. Furthermore, the map
$X^{\prime}/G\rightarrow X$ is purely inseparable since $X$ is normal.

Using [Ayo07, 2.1.165], for any integer $n$ we have a canonical isomorphism

$H^{n}(\omega^{0}h_{k}(X))\rightarrow H^{n}(\omega^{0}h_{k}(X^{\prime}))^{G}.$

Furthermore, by cdh-descent, and by using the induction hypothesis, the
canonical map

$H^{n}(\omega^{0}h_{k}(X^{\prime}))\rightarrow H^{n}(\omega^{0}h_{k}(Y))$

is an isomorphism if $n>2d$.

Now, by Poincaré Duality in étale cohomology [Del77] and by Theorem 3.1.4, we
have

$H^{2d+1}(\omega^{0}h_{k}(Y))=\alpha_{!}\hskip
2.84526pt\left[\mathbb{Q}/\mathbb{Z}(-d)\otimes_{\mathbb{Z}}R[\pi_{0}(Y_{\overline{k}})]\right]$

and the $G$-action on $H^{2d+1}(\omega^{0}h_{k}(Y))$ is induced by the action
of $G$ on the connected components of $Y_{\overline{k}}$.

Since the map $Y\rightarrow X^{\prime}$ is an isomorphism in codimension $1$,
the map

$\pi_{0}(Y_{\overline{k}})\rightarrow\pi_{0}(X^{\prime}_{\overline{k}})$

is a bijection. Thus, we get

$H^{2d+1}(\omega^{0}h_{k}(Y))=\alpha_{!}\hskip
2.84526pt\left[\mathbb{Q}/\mathbb{Z}(-d)\otimes_{\mathbb{Z}}R[\pi_{0}(X^{\prime}_{\overline{k}})]\right]^{G}$

If $x\in\mathbb{Q}/\mathbb{Z}(-d)$ and if $S$ is a connected component of the
scheme $X^{\prime}_{\overline{k}}$, denote by $[x]_{S}$ the element $x\otimes
e_{S}$ of
$\mathbb{Q}/\mathbb{Z}(-d)\otimes_{\mathbb{Z}}R[\pi_{0}(X^{\prime}_{\overline{k}})]$
where $e_{S}$ is the element of the basis of
$R[\pi_{0}(X^{\prime}_{\overline{k}})]$ attached to $S$. Then, the $G$-fixed
points are exactly the elements of the form:

$\sum\limits_{\begin{subarray}{c}T\in\pi_{0}(X_{\overline{k}}),S\in\pi_{0}(X^{\prime}_{\overline{k}})\\\
p(S)=T\end{subarray}}[x_{T}]_{S}$

where $x_{T}\in\mathbb{Q}/\mathbb{Z}(-d)$. Therefore, this set of $G$-fixed
points is isomorphic as a $G_{k}$-module to

$\mathbb{Q}/\mathbb{Z}(-d)\otimes_{\mathbb{Z}}R[\pi_{0}(X_{\overline{k}})]$

which is not an Artin representation. ∎

We now explore more consequences of Theorem 3.1.4.

######  Corollary 3.1.11.

Let $R$ be a localization of the ring of integers of a number field, let $k$
be a field of characteristic exponent $p$ and let $f\colon X\rightarrow S$ be
a proper morphism. Assume that

$X=\bigcup\limits_{i\in I}X_{i}$

where $I$ is a finite set and $X_{i}$ is a closed subscheme of $X$.

For $J\subseteq I$, let

$X_{J}=\bigcap\limits_{j\in J}X_{j}$

and let $f_{J}\colon X_{J}\rightarrow\operatorname{Spec}(k)$ be the structural
morphism. Assume that for any such $J$, the scheme $X_{J}$ is nil-regular.

  1. (1)

The Artin motive $\omega^{0}f_{*}\mathbbm{1}_{X}$ is the image through the
functor $\alpha_{!}$ of the following limit

$\lim_{[n]\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\ |J|=n+1\end{subarray}}(f_{J})_{*}\underline{R[1/p]}$

where the degeneracy maps are given by the appropriate closed immersions (see
[DDØ21, 5.1.3]).

  2. (2)

Let $Y$ be a regular scheme and $i\colon X\rightarrow Y$ be a closed
immersion. Let $r$ be a non-negative integer. Assume that for all $J\subseteq
I$, the subscheme $X_{J}$ is of codimension $|J|$ in $Y$. Then, the Artin
motive $\omega^{0}f_{*}i^{!}\mathbbm{1}_{Y}(-r)$ is the image through the
functor $\alpha_{!}$ of the following colimit

$\operatorname{colim}_{[n]\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\
|J|=n+1\end{subarray}}(f_{J})_{*}\left(\underline{R[1/p]\otimes\mathbb{Q}/\mathbb{Z}}\right)(-n-1-r)[-2n-3]$

###### Proof.

We begin with the first assertion. First, [DDØ21, 5.7] asserts that the
canonical map

$f_{*}\mathbbm{1}_{X}\rightarrow\lim_{[n]\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\ |J|=n+1\end{subarray}}(f_{J})_{*}\mathbbm{1}_{X_{J}}$

is an equivalence.

Since $\omega^{0}$ is an exact functor, it is compatible with finite limits.
Therefore, the canonical map

$\omega^{0}f_{*}\mathbbm{1}_{X}\rightarrow\lim_{[n]\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\ |J|=n+1\end{subarray}}\omega^{0}(f_{J})_{*}\mathbbm{1}_{X_{J}}$

is an equivalence. The first assertion then follows from Theorem 3.1.4.

We now prove the second assertion. If $J\subseteq I$, let $i_{J}\colon
X_{J}\rightarrow X$ be the inclusion. Again, [DDØ21, 5.7] asserts that the
canonical map

$i^{!}\mathbbm{1}_{Y}\rightarrow\operatorname{colim}_{[n]\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\ |J|=n+1\end{subarray}}(i_{J})_{*}(i\circ i_{J})^{!}\mathbbm{1}_{X}$

is an equivalence.

Since $\omega^{0}$ is an exact functor, it is compatible with finite colimits,
and therefore the canonical map

$\operatorname{colim}_{[n]\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\ |J|=n+1\end{subarray}}\omega^{0}f_{*}(i_{J})_{*}(i\circ
i_{J})^{!}\mathbbm{1}_{X}(-r)\rightarrow\omega^{0}f_{*}i^{!}\mathbbm{1}_{Y}(-r)$

is an equivalence.

Using absolute purity property [CD16, 5.6.2], we get that
$\omega^{0}f_{*}i^{!}\mathbbm{1}_{Y}(-r)$ is equivalent to

$\operatorname{colim}_{[n]\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\
|J|=n+1\end{subarray}}\omega^{0}f_{*}(i_{J})_{*}\mathbbm{1}_{X_{J}}(-n-1-r)[-2n-2].$

The result then follows from Theorem 3.1.4. ∎

With the notations of Corollary 3.1.11, we now want to describe the cohomology
of $\omega^{0}f_{*}\mathbbm{1}_{X}$ seen as the image of a discrete
$G_{k}$-module through the functor $\alpha_{!}$. Using [Lur17, 1.2.2.7],
Corollary 3.1.11 provides:

######  Corollary 3.1.12.

Keep the notations of Corollary 3.1.11.

  1. (1)

The limit of (1) of Corollary 3.1.11 induces a spectral sequence:

$\alpha_{!}E_{1}^{n,m}\Rightarrow H^{n+m}(\omega^{0}f_{*}\mathbbm{1}_{X})$

where

$E_{1}^{n,m}=\bigoplus\limits_{\begin{subarray}{c}J\subseteq I\\\
|J|=n+1\end{subarray}}H^{m}_{\operatorname{\acute{e}t}}((X_{J})_{\overline{k}},R[1/p])=\begin{cases}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\
|J|=n+1\end{subarray}}R[1/p]\left[\pi_{0}((X_{J})_{\overline{k}})\right]&\text{if
}m=0\\\ \bigoplus\limits_{\begin{subarray}{c}J\subseteq I\\\
|J|=n+1\end{subarray}}\mu^{m-1}(X_{J},R)&\text{if }m\geqslant 2\\\
0&\text{otherwise}\end{cases}.$

  2. (2)

the colimit of (2) of Corollary 3.1.11 induces a spectral sequence:

$\alpha_{!}\hskip 2.84526ptE^{n,m}_{1}=\alpha_{!}\hskip
2.84526pt\bigoplus\limits_{\begin{subarray}{c}J\subseteq I\\\
|J|=n+1\end{subarray}}\mu^{m-n-3}(X_{J})(-n-1-r)\Longrightarrow
H^{n+m}(\omega^{0}f_{*}i^{!}\mathbbm{1}_{Y}(-r)).$

###### Proof.

The Artin motive $\omega^{0}f_{*}\mathbbm{1}_{X}$ is the image through the
functor $\alpha_{!}$ of the limit

$\lim\limits_{n\in\Delta^{\mathrm{inj}}}\bigoplus\limits_{\begin{subarray}{c}J\subseteq
I\\\ |J|=n+1\end{subarray}}(f_{J})_{*}\underline{R[1/p]}.$

We can truncate this diagram in each degree to write
$\omega^{0}f_{*}\mathbbm{1}_{X}$ as the limit of a sequential diagram
$(M^{n})_{n\geqslant 0}$ such that the fiber of the map $M^{n}\rightarrow
M^{n-1}$ is $\alpha_{!}\hskip
2.84526pt\bigoplus\limits_{\begin{subarray}{c}J\subseteq I\\\
|J|=n+1\end{subarray}}(f_{J})_{*}\underline{R[1/p]}.$

By [Lur17, 1.2.2.14], this sequential diagram gives rise to a spectral
sequence such that

$\alpha_{!}\hskip 2.84526ptE_{1}^{n,m}=\alpha_{!}\hskip
2.84526pt\bigoplus\limits_{\begin{subarray}{c}J\subseteq I\\\
|J|=n+1\end{subarray}}H^{m}_{\operatorname{\acute{e}t}}((X_{J})_{\overline{k}},R[1/p])\Longrightarrow
H^{n+m}(\omega^{0}f_{*}\mathbbm{1}_{X}).$

This proves the first assertion. We now prove the second assertion.

As in the proof of the first assertion, we can compute the motive
$\omega^{0}f_{*}i^{!}\mathbbm{1}_{Y}$ using Corollary 3.1.11 as the colimit of
a sequential diagram $(N_{p})_{p\geqslant 0}$ such that the cofiber of the map
$N_{p-1}\rightarrow N_{p}$ is
$\rho_{!}\bigoplus\limits_{\begin{subarray}{c}J\subseteq I\\\
|J|=p+1\end{subarray}}(f_{J})_{*}\left(\underline{R[1/p]\otimes\mathbb{Q}/\mathbb{Z}}\right)(-n-1-r)[-2n-3].$

Now, by [Lur17, 1.2.2.14] this sequential diagram gives rise to a spectral
sequence such that

$\alpha_{!}\hskip 2.84526ptE^{n,m}_{1}=\alpha_{!}\hskip
2.84526pt\bigoplus\limits_{\begin{subarray}{c}J\subseteq I\\\
|J|=n+1\end{subarray}}\mu^{m-n-3}(X_{J})(-n-1-r)\Longrightarrow
H^{n+m}(\omega^{0}f_{*}i^{!}\mathbbm{1}_{Y})$

which is exactly the second assertion. ∎

###  3.2. Artin residues over $1$-dimensional schemes

In this section, we study an analog of the nearby cycle functor (or more
accurately residue functor) in our setting. Namely, let $R$ be a localization
of the ring of integers of a number field, let $C$ be an excellent
$1$-dimensional scheme, let $i\colon F\rightarrow C$ be a closed immersion of
positive codimension and let $j\colon U\rightarrow C$ be the complementary
open immersion; we study the functor

$\omega^{0}i^{*}j_{*}:\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(U,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R).$

First, notice that the scheme $F$ is $0$-dimensional. Therefore, the scheme
$F$ is the disjoint union of its points. Therefore, denoting by

$i_{x}\colon\\{x\\}\rightarrow F$

the inclusion for $x\in F$, we get

$\omega^{0}i^{*}j_{*}=\bigoplus\limits_{x\in
F}(i_{x})_{*}i_{x}^{*}\omega^{0}i^{*}j_{*}.$

Now, since the map $i_{x}$ is an open immersion, by Proposition 1.4.4, the
$2$-morphism $i_{x}^{*}\omega^{0}\rightarrow\omega^{0}i_{x}^{*}$ is an
equivalence. Therefore, the functor $\omega^{0}i^{*}j_{*}$ is equivalent to
the functor

$\bigoplus\limits_{x\in F}(i_{x})_{*}\omega^{0}(i\circ i_{x})^{*}j_{*}.$

Hence, studying the case where $F$ is a point will yield a description in the
general case. Assume from now on that $F$ is reduced to a single point. Let
$x$ be the only point of $F$ and let $p$ be the characteristic exponent of
$k(x)$.

Now, recall that Theorem 2.1.1 states that if $S$ is a regular scheme, and if
all the residue characteristic exponents of $S$ are invertible in $R$, the
functor

$\rho_{!}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)\rightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$

is an equivalence.

If the scheme $U$ is regular and if all the residue characteristic exponents
of $U$ are invertible in $R$, the functor $\omega^{0}i^{*}j_{*}$ above can be
seen as a functor

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(U,R)\rightarrow\operatorname{\mathcalboondox{Sh}}(k(x)_{\operatorname{\acute{e}t}},R[1/p]).$

Since the functor $\rho_{!}$ is the left adjoint functor of a premotivic
adjunction, we have an equivalence
$\rho_{!}i^{*}\overset{\sim}{\rightarrow}i^{*}\rho_{!}$ and an exchange
transformation $\rho_{!}j_{*}\rightarrow j_{*}\rho_{!}$.

This gives a transformation

$\rho_{!}i^{*}j_{*}\rightarrow i^{*}j_{*}\rho_{!}$

of functors

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(U,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(F,R).$

Furthermore, if $M$ is a lisse étale sheaf over $U$, then, the motive
$\rho_{!}i^{*}j_{*}M$ is Artin by Proposition 1.4.3. Therefore, we have a
transformation

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

of functors

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(U,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R).$

Finally, let

$\Theta=\overline{\Theta}\otimes_{R}R[1/p].$

Since tensoring with $R[1/p]$ induces an equivalence

$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(F,R[1/p]),$

the functor $\Theta$ can be seen as a transformation

$\rho_{!}i^{*}j_{*}\otimes_{R}R[1/p]\rightarrow\omega^{0}i^{*}j_{*}\rho_{!}.$

######  Proposition 3.2.1.

Let $R$ be a localization of the ring of integers of a number field, let $C$
be an excellent $1$-dimensional scheme, let $x$ be a closed point of $C$, let
$i\colon\\{x\\}\rightarrow C$ be the closed immersion, let $j\colon
U\rightarrow C$ be the complementary open immersion and let $p$ be the
characteristic exponent of $k(x)$.

Assume that the scheme $U$ is nil-regular and the residue characteristics of
$U$ are invertible in $R$.

Then, the transformation

$\Theta\colon\rho_{!}i^{*}j_{*}\otimes_{R}R[1/p]\rightarrow\omega^{0}i^{*}j_{*}\rho_{!}$

is an equivalence. In particular, the square

${\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(U,R)}$${\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(U,R)}$${\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(F,R[1/p])}$${\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(F,R)}$$\scriptstyle{\rho_{!}}$$\scriptstyle{i^{*}j_{*}\otimes_{R}R{[1/p]}}$$\scriptstyle{\omega^{0}i^{*}j_{*}}$$\scriptstyle{\rho_{!}}$

is commutative.

###### Proof.

Tensoring with $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Q}$ commutes with the six
functors formalism and with $\rho_{!}$ by Proposition 1.2.7, it also commutes
with the functor $\omega^{0}$ by Proposition 1.4.8. Using the conservativity
property (Lemma 1.2.5), we can therefore assume that the ring $R$ is either a
$\mathbb{Q}$-algebra or an $n$-torsion algebra with $n$ an integer prime to
$p$.

In the $n$-torsion case, the proposition follows from the fact that
$\omega^{0}$ is an equivalence by Corollary 1.4.10 and from the rigidity
theorem and the fact that the six functors commute with the functor $\rho_{!}$
in the torsion case by Proposition 1.3.3.

Assume now that the ring $R$ is a $\mathbb{Q}$-algebra.

Let $\nu\colon\widetilde{C}\rightarrow C$ be the normalization of $C$. Write

${U_{red}}$${\widetilde{C}}$${F}$${U}$${C}$${\\{x\\}}$$\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
$U_{red}$ is the reduced subscheme associated to $U$. Therefore, the functor
$(\nu_{U})_{*}$ is an equivalence and commutes with the functor $\rho_{!}$.
Hence, it suffices to show that

$\Theta(\nu_{U})_{*}:\rho_{!}i^{*}j_{*}(\nu_{U})_{*}\rightarrow\omega^{0}i^{*}j_{*}(\nu_{U})_{*}\rho_{!}$

is an equivalence.

Therefore, it suffices to show that the map
$\rho_{!}i^{*}\nu_{*}\gamma_{*}\rightarrow\omega^{0}i^{*}\nu_{*}\gamma_{*}\rho_{!}$
is an equivalence.

Using proper base change, this is equivalent to the fact that the map

$\rho_{!}(\nu_{F})_{*}\iota^{*}\gamma_{*}\rightarrow\omega^{0}(\nu_{F})_{*}\iota^{*}\gamma_{*}\rho_{!}$

is an equivalence. Since the map induced by $\nu_{F}$ over the reduced schemes
is finite and étale, the functor $(\nu_{F})_{*}$ commutes with the functor
$\rho_{!}$ by Proposition 1.2.1 and with the functor $\omega^{0}$ by
Proposition 1.4.4. Thus, it suffices to show that the map

$\rho_{!}\iota^{*}\gamma_{*}\rightarrow\omega^{0}\iota^{*}\gamma_{*}\rho_{!}$

is an equivalence.

Moreover, since the scheme $F$ is $0$-dimensional, we can assume as before
that $F$ is a point. Therefore, we can assume that the $1$-dimensional scheme
$C$ is normal and therefore regular.

Since by Proposition 1.5.9 the stable category
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(U,R)$ is generated
as a thick subcategory of itself by the sheaves of the form
$R_{U}(V)=f_{*}\underline{R}$ for $f\colon V\rightarrow U$ finite and étale
and since the functors $\rho_{!}i^{*}j_{*}$ and $\omega^{0}i^{*}j_{*}$ are
exact and therefore compatible with finite limits, finite colimits and
retracts, it suffices to show that the map

$\Theta(R_{U}(V))\colon\rho_{!}i^{*}j_{*}f_{*}\underline{R}\rightarrow\omega^{0}i^{*}j_{*}f_{*}\mathbbm{1}_{V}$

is an equivalence.

To show that this map is an equivalence, we introduce $\overline{f}\colon
C_{V}\rightarrow C$ the normalization of $C$ in $V$. Write

${V}$${C_{V}}$${Z}$${U}$${C}$${\\{x\\}}$$\scriptstyle{f}$$\scriptstyle{\gamma}$$\scriptstyle{\overline{f}}$$\scriptstyle{p}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\scriptstyle{i}$

the commutative diagram such that both squares are cartesian. The map
$\Theta(R_{U}(V))$ is an equivalence if and only if the map

$\rho_{!}i^{*}\overline{f}_{*}\gamma_{*}\underline{R}\rightarrow\omega^{0}i^{*}\overline{f}_{*}\gamma_{*}\mathbbm{1}_{V}$

is an equivalence. The morphism $p$ induces a finite and étale map over the
reduced schemes and therefore the functor $p_{*}$ commutes with the functor
$\rho_{!}$ by Proposition 1.2.1 and with the functor $\omega^{0}$ by
Proposition 1.4.4. Hence, using proper base change, it suffices to show that
the map

$\rho_{!}\iota^{*}\gamma_{*}\underline{R}\rightarrow\omega^{0}\iota^{*}\gamma_{*}\mathbbm{1}_{V}$

is an equivalence.

Hence, we can assume that $V=U$. Therefore, we need to prove that the map

$\Theta(\underline{R})\colon\rho_{!}i^{*}j_{*}\underline{R}\rightarrow\omega^{0}i^{*}j_{*}\mathbbm{1}_{U}$

is an equivalence. Using Proposition 1.4.4, the exchange transformation

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

is an equivalence. Furthermore, by Proposition 1.2.1, the exchange
transformation $\rho_{!}i^{*}\rightarrow i^{*}\rho_{!}$ is an equivalence.
Therefore, it suffices to show that the exchange map

$\rho_{!}j_{*}\underline{R}\rightarrow\omega^{0}j_{*}\mathbbm{1}_{U}$

is an equivalence which is a direct consequence of the two lemmas below. ∎

######  Lemma 3.2.2.

Let $R$ be a $\mathbb{Q}$ algebra. Let $S$ be a regular scheme, and $j\colon
U\rightarrow S$ an open immersion with complement a simple normal crossing
divisor. Then, the canoncal map

$\mathbbm{1}_{S}\rightarrow\omega^{0}j_{*}\mathbbm{1}_{U}$

is an equivalence.

###### Proof.

This result generalizes [AZ12, 2.11]. Let $i\colon D\rightarrow S$ be the
closed immersion which is complementary to $j$. Using Proposition 1.4.7, we
have an exact triangle

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

It therefore suffices to show that

$\omega^{0}i^{!}\mathbbm{1}_{S}=0.$

Using [DDØ21, 5.7], we can reduce to the case when $D$ is a regular subscheme
of codimension $c>0$. In this setting, we have

$\omega^{0}i^{!}\mathbbm{1}_{S}=\omega^{0}(\mathbbm{1}_{D}(-c))[-2c]$

and the latter vanishes by [Leh19b, 3.9]. ∎

######  Lemma 3.2.3.

Let $R$ be a $\mathbb{Q}$ algebra, let $S$ be a normal scheme, and let
$j\colon U\rightarrow S$ be an open immersion. Then, the canonical map

$\underline{R}\rightarrow j_{*}\underline{R}$

is an equivalence.

###### Proof.

The canonical map $\underline{R}\rightarrow j_{*}\underline{R}$ is an
equivalence if and only if for any geometric point $\xi$ of $S$, the canonical
map

$R\rightarrow\xi^{*}j_{*}\underline{R}$

is an equivalence. If $n$ is an integer, [AGV73, VIII.5] yields

$H^{n}(\xi^{*}j_{*}\underline{R})=H^{n}_{\operatorname{\acute{e}t}}(\operatorname{Spec}(\mathcal{O}^{sh}_{S,x})\times_{S}U,R)$

where $\mathcal{O}^{sh}_{S,x}$ is the strict henselization of the local ring
$\mathcal{O}_{S,x}$ of $S$ at $x$.

Since the scheme $S$ is normal, so is the scheme
$\operatorname{Spec}(\mathcal{O}^{sh}_{S,x})\times_{S}U$. Moreover, this
scheme is also connected. Therefore Proposition 1.7.2 yields the result. ∎

######  Remark 3.2.4.

Proposition 3.2.1 is specific to curves. Assume that $R=\mathbb{Q}$. A
reasonable equivalent to Proposition 3.2.1 would then be that if $S$ is an
excellent scheme, $i\colon F\rightarrow S$ a closed immersion and $j\colon
U\rightarrow S$ is the complementary open immersion, with $F$ a Cartier
divisor (so that the morphism $j$ is affine) and $U$ regular, then, the
$2$-morphism

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

is an equivalence.

However, this statement is not true. We now give a counterexample. Let $k$ be
a field. Take $S=\operatorname{Spec}(k[x,y,z]/(x^{2}+y^{2}-z^{2}))$ to be the
affine cone over the rational curve of degree $2$ in $\mathbb{P}^{2}$ given by
the homogeneous equation $x^{2}+y^{2}=z^{2}$. The scheme $S$ has a single
singular point $P=(0,0,0)$.

Let $F$ be the Cartier divisor of $S$ given by the equation $x=0$, so that $F$
is the union of the two lines

$L_{1}\colon y-z=0\text{ and }L_{2}\colon y+z=0$

which contain $P$ and therefore the complement $U$ of $F$ is regular.

The blow up $B_{P}(S)$ of $S$ at $P$ is a resolution of singularities of $S$
and the exceptional divisor $E$ above $F$ is the reunion of
$\mathbb{P}^{1}_{k}$ and of the strict transform $L_{i}^{\prime}$ of the line
$L_{i}$ for $i\in\\{1,2\\}$. Each $L_{i}^{\prime}$ crosses
$\mathbb{P}^{1}_{k}$ at a single point and $L_{1}^{\prime}$ and
$L_{2}^{\prime}$ do not cross each other.

We therefore have a commutative diagram

${U}$${\widetilde{S}=B_{P}(S)}$${E=L^{\prime}_{1}\cup\mathbb{P}^{1}_{k}\cup
L^{\prime}_{2}}$${U}$${S}$${F=L_{1}\cup
L_{2}}$$\scriptstyle{\gamma}$$\scriptstyle{f}$$\scriptstyle{p}$$\scriptstyle{\iota}$$\scriptstyle{j}$$\scriptstyle{i}$

where both squares are cartesian.

Now, Lemma 3.2.3 yields

$\rho_{!}i^{*}j_{*}\underline{R}=\mathbbm{1}_{F}.$

On the other hand, we have

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

by Proposition 1.4.4. But Lemma 3.2.2 ensures that

$\omega^{0}\gamma_{*}\mathbbm{1}_{U}=\mathbbm{1}_{\widetilde{S}}$

and thus, we get

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

Finally, notice that $\omega^{0}p_{*}\mathbbm{1}_{E}$ does not coincide with
$\mathbbm{1}_{F}$ since by Theorem 3.1.4 $\omega^{0}i_{x}^{!}\mathbbm{1}_{F}$
is trivial while $\omega^{0}i_{x}^{!}\omega^{0}p_{*}\mathbbm{1}_{E}$ is
equivalent to $\mathbbm{1}_{x}$.

##  4\. Ordinary homotopy $\mathrm{t}$-structure on Artin étale motives

###  4.1. Definition for non-constructible Artin motives

######  Definition 4.1.1.

Let $S$ be a scheme and let $R$ be a ring. The _ordinary homotopy_ t-structure
on $\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$ is the
t-structure generated in the sense of Proposition 1.1.6 by the family of the
motives $M_{S}^{\operatorname{BM}}(X)$ with $X$ quasi-finite over $S$.

We denote with the small letters ${\mathrm{ord}}$ the notions related to the
perverse t-structure on
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$. For
instance, we denote by ${}^{\mathrm{ord}}\tau_{\leqslant 0}$ the left
truncation with respect to the ordinary homotopy t-structure and by
$t_{\mathrm{ord}}$ the ordinary homotopy t-structure itself.

Using the same method as Proposition 1.4.1, this t-structure is also generated
by family of the motives $M_{S}^{\operatorname{BM}}(X)$ with $X$ finite (resp.
étale) over $S$. Thus, an object $M$ of
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$ is in
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)^{\geqslant
n}$ if and only if for any étale $S$-scheme $X$, the complex
$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M_{S}(X),M)$
is $(n-1)$-connected.

######  Proposition 4.1.2.

Let $R$ be a ring. Let $f$ be a quasi-finite morphism of schemes and let $g$
be a morphism which is essentially of finite type. Then,

  1. (1)

The adjunction $(f_{!},\omega^{0}f^{!})$ is a $t_{\mathrm{ord}}$-adjunction.

  2. (2)

If $M$ is a $t_{\mathrm{ord}}$-non-positive Artin motive, the functor
$-\otimes_{S}M$ is right $t_{\mathrm{ord}}$-exact.

  3. (3)

The adjunction $(g^{*},\omega^{0}g_{*})$ is a $t_{\mathrm{ord}}$-adjunction.

###### Proof.

The full subcategory of those Artin motives $N$ such that $f_{!}N$,
$g^{*}N[d]$ and $N\otimes M$ are $t_{\mathrm{ord}}$-non-positive is closed
under extensions, small coproducts and negative shifts and contains the set of
generators of the t-structure. Thus, by definition of $t_{\mathrm{ord}}$, this
subcategory contains any $t_{\mathrm{ord}}$-non-positive object. Hence, the
functors $f_{!}$, $g^{*}[d]$ and $-\otimes M$ are right
$t_{\mathrm{ord}}$-exact. ∎

######  Corollary 4.1.3.

Let $R$ be a ring and let $f$ be a morphism of schemes. Then,

  1. (1)

If $f$ is étale, the functor $f^{*}=f^{!}=\omega^{0}f^{!}$ is
$t_{\mathrm{ord}}$-exact.

  2. (2)

If $f$ is finite, the functor $f_{!}=f_{*}=\omega^{0}f_{*}$ is
$t_{\mathrm{ord}}$-exact.

######  Proposition 4.1.4.

Let $S$ be a scheme, let $R$ be a ring, let $i\colon F\rightarrow S$ be a
closed immersion and $j\colon U\rightarrow S$ be the open complement. Then,
the sequence

$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(F,R)\overset{i_{*}}{\rightarrow}\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)\overset{j^{*}}{\rightarrow}\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(U,R)$

satisfies the axioms of the gluing formalism (see [BBDG18, 1.4.3]), i.e

  1. (1)

The functors $i_{*}$ and $j^{*}$ both have exact left and right adjoints
$i^{*}$, $\omega^{0}i^{!}$, $j_{!}$ and $\omega^{0}j_{*}$.

  2. (2)

We have $j^{*}i_{*}=0$.

  3. (3)

The functors $i_{*}$, $j_{*}$ and $j_{!}$ are fully faithful.

  4. (4)

We have exact triangles

$j_{!}j^{*}\rightarrow 1\rightarrow i_{*}i^{*}$

and

$i_{*}\omega^{0}i^{!}\rightarrow 1\rightarrow\omega^{0}j_{*}j^{*}.$

Moreover, the ordinary homotopy t-structure on
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$ is obtained by gluing the
t-structures of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(U,\mathbb{Z}_{\ell})$ and
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(F,R)$ (see [BBDG18, 1.4.9]) i.e.
for any object $M$ of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$, we
have

  1. (1)

$M\geqslant_{{\mathrm{ord}}}0$ if and only if
$j^{*}M\geqslant_{{\mathrm{ord}}}0$ and
$\omega^{0}i^{!}M\geqslant_{{\mathrm{ord}}}0$.

  2. (2)

$M\leqslant_{{\mathrm{ord}}}0$ if and only if
$j^{*}M\leqslant_{{\mathrm{ord}}}0$ and $i^{*}M\leqslant_{{\mathrm{ord}}}0$.

###### Proof.

This follows from Proposition 4.1.2, Corollary 4.1.3 and from the usual
properties of the six functors. ∎

######  Corollary 4.1.5.

Let $S$ be a scheme, let $R$ be a ring and let $M$ be an object of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$. Then, we have

  1. (1)

$M\geqslant_{{\mathrm{ord}}}0$ if and only if there is a stratification of $S$
such that for any stratum $i\colon T\hookrightarrow S$, we have
$\omega^{0}i^{!}M\geqslant_{{\mathrm{ord}}}0$.

  2. (2)

$M\leqslant_{{\mathrm{ord}}}0$ if and only if there is a stratification of $S$
such that for any stratum $i\colon T\hookrightarrow S$, we have
$i^{*}M\leqslant_{{\mathrm{ord}}}0$.

We now explore the compatibility of the ordinary homotopy t-structure with
respect to change of coefficients.

######  Proposition 4.1.6.

Let $S$ be a scheme, let $R$ be a ring, let $n$ be an integer and let $A$ be a
localization of $\mathbb{Z}$. Denote by $\sigma_{n}\colon R\rightarrow R/nR$
and by $\sigma_{A}\colon R\rightarrow R\otimes_{\mathbb{Z}}A$ the natural ring
morphisms. Recall the notations of Section 1.2.7. Then,

  1. (1)

The functors $(\sigma_{A})_{*}$ and $(\sigma_{n})_{*}$ are
$t_{\mathrm{ord}}$-exact.

  2. (2)

the functor $\sigma_{A}^{*}$ is $t_{\mathrm{ord}}$-exact.

###### Proof.

Let $R^{\prime}=R/nR$ or $R\otimes_{\mathbb{Z}}A$ and let $\sigma\colon
R\rightarrow R^{\prime}$ be the natural ring morphism. Since the six functors
commute with change of coefficients to $R^{\prime}$, the natural map

$M_{S}^{BM}(X,R)\otimes_{R}R^{\prime}\rightarrow M_{S}^{BM}(X,R^{\prime})$

is an equivalence for any quasi-finite $S$-scheme $X$. Hence, the colimit
preserving exact functor $\sigma^{*}$ sends the generators of
$t_{\mathrm{ord}}$ to $t_{\mathrm{ord}}$-non-positive objects and is therefore
right $t_{\mathrm{ord}}$-exact. By adjunction, the functor $\sigma_{*}$ is
therefore left $t_{\mathrm{ord}}$-exact.

First, notice that if $X$ is quasi-finite over $S$, the motive
$M_{S}^{BM}(X,R/nR)$ is the cofiber of the map

$M_{S}^{BM}(X,R)\overset{\times n}{\rightarrow}M_{S}^{BM}(X,R).$

Therefore the motive $M_{S}^{BM}(X,R/nR)$ is a colimit of
$t_{\mathrm{ord}}$-non-positive object and is therefore
$t_{\mathrm{ord}}$-non-positive. Since the subcategory of
$t_{\mathrm{ord}}$-non-positive objects of the stable category
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R/nR)$ is the smallest
subcategory closed under small colimits and extensions containing the
$M_{S}^{BM}(X,R/nR)$ when $X$ runs through the set of quasi-finite
$S$-schemes, and since the exact functor $(\sigma_{n})_{*}$ preserves small
colimits, the functor $(\sigma_{n})_{*}$ is therefore right
$t_{\mathrm{ord}}$-exact.

On the other hand, write $A=\Sigma^{-1}\mathbb{Z}$ with $\Sigma$ a
multiplicative system. Then, the motive $M_{S}^{BM}(X,R\otimes_{\mathbb{Z}}A)$
is the colimit of the $M_{S}^{BM}(X,R)$ with transition maps the
multiplications by $s$ for $s\in\Sigma$. Hence, it is $t_{\mathrm{ord}}$-non-
positive. Since the functor $(\sigma_{A})_{*}$ is also colimit preserving,
this yields its $t_{\mathrm{ord}}$-exactness in a similar fashion.

It now suffices to show that the exact functor $\sigma_{A}^{*}$ is left-t-
exact. Take $M$ a $t_{\mathrm{ord}}$-non-negative object of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)$. Let $X$ be
a quasi-finite $S$-scheme. Then, we have

$\operatorname{Map}(M_{S}^{BM}(X,R\otimes_{\mathbb{Z}}A),\sigma_{A}^{*}M)=\operatorname{Map}(\sigma_{A}^{*}M_{S}^{BM}(X,R),\sigma_{A}^{*}M)$

But, by Proposition 1.2.3, we have

$\operatorname{Map}(\sigma_{A}^{*}M_{S}^{BM}(X,R),\sigma_{A}^{*}M)=\operatorname{Map}(M_{S}^{BM}(X,R),M)\otimes_{R}(R\otimes_{\mathbb{Z}}A).$

Thus, the chain complex
$\operatorname{Map}(M_{S}^{BM}(X,R\otimes_{Z}A),\sigma_{A}^{*}M)$ is
$(-1)$-connected and the motive $\sigma_{A}^{*}(M)$ is t-positive with respect
to the ordinary homotopy t-structure. Therefore, the functor $\sigma_{A}^{*}$
is t-exact. ∎

######  Proposition 4.1.7.

Let $S$ be a scheme and let $R$ be a ring. Recall Proposition 1.4.9. Then, the
inclusion functor

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

is t-exact when the left hand side is endowed with the ordinary t-structure
(see Definition 1.3.7) and when the right hand side is endowed with the
ordinary homotopy t-structure.

###### Proof.

If $X$ is étale over $S$, the functor $\rho_{!}$ maps the étale sheaf
$R_{S}(X)$ to the motive $M_{S}(X)$ which is $t_{\mathrm{ord}}$-non-positive.
Recall that Proposition 1.3.5 asserts that the sheaves of the form $R_{S}(X)$
with $X$ étale over $S$ generate the ordinary t-structure on the stable
category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$. Since
the exact functor

$\rho_{!}\colon\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$

of Proposition 1.4.3 is compatible with small colimits, it is therefore right
t-exact.

If $p$ is a prime number, we denote by $j_{p}\colon S[1/p]\rightarrow S$ the
open immersion. Recall that by Proposition 1.3.3, the functor $j_{p}^{*}$
induces an equivalence

$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S[1/p],R)$

with inverse $(j_{p})_{!}=(j_{p})_{*}$.

Now, let $M$ be a torsion étale motive. Assume first that $M$ is t-non-
positive with respect to the ordinary t-structure. By definition, this means
that

$M=\bigoplus_{p\text{ prime}}(j_{p})_{!}\rho_{!}N_{p},$

where for any prime number $p$, $N_{p}$ is a $p^{\infty}$-torsion étale sheaf
which is t-non-positive with respect to the ordinary t-structure .

By right t-exactness of the functor $\rho_{!}$, for any prime number $p$, the
motive $\rho_{!}N_{p}$ is $t_{\mathrm{ord}}$-non-positive. Hence, using
Proposition 4.1.2, so is the motive $(j_{p})_{!}\rho_{!}N_{p}$. Therefore, the
motive $M$ is $t_{\mathrm{ord}}$-non-positive.

Assume now that $M$ is t-non-negative with respect to the ordinary
t-structure. Let $X$ be an étale $S$-scheme. By Proposition 1.2.7, we get

$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M_{S}(X),M)=\bigoplus_{p\text{
prime}}\operatorname{colim}_{n}\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M_{S}(X),M\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1]).$

If $p$ is a prime number and if $n$ is an integer, denote by $M_{p^{n}}$ the
motive $M\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1]$. We have

$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M_{S}(X),M_{p^{n}})=\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R/nR)}(M_{S}(X,R/nR),M_{p^{n}}).$

Since by [CD16, A.3.4], we have

$M_{p^{n}}=(j_{p})_{*}j_{p}^{*}M_{p^{n}},$

we get

$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R/nR)}(M_{S}(X,R/nR),M_{p^{n}})=\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S[1/p]_{\operatorname{\acute{e}t}},R/nR)}(M_{S[1/p]}(X[1/p]),j_{p}^{*}(M_{p^{n}})).$

And therefore, we get

$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M_{S}(X),M_{p^{n}})=\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S[1/p]_{\operatorname{\acute{e}t}},R)}(R_{S[1/p]}(X[1/p]),\rho^{!}j_{p}^{*}(M_{p^{n}})).$

If $p$ is a prime number and if $n$ is an integer, the $p^{\infty}$-torsion
étale motive $M\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1]$ is t-non-
negative with respect to the ordinary t-structure. Therefore, by definition of
the ordinary t-structure, the $p^{\infty}$-torsion étale sheaf
$\rho^{!}j_{p}^{*}(M_{p^{n}})$ is t-non-negative with respect to the ordinary
t-structure. Therefore, the complex
$\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S[1/p]_{\operatorname{\acute{e}t}},R)}(R_{S[1/p]}(X[1/p]),\rho^{!}j_{p}^{*}(M_{p^{n}}))$
is $(-1)$-connected. Hence, the complex
$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M_{S}(X),M)$
is also $(-1)$-connected. Hence, the motive $M$ is $t_{\mathrm{ord}}$-non-
negative. ∎

###  4.2. The ordinary homotopy t-structure on constructible Artin motives and
the t-exactness of the $\ell$-adic realization

We begin this section with the case of smooth Artin motives when the residue
characteristic exponents of the base scheme are invertible in the ring of
coefficients.

######  Proposition 4.2.1.

Let $R$ be a regular good ring and let $S$ be a regular scheme. Assume that
the residue characteristic exponents of $S$ are invertible in $R$.

Then, the functor $\rho_{!}$ of Proposition 1.4.3 induces a fully faithful
t-exact functor

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

when the left hand side is endowed with the ordinary t-structure and the right
hand side is endowed with the ordinary homotopy t-structure.

In particular, the ordinary t-structure on the stable category
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)$ induces a
t-structure on the stable subcategories
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$ and
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)$.

###### Proof.

In the proof of Proposition 4.1.7, we showed that the exact functor

$\rho_{!}\colon\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$

is right t-exact. Therefore, it suffices to show that the exact functor

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

is left t-exact. Let $N$ be an Ind-lisse étale sheaf and let $X$ be an étale
$S$-scheme. By Theorem 2.1.1, we have

$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S;R)}(M_{S}(X),\rho_{!}N)=\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}(R_{S}(X),N).$

Hence, if the sheaf $N$ is t-non-negative, by Proposition 1.1.7, so is the
motive $\rho_{!}N$. Therefore, the functor

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

is t-exact.

By Theorem 2.1.1, the functor $\rho_{!}$ is fully faithful with essential
image the stable subcategory
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$.
Therefore, the ordinary 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
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$.

Finally, since by Proposition 1.5.8, the ordinary t-structure on the stable
category $\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$
induces a t-structure on the stable subcategory
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$, the ordinary
homotopy t-structure on the stable category
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$ induces a
t-structure on the stable subcategory
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)$ by
Theorem 2.1.1. ∎

We want to show that the ordinary homotopy t-structure induces a t-structure
on constructible Artin motives and to study the properties of the induced
t-structure. In the case of Artin motives with rational coefficients, this was
done in [Leh19a]. We will use Proposition 4.1.7 to extend Pepin Lehalleur’s
result to arbitrary regular good rings. The first step is to extend Pepin
Lehalleur’s result on the t-exactness of the functor $f^{*}$ for any morphism
of schemes $f$ to a result with intgeral coefficients. To that end, we need to
introduce the following auxiliary notion.

######  Definition 4.2.2.

Let $S$ be a scheme and let $R$ be a ring. An Artin étale motive $M$ over $S$
with coefficients in $R$ is _$\mathbb{Q}$ -constructible_ if the Artin motive
$M\otimes_{\mathbb{Z}}\mathbb{Q}$ is constructible. We denote by
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},\mathbb{Q}-c}^{A}(S,R)$
the thick stable subcategory of the stable category
$\operatorname{\mathcal{DM}}_{et}^{A}(S,R)$ made of those objects which are
$\mathbb{Q}$-constructible.

Recall the following definition from [Leh19a].

######  Definition 4.2.3.

We say that a scheme $S$ allows resolution of singularities by alterations if
for any separated $S$-scheme $X$ of finite type and any nowhere dense closed
subset $Z$ of $X$, there is a proper alteration $g\colon X^{\prime}\rightarrow
X$ with $X^{\prime}$ regular and such that $g^{-1}(Z)$ is a strict normal
crossing divisor.

######  Proposition 4.2.4.

Let $S$ be an excellent scheme allowing resolution of singularities by
alterations and let $R$ be a regular good ring. The ordinary homotopy
t-structure on the stable category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$ induces a
t-structure on the stable subcategory
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},\mathbb{Q}-c}^{A}(S,R)$.

###### Proof.

Assume first that the ring $R$ is a $\mathbb{Q}$-algebra. The proposition
precisely asserts that the ordinary homotopy t-structure induces a t-structure
on the stable category
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$.

But by [Leh19b, 4.17], the inclusion functor

$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)\rightarrow\operatorname{\mathcal{DM}}^{1}_{\operatorname{\acute{e}t}}(S,R)$

is t-exact when the left hand side is endowed with the ordinary homotopy
t-structure and the right hand side is endowed with the standard motivic
t-structure of Pepin Lehalleur defined in [Leh19b, 4.4]. By [Leh19a, 4.1], the
truncation ${}^{\mathrm{std}}\tau_{\leqslant 0}$ with respect to the standard
t-structure preserves compact objects. Therefore, the functor
${}^{\mathrm{std}}\tau_{\leqslant 0}={}^{{\mathrm{ord}}}\tau_{\leqslant 0}$
preserves
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$. Hence,
using [BBDG18, 1.3.19], the ordinary homotopy t-structure induces a
t-structure on the stable category
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$.

Take now a general ring $R$. Let $M$ be a $\mathbb{Q}$-constructible Artin
motive. Proposition 4.1.6 implies that

${}^{{\mathrm{ord}}}\tau_{\leqslant
0}(M)\otimes_{\mathbb{Z}}\mathbb{Q}={}^{{\mathrm{ord}}}\tau_{\leqslant
0}(M\otimes_{\mathbb{Z}}\mathbb{Q}).$

But the Artin motive ${}^{{\mathrm{ord}}}\tau_{\leqslant
0}(M\otimes_{\mathbb{Z}}\mathbb{Q})$ is constructible using the result with
coefficients in $R\otimes_{\mathbb{Z}}\mathbb{Q}$. Therefore, the Artin motive
${}^{{\mathrm{ord}}}\tau_{\leqslant 0}(M)$ is $\mathbb{Q}$-constructible. The
result then follows from [BBDG18, 1.3.19]. ∎

We can now formulate and prove an analog of Pepin Lehalleur’s result on the
t-exactness of the functor $f^{*}$ to for any morphism of schemes $f$.

######  Proposition 4.2.5.

Let $S$ be an excellent scheme allowing resolution of singularities by
alterations, let $f\colon T\rightarrow S$ be a morphism of schemes and let $R$
be a ring.

Then, the functor

$f^{*}\colon\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},\mathbb{Q}-c}(S,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},\mathbb{Q}-c}(T,R)$

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

###### Proof.

Proposition 4.1.2 implies that the functor $f^{*}$ is right
$t_{\mathrm{ord}}$-exact. Hence, it suffices to show that it is left
$t_{\mathrm{ord}}$-exact.

Let $M$ be a $\mathbb{Q}$-constructible motive over $S$ which is
$t_{\mathrm{ord}}$-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
$t_{\mathrm{ord}}$-non-negative. Since the subcategory of
$t_{\mathrm{ord}}$-non-negative objects is closed under limits, the motive
$M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1]$ is also
$t_{\mathrm{ord}}$-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 [Leh19a, 4.1], the motive $f^{*}(M\otimes_{\mathbb{Z}}\mathbb{Q})$ is
$t_{\mathrm{ord}}$-non-negative and by Proposition 1.3.9, the motive
$f^{*}(M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1])$ is
$t_{\mathrm{ord}}$-non-negative. Since the subcategory of t-non-negative is
closed under extensions, the motive $f^{*}(M)$ is also $t_{\mathrm{ord}}$-non-
negative. ∎

We can now prove the main result of this section which extends [Leh19a, 4.1].

######  Theorem 4.2.6.

Let $S$ be a scheme allowing resolution of singularities by alterations and
let $R$ be a regular good ring. Then,

  1. (1)

The ordinary homotopy t-structure induces a t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$. We still
call this t-structure the _ordinary homotopy_ t-structure.

  2. (2)

The objects of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$ are
bounded with respect to the ordinary homotopy t-structure which is therefore
non-degenerate.

  3. (3)

Let $f\colon T\rightarrow S$ be a morphism of schemes. Then, the functor

$f^{*}\colon\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(S,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(T,R)$

is $t_{\mathrm{ord}}$-exact.

  4. (4)

If $x$ is a point of $S$, denote by $i_{x}\colon\\{x\\}\rightarrow S$ the
inclusion. The family

$\left(i^{*}_{x}:\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(S,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(k(x),R)\mid
x\in S\right)$

is a conservative family of $t_{\mathrm{ord}}$-exact functors.

  5. (5)

Assume that the ring $R$ is 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
ordinary homotopy t-structure and the right hand side is endowed with the
ordinary t-structure.

###### Proof.

First notice that if the first assertion is true, the third assertion makes
sense and follows from Proposition 4.2.5. Furthermore, the third assertion
implies the fourth assertion since the family $(i_{x}^{*})_{x\in S}$ is
conservative by [CD19, 4.3.17]. Using Lemma 1.1.8, the fourth assertion
implies the fifth assertion. Finally the second assertion follows from Lemma
4.2.7 below. Hence, it suffices to show the first assertion.

If $\mathfrak{p}$ is a prime ideal of $\mathbb{Z}$ and if $X$ is a scheme, we
say that an Artin motive is _$\mathfrak{p}$ -constructible_ when, with the
notations of Section 1.1.5, its image in the category
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(X,R)_{\mathfrak{p}}$
belongs to
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(X,R)_{\mathfrak{p}}$.
Notice that

  * •

if $f$ is a morphism of schemes, the functor $f^{*}$ preserve
$\mathfrak{p}$-constructible Artin motives.

  * •

if $f$ is a quasi-finite morphism of schemes, the functor $f_{!}$ preserve
$\mathfrak{p}$-constructible Artin motives.

  * •

the category of $\mathfrak{p}$-constructible motives over $X$ is a thick
subcategory of
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(X,R)$ for any
scheme $X$.

Let $M$ be a constructible Artin motive over $S$. We want to show that the
Artin motive ${}^{{\mathrm{ord}}}\tau_{\leqslant 0}(M)$ is constructible. By
Proposition 1.1.12, it suffices to show that it is
$\mathfrak{p}$-constructible for any maximal ideal $\mathfrak{p}$ of
$\mathbb{Z}$.

Fix a maximal ideal $\mathfrak{p}$ of $\mathbb{Z}$. Let $p$ be a generator of
$\mathfrak{p}$, let $S_{p}=S\times_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$, let
$i_{p}\colon S_{p}\rightarrow S$ be the immersion and let $j_{p}\colon
S[1/p]\rightarrow S$ be the complementary open immersion.

We have an exact triangle

$(j_{p})_{!}j_{p}^{*}\hskip 2.84526pt{}^{{\mathrm{ord}}}\tau_{\leqslant
0}(M)\rightarrow{}^{{\mathrm{ord}}}\tau_{\leqslant
0}(M)\rightarrow(i_{p})_{*}i_{p}^{*}\hskip
2.84526pt{}^{{\mathrm{ord}}}\tau_{\leqslant 0}(M).$

The functors $(i_{p})_{*}$ and $(j_{p})_{!}$ preserve
$\mathfrak{p}$-constructible Artin motives. Therefore, it suffices to show
that the Artin motives $j_{p}^{*}\hskip
2.84526pt{}^{{\mathrm{ord}}}\tau_{\leqslant 0}(M)$ and $i_{p}^{*}\hskip
2.84526pt{}^{{\mathrm{ord}}}\tau_{\leqslant 0}(M)$ are
$\mathfrak{p}$-constructible.

Using Proposition 4.2.5, we have

$j_{p}^{*}\hskip 2.84526pt{}^{{\mathrm{ord}}}\tau_{\leqslant
0}(M)={}^{{\mathrm{ord}}}\tau_{\leqslant 0}(j_{p}^{*}M)$

and

$i_{p}^{*}\hskip 2.84526pt{}^{{\mathrm{ord}}}\tau_{\leqslant
0}(M)={}^{{\mathrm{ord}}}\tau_{\leqslant 0}(i_{p}^{*}M).$

Hence, we can assume that either $p$ is invertible on $S$ or that $S$ is of
characteristic $p$.

Proposition 2.3.1 then gives a stratification of $S$ with regular strata such
that for any stratum $i\colon T\rightarrow S$, the motive $i^{*}M$ is a
constructible smooth Artin motive. Assume that for any stratum $i\colon
T\rightarrow S$, the motive ${}^{{\mathrm{ord}}}\tau_{\leqslant 0}(i^{*}M)$ is
$\mathfrak{p}$-constructible. Using the localization triangle and the
stability of $\mathfrak{p}$-constructible objects under the functors $f^{*}$
and $f_{!}$ for $f$ quasi-finite, we can prove by induction on the number of
strata that the motive ${}^{{\mathrm{ord}}}\tau_{\leqslant 0}(M)$ is
constructible. Hence, we can assume that the scheme $S$ is regular and that
the motive $M$ is a constructible smooth Artin motive.

If $p$ is invertible on $S$, the result follows from Lemma 4.2.8 below.

Assume that $S$ is of characteristic $p$. It suffices to show that the Artin
motive ${}^{{\mathrm{ord}}}\tau_{\leqslant 0}(M)$ is a constructible smooth
Artin motive. Using [CD16, A.3.4], we can assume that $p$ is invertible in
$R$. The result then follows from Proposition 4.2.1. ∎

######  Lemma 4.2.7.

Let $S$ be a scheme and let $R$ be a ring. Let $X$ be finite over $S$. Then,
the motive $h_{S}(X)$ lies in the heart of the ordinary homotopy t-structure.

###### Proof.

If $f$ is finite, the functor $f_{*}$ is $t_{\mathrm{ord}}$-exact by Corollary
4.1.3 and therefore, we may assume that $X=S$. By definition, we have

$\mathbbm{1}_{S}\leqslant_{\mathrm{ord}}0.$

Let $U$ be étale over $S$, and let $n$ be an integer, by definition, we have

$\pi_{n}\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M_{S}(U),\mathbbm{1}_{S})=H^{n,0}_{\mathcal{M},\operatorname{\acute{e}t}}(U,R).$

But if the integer $n$ is negative, this group vanishes by Lemma 1.7.1 and the
result follows. ∎

######  Lemma 4.2.8.

Let $R$ be a regular good ring, let $S$ be a regular scheme, let
$\mathfrak{p}$ be a maximal ideal of $\mathbb{Z}$ and let $p$ be a generator
of $\mathfrak{p}$. Assume that $p$ is invertible in $S$.

  1. (1)

The functor $\rho_{!}$ of Proposition 1.4.3 induces monoidal equivalences

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}\longrightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)_{\mathfrak{p}}\longrightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}.$

  2. (2)

The functor $\rho_{!}$ of Proposition 1.4.3 induces a t-exact fully faithful
functor

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

when the left hand side is endowed with the t-structure obtained by applying
Proposition 1.1.10 to the ordinary t-structure and the right hand side is
endowed with the t-structure obtained by applying Proposition 1.1.10 to the
ordinary homotopy t-structure.

###### Proof.

The category
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
is generated as a localizing subcategory of itself by the images of the
$M_{S}(X)$ when $X$ runs through the finite étale schemes over $S$. But we
have

$M_{S}(X)=\rho_{!}R_{S}(X).$

Since the functor

$(\rho_{!})_{\mathfrak{p}}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}\longrightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$

is compatible with colimits by Proposition 1.1.10, it is therefore essentially
surjective if we assume it to be fully faithful.

To show that the functor $(\rho_{!})_{\mathfrak{p}}$ is fully faithful, it
suffices to show that the canonical $2$-morphism
$\operatorname{Id}\rightarrow\rho_{\mathfrak{p}}^{!}(\rho_{!})_{\mathfrak{p}}$
is an equivalence.

Therefore, it suffices to show that for any object $M$ of the stable category
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}$
and for any étale $S$-scheme $X$, the map

$\rho_{X,M}\colon\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)_{\mathfrak{p}}}(R_{S}(X),M)\rightarrow\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)_{\mathfrak{p}}}(R_{S}(X),\rho_{\mathfrak{p}}^{!}(\rho_{!})_{\mathfrak{p}}M)$

is an equivalence.

Let $X$ be an étale $S$-scheme. Any object $M$ of
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}$
is a retract of an object of the naive $\mathfrak{p}$-localization
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}^{\mathrm{na}}$.
To show that the map $\rho_{X,M}$ is an equivalence, we can therefore assume
that $M$ belongs to the essential image of the canonical functor

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)\rightarrow\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}.$

In this case, the map $\rho_{X,M}$ identifies with the map

$\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}(R_{S}(X),M)\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}}\rightarrow\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}(R_{S}(X),\rho^{!}\rho_{!}M)\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}}.$

which by Proposition 1.2.3 identifies with the map

$\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}})}((R\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}})_{S}(X),M\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}})\rightarrow\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}})}((R\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}})_{S}(X),\rho^{!}\rho_{!}M\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}}).$

which is an equivalence by Theorem 2.1.1.

Hence, the map

$(\rho_{!})_{\mathfrak{p}}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}\longrightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$

is an equivalence. To show that the induced map

$(\rho_{!})_{\mathfrak{p}}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)_{\mathfrak{p}}\longrightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$

is also an equivalence, notice that
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)_{\mathfrak{p}}$
is the thick subcategory of
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}$
generated by the $M_{S}(X)$ with $X$ finite étale over $S$ and that
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)_{\mathfrak{p}}$
is the thick subcategory of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)_{\mathfrak{p}}$
generated by the $R_{S}(X)$ with $X$ finite étale over $S$ by Proposition
1.5.14. This proves the first assertion.

We now prove the second assertion. Recall that by Proposition 1.3.5, the
ordinary t-structure on
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ is
generated by the sheaves of the form $R_{S}(X)$ with $X$ étale over $S$ and
that we proved that therefore, the exact functor

$\rho_{!}\colon\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$

is right t-exact. Thus, the functor

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

is right t-exact by Proposition 1.1.11.

The t-structure on
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)_{\mathfrak{p}}$
obtained by applying Proposition 1.1.10 to the ordinary t-structure is still
generated by the sheaves of the form $R_{S}(X)$ with $X$ étale over $S$ since
by definition the t-non-positive objects are the retracts of the objects of
the essential image in the naive $\mathfrak{p}$-localization of the t-non-
positive objects of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$.

Similarly, the t-structure on
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)_{\mathfrak{p}}$
obtained by applying Proposition 1.1.10 to the ordinary t-structure is
generated by the motives of the form $M_{S}(X)$ with $X$ étale over $S$.

Let $N$ be an object of
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)_{\mathfrak{p}}$
and let $X$ be an étale $S$-scheme. The first assertion ensures that

$\operatorname{Map}_{\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)_{\mathfrak{p}}}(M_{S}(X),(\rho_{!})_{\mathfrak{p}}N)=\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}(R_{S}(X),N).$

Hence, if the sheaf $N$ is t-non-negative, so is the motive
$(\rho_{!})_{\mathfrak{p}}N$. Therefore, the functor

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

is t-exact. ∎

######  Definition 4.2.9.

Let $S$ be a scheme allowing resolution of singularities by alterations and
let $R$ be a regular good ring. The _abelian category of ordinary Artin
motives_ $\mathrm{M}^{A}_{{\mathrm{ord}}}(S,R)$ is the heart of the ordinary
homotopy t-structure of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(S,R)$.

######  Corollary 4.2.10.

Let $S$ be a scheme allowing resolution of singularities by alterations and
let $R$ be a regular good ring. The following properties hold.

  1. (1)

Let $f\colon T\rightarrow S$ be a morphism of schemes. Then, the functor
$f^{*}$ induces an exact functor

$f^{*}\colon\mathrm{M}^{A}_{{\mathrm{ord}}}(S,R)\rightarrow\mathrm{M}^{A}_{{\mathrm{ord}}}(T,R).$

  2. (2)

If $x$ is a point of $S$, denote by $i_{x}\colon\\{x\\}\rightarrow S$ the
inclusion and by $p(x)$ the characteristic exponent of $k(x)$. The family

$\left(\alpha^{!}i^{*}_{x}:\mathrm{M}^{A}_{{\mathrm{ord}}}(S,R)\rightarrow\operatorname{Rep}^{A}(G_{k(x)},R[1/p(x)])\mid
x\in S\right)$

is a conservative family of exact functors.

  3. (3)

Assume that the ring $R$ is 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 induces an exact and conservative functor

$\overline{\rho}_{v}\colon\mathrm{M}^{A}_{{\mathrm{ord}}}(S,R)\rightarrow\operatorname{Sh}^{A}(S[1/\ell],R_{v}).$

##  5\. Perverse homotopy $\mathrm{t}$-structure on Artin étale motives

From now on, all schemes are assumed to be endowed with a dimension function
$\delta$ (see Definition 1.3.11). If $S$ is a scheme and if $\Lambda$ is an
$\ell$-adic field or its ring of integers, we denote with the small letters
$p$ the notions related to the perverse t-structure on
$\mathcal{D}^{b}_{c}(S,\Lambda)$. For instance, we denote by
${}^{p}\tau_{\leqslant 0}$ the left truncation with respect to the perverse
t-structure and $t_{p}$ denotes the perverse t-structure itself.

One of the main goals of this paper is to construct the perverse motivic
t-structure on constructible Artin étale motives in the following sense

######  Definition 5.0.1.

Let $S$ be a scheme and $R$ be a localization of the ring of integers of a
number field $K$. Let $t_{0}$ be a t-structure on
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(S,R)$. Recall
the reduced $v$-adic realization functor (see Definition 2.2.1). We say that

  1. (1)

the t-structure $t_{0}$ is the _perverse motivic t-structure_ if for any
constructible Artin étale motive $M$, the motive $M$ is $t_{0}$-non-positive
if and only if for all non-archimedian valuation $v$ on $K$ which is non-
negative on $R$, the complex $\overline{\rho}_{v}(M)$ is $t_{p}$-non-positive
(see Section 1.6.3).

  2. (2)

the t-structure $t_{0}$ is the _perverse motivic t-structure in the strong
sense_ if for any non-archimedian valuation $v$ on $K$ which is non-negative
on $R$, the functor

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

is t-exact when the left hand side is endowed with the t-structure $t_{0}$ and
the right hand side is endowed with the perverse t-structure.

######  Remark 5.0.2.

  1. (1)

If the perverse motivic t-structure exists, then it is unique.

  2. (2)

If $t$ is the perverse motivic t-structure in the strong sense, Proposition
2.2.2 and Lemma 1.1.8 ensure that it is the perverse motivic t-structure.

The most suitable candidate to be the perverse motivic t-structure will be the
_perverse homotopy t-structure_ (see Proposition 5.2.1 below) which we will
construct and study in this section.

###  5.1. Definition for non-constructible Artin motives

######  Definition 5.1.1.

Let $S$ be a scheme and let $R$ be a ring. The _perverse homotopy t-structure_
on $\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$ is the
t-structure generated in the sense of Proposition 1.1.6 by the family of the
motives $M_{S}^{\operatorname{BM}}(X)[\delta(X)]$ with $X$ quasi-finite over
$S$.

We denote with the small letters $hp$ the notions related to the perverse
t-structure on
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$. For
instance, we denote by ${}^{hp}\tau_{\leqslant 0}$ the left truncation with
respect to the ordinary homotopy t-structure and $t_{hp}$ denotes the perverse
t-structure itself.

Using the same method as Proposition 1.4.1, this t-structure is also generated
by family of the motives $h_{S}(X)[\delta(X)]$ with $X$ finite over $S$. Thus,
an object $M$ of
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$ belongs to
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)^{hp\geqslant
n}$ if and only if for any finite $S$-scheme $X$ the complex
$\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(h_{S}(X),M)$
is $(n-\delta(X)-1)$-connected.

######  Proposition 5.1.2.

Let $R$ be a ring. Let $f$ be a quasi-finite morphism of schemes and let $g$
be a morphism which is essentially of finite type.

  1. (1)

The adjunction $(f_{!},\omega^{0}f^{!})$ is a $t_{hp}$-adjunction.

  2. (2)

If $M$ is a $t_{hp}$-non-positive Artin motive, then, the functor
$-\otimes_{S}M$ is right $t_{hp}$-exact.

  3. (3)

If $\dim(g)\geqslant d$, then, the adjunction $(g^{*}[d],\omega^{0}g_{*}[d])$
is a $t_{hp}$-adjunction.

###### Proof.

The proof is the same as the proof of Proposition 4.1.2 ∎

######  Corollary 5.1.3.

Let $R$ be a ring and let $f$ be a morphism of schemes. Then,

  1. (1)

If $f$ is étale, the functor $f^{*}=f^{!}=\omega^{0}f^{!}$ is $t_{hp}$-exact.

  2. (2)

If $f$ is finite, the functor $f_{!}=f_{*}=\omega^{0}f_{*}$ is $t_{hp}$-exact.

######  Proposition 5.1.4.

Let $S$ be a scheme, let $R$ be a ring, let $i\colon F\rightarrow S$ be a
closed immersion and let $j\colon U\rightarrow S$ be the open complement.
Then, the perverse homotopy t-structure on
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$ is obtained by gluing the
t-structures of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(U,\mathbb{Z}_{\ell})$ and
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(F,R)$ (see [BBDG18, 1.4.9]) i.e.
for any object $M$ of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$, we
have

  1. (1)

$M\geqslant_{hp}0$ if and only if $j^{*}M\geqslant_{hp}0$ and
$\omega^{0}i^{!}M\geqslant_{hp}0$.

  2. (2)

$M\leqslant_{hp}0$ if and only if $j^{*}M\leqslant_{hp}0$ and
$i^{*}M\leqslant_{hp}0$.

###### Proof.

This follows from Propositions 4.1.4, 5.1.2 and 5.1.3. ∎

######  Corollary 5.1.5.

Let $S$ be a scheme, let $R$ be a ring and let $M$ be an object of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$. Then, we have

  1. (1)

$M\geqslant_{hp}0$ if and only if there is a stratification of $S$ such that
for any stratum $i\colon T\hookrightarrow S$, we have
$\omega^{0}i^{!}M\geqslant_{hp}0$.

  2. (2)

$M\leqslant_{hp}0$ if and only if there is a stratification of $S$ such that
for any stratum $i\colon T\hookrightarrow S$, we have $i^{*}M\leqslant_{hp}0$.

######  Proposition 5.1.6.

Let $S$ be a scheme and let $R$ be a ring. Then, the constructible objects are
bounded with respect to the perverse homotopy t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)$.

###### Proof.

The subcategory of bounded objects is thick. Therefore, it suffices to show
that the motives of the form $f_{*}\mathbbm{1}_{X}$ with $f\colon X\rightarrow
S$ finite are bounded. By Corollary 5.1.3, the functor $f_{*}$ is
$t_{hp}$-exact when $f$ is finite. Therefore, it suffices to show that the
object $\mathbbm{1}_{S}$ is bounded with respect to $t_{hp}$ for any scheme
$S$. We already know that it lies in degree at most $\delta(S)$ by definition
of $t_{hp}$.

Let $Y$ be a finite $S$-scheme. The objects $h_{S}(Y)$ and $\mathbbm{1}_{S}$
belong to the heart of the ordinary homotopy t-structure by Lemma 4.2.7.
Therefore, the complex $\operatorname{Map}(h_{S}(Y),\mathbbm{1}_{S})$ is
$(-1)$-connected. Therefore, if $c$ is a lower bound for $\delta$, the complex
$\operatorname{Map}(h_{S}(X),\mathbbm{1}_{S}[c])$ is $(\delta(X)-1)$-connected
and therefore, the object $\mathbbm{1}_{S}$ is bounded below by $c$. ∎

######  Proposition 5.1.7.

Let $S$ be a scheme, let $R$ be a ring, let $n$ be an integer and let $A$ be a
localization of $\mathbb{Z}$. Denote by $\sigma_{n}\colon R\rightarrow R/nR$
and by $\sigma_{A}\colon R\rightarrow R\otimes_{\mathbb{Z}}A$ the natural ring
morphisms. Recall the notations of Section 1.2.7. Then,

  1. (1)

The functors $(\sigma_{A})_{*}$ and $(\sigma_{n})_{*}$ are $t_{hp}$-exact.

  2. (2)

the functor $\sigma_{A}^{*}$ is $t_{hp}$-exact.

###### Proof.

The proof is the same as the proof of Proposition 4.1.6. ∎

We now describe the perverse homotopy t-structure in the case when the base
scheme is the spectrum of a field.

######  Proposition 5.1.8.

Let $k$ be a field of characteristic exponent $p$ and let $R$ be a regular
goood ring. Then, the following properties hold.

  1. (1)

The ordinary homotopy and the perverse homotopy t-structure on the stable
category $\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(k,R)$
coincide up to a shift of $\delta(k)$.

  2. (2)

The perverse homotopy t-structure induces a t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(k,R)$.

  3. (3)

If $R$ is good, then, the functors

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

of Corollary 2.1.7 are t-exact equivalences when the right hand sides are
endowed with the perverse homotopy t-structure and the left hand sides are
endowed with the ordinary t-structure shifted by $\delta(k)$.

  4. (4)

If $R$ is a localization of the ring of integers of a number field $K$ and $v$
is a non-archimedian valuation on $K$, the $v$-adic realization functor

$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(k,R)\rightarrow\mathcal{D}^{b}_{c}(k,R_{v})$

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.

Notice that $t_{\mathrm{ord}}$ and $t_{hp}$ have the same set of generators up
to a shift of $\delta(k)$. This proves the first point. The other points
follow from the first point and from the properties of $t_{\mathrm{ord}}$. ∎

######  Proposition 5.1.9.

Let $S$ be a scheme and let $R$ be a ring. The inclusion functor

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

is t-exact when the left hand side is endowed with the perverse t-structure
(see Definition 1.3.15) and the right hand side is endowed with the perverse
homotopy t-structure.

###### Proof.

If $f\colon X\rightarrow S$ is quasi-finite, Proposition 1.2.1 implies that
the functor $\rho_{!}$ maps the étale sheaf $f_{!}\underline{R}$ to the motive
$M_{S}^{BM}(X)$. Recall that Proposition 1.3.13 asserts that the sheaves of
the form $f_{!}\underline{R}[\delta(X)]$ with $f\colon X\rightarrow S$ quasi-
finite generate the perverse t-structure on the stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$. Since
the exact functor

$\rho_{!}\colon\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)$

is compatible with small colimits, it is therefore right t-exact.

We can prove the left t-exactness using the same proof as the proof of the
left t-exactness in Proposition 4.1.7. We also give a shorter proof which uses
Proposition 5.2.1(1)(a) below.

Let $M$ be a torsion étale motive which is perverse t-non-negative.
Proposition 1.3.17 ensures that $M$ is bounded below (with respect to the
ordinary t-structure) and that for any point $x$ of $S$, we have

$i_{x}^{!}M\geqslant-\delta(x)$

with respect to the ordinary t-structure. Corollary 1.4.10 then ensures that

$i_{x}^{!}M=\omega^{0}i_{x}^{!}M.$

Therefore, using Proposition 4.1.7, we have

$\omega^{0}i_{x}^{!}M\geqslant_{\mathrm{ord}}-\delta(x).$

Proposition 5.2.1(1)(a) below ensures that the Artin motive $M$ is therefore
$t_{hp}$-non-negative. ∎

###  5.2. Locality and consequences

The following proposition summarizes the main features of the perverse
homotopy t-structure. They are analogous to the properties of the perverse
t-structure on $\ell$-adic sheaves.

######  Proposition 5.2.1.

Let $S$ be a scheme and $R$ be a regular good ring. If $x$ is a point of $S$,
denote by $i_{x}\colon\\{x\\}\rightarrow S$ the inclusion. Then, the following
properties hold.

  1. (1)

Let $M$ be an Artin motive over $S$ with coefficients in $R$. Then,

    1. (a)

The Artin motive $M$ is $t_{hp}$-non-negative if and only if it is bounded
below with respect to $t_{\mathrm{ord}}$ and for any point $x$ of $S$, we have

$\omega^{0}i_{x}^{!}M\geqslant_{hp}0,$

i.e. $\omega^{0}i_{x}^{!}M\geqslant_{\mathrm{ord}}-\delta(x)$.

    2. (b)

Assume that the Artin motive $M$ is constructible. Then, it is $t_{hp}$-non-
positive if and only if for any point $x$ of $S$, we have

$i_{x}^{*}M\leqslant_{hp}0,$

i.e. $i_{x}^{*}M\leqslant_{\mathrm{ord}}-\delta(x)$.

  2. (2)

Assume that the scheme $S$ is regular. Then,

    1. (a)

Assume that the regular scheme $S$ is connected. Then, the functor $\rho_{!}$
of Proposition 1.4.3 induces a t-exact functor

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

when the left hand side is endowed with the ordinary t-structure shifted by
$\delta(S)$ and the right hand side is endowed with the perverse homotopy
t-structure.

In particular, when the residue characteristic exponents of $S$ are invertible
in $R$, 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
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)$.

    2. (b)

If $R$ is a localization of the ring of integers of a number field $K$, if the
residue characteristic exponents of $S$ are invertible in $R$, and if $v$ is a
non-archimedian valuation on $K$, the $v$-adic realization functor

$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)\rightarrow\mathcal{D}^{b}_{c}(S,R_{v})$

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.

  3. (3)

Assume that $S$ is excellent and that the ring $R$ a localization of the ring
of integers of a number field $K$. Then, the perverse motivic t-structure
exists if and only if the perverse homotopy t-structure induces a t-structure
on $\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$. In
that case, the perverse homotopy t-structure is the perverse motivic
t-structure.

###### Proof.

Notice that Assertion (2)(b) follows from Assertion (2)(a) and Theorem 4.2.6.
We now prove the other assertions.

Proof of Assertion (1)(a):

If $M\geqslant_{hp}0$, we already know from Proposition 5.1.2 that for any
point $x$ of $S$, we have $\omega^{0}i_{x}^{!}M\geqslant_{hp}0$. Furthermore,
if $c$ is a lower bound for the function $\delta$ on $S$, the complex
$\operatorname{Map}(M_{S}(X),M)$ is $(c-1)$-connected for any étale $S$-scheme
$X$. Therefore, the Artin motive $M$ is bounded below with respect to the
ordinary homotopy t-structure.

Conversely, assume that $M$ is $t_{\mathrm{ord}}$-bounded below and that for
any point $x$ of $S$, the Artin motive $\omega^{0}i_{x}^{!}M$ is $t_{hp}$-non-
negative. Let $f\colon X\rightarrow S$ be quasi-finite. As in the proof of
Proposition 1.3.13, the $\delta$-niveau spectral sequence of [BD17, 3.1.5] and
the strong continuity property of Proposition 1.3.10 yields

$E^{1}_{p,q}\Rightarrow\pi_{-p-q}\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M^{\operatorname{BM}}_{S}(X),M)$

with

$E^{1}_{p,q}=\bigoplus_{y\in X,\hskip
2.84526pt\delta(y)=p}\pi_{-p-q}\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(k(f(y)),R)}(M_{f(y)}^{\operatorname{BM}}(y),\omega^{0}i_{f(y)}^{!}M).$

Since for any point $x$ of $S$, we have $\omega^{0}i_{x}^{!}M\geqslant_{hp}0$,
the complex

$\operatorname{Map}(M_{f(y)}^{\operatorname{BM}}(y),\omega^{0}i_{f(y)}^{!}M)$

is $(-\delta(X)-1)$-connected for any point $y$ of $X$ and thus, if
$n>\delta(X),$ the $R$-module
$\pi_{-n}\operatorname{Map}_{\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)}(M^{\operatorname{BM}}_{S}(X),M)$
vanishes. Therefore, the Artin motive $M$ is $t_{hp}$-non-negative.

Proof of Assertion (2)(a):

Proposition 1.5.16 implies that the functor

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

is right t-exact when the left hand side is endowed with the ordinary
t-structure shifted by $-\delta(S)$ and the right hand side is endowed with
the perverse homotopy t-structure. Since the scheme $S$ is regular and
connected $\delta(x)=-\operatorname{codim}_{S}(x)$ is a dimension function on
$S$. In particular, we can assume that $\delta(S)=0$.

By dévissage, it therefore suffices to show that any étale sheaf which belongs
to $\operatorname{Loc}_{S}(R)$ is send through the functor $\rho_{!}$ to a
$t_{hp}$-non-negative object. Let $M$ be such an étale sheaf. We have an exact
triangle

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

The sheaf $M\otimes_{\mathbb{Z}}\mathbb{Q}$ lies in
$\operatorname{Loc}_{S}(R\otimes_{\mathbb{Z}}\mathbb{Q})$.

Furthermore, if $n$ is a positive integer, the sheaf
$M\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}[-1]$ is t-non-negative with
respect to the ordinary t-structure. Let $x$ be a point of $S$. Proposition
1.2.7 yields

$i_{x}^{!}(M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1])=\operatorname{colim}_{n}i_{x}^{!}(M\otimes\mathbb{Z}/n\mathbb{Z}[-1]).$

Therefore, since the scheme $S$ is regular, and since the sheaf
$M\otimes\mathbb{Z}/n\mathbb{Z}[-1]$ is lisse, we get by absolute purity
[LYF14, XVI.3.1.1] that

$i_{x}^{!}(M\otimes\mathbb{Z}/n\mathbb{Z}[-1])=i_{x}^{*}(M\otimes\mathbb{Z}/n\mathbb{Z}[-1])(\delta(x))[2\delta(x)].$

Therefore, we get

$i_{x}^{!}(M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1])=i_{x}^{*}(M\otimes\mathbb{Q}/\mathbb{Z}[-1])(\delta(x))[2\delta(x)].$

Finally, we get

$\omega^{0}i_{x}^{!}\rho_{!}(M\otimes\mathbb{Q}/\mathbb{Z}[-1])=\rho_{!}\left[i_{x}^{*}(M\otimes\mathbb{Q}/\mathbb{Z}[-1])(\delta(x))\right][2\delta(x)]$

where the left hand side in degree $-2\delta(x)$ or more. It is therefore in
degree $-\delta(x)$ or more and therefore, the motive
$\rho_{!}(M\otimes\mathbb{Q}/\mathbb{Z}[-1])$ is t-non-negative.

Hence, the torsion étale sheaf
$M\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[-1]$ is perverse t-non-negative
and therefore it is perverse homotopy t-non-negative by Proposition 5.1.9.

We can therefore assume that the ring $R$ is a $\mathbb{Q}$-algebra. In that
case, Corollary 3.1.8 ensures that $\omega^{0}i_{x}^{!}(M)$ vanishes when $x$
is not the generic point of $S$. When $x$ is the generic point of $S$, we get

$\omega^{0}i_{x}^{!}(M)=i_{x}^{*}(M).$

Therefore, the sheaf $\omega^{0}i_{x}^{!}(M)$ is in degree $0=-\delta(x)$.
Hence, Assertion (1)(a) ensures that $M$ is perverse homotopy t-non-negative

Proof of Assertion (1)(b):

If $M\leqslant_{hp}0$, we already know from Proposition 5.1.2 that for any
point $x$ of $S$, we have $i_{x}^{*}M\leqslant_{hp}0$.

We now prove the converse. Corollary 1.2.6 and Proposition 5.1.7 ensure that
the family $(-\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)})_{p\text{ prime}}$ is
conservative family which is made of $t_{hp}$-exact functors. Using Lemma
1.1.8, we can therefore assume that the ring $R$ is a
$\mathbb{Z}_{(p)}$-algebra where $p$ is a prime number.

Write

$S=S[1/p]\sqcup S_{p}$

where $S_{p}=S\times_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$. The subscheme
$S[1/p]$ of $S$ is open while the subscheme $S_{p}$ is closed.

Let $M$ be a constructible Artin motive and such that for all point $x$ of
$S$, we have

$i_{x}^{*}M\leqslant_{hp}0.$

Corollary 2.3.4 ensures that we have a stratification $\mathcal{S}[1/p]$ of
$S[1/p]$ with regular strata and a stratification $\mathcal{S}_{p}$ of $S_{p}$
with regular strata, lisse étale sheaves
$M_{T}\in\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(T,R[1/p])$
for $T\in\mathcal{S}_{p}$ and lisse étale sheaves
$M_{T}\in\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(T,R)$ for
$T\in\mathcal{S}[1/p]$ such that for all any stratum $T$, we have

$M|_{T}=\rho_{!}M_{T}.$

Lemma 5.2.2 below then ensures that $M|_{T}\leqslant_{hp}0$ for each
$T\in\mathcal{S}$. Therefore, using Corollary 5.1.5, we get

$M\leqslant_{hp}0.$

Proof of Assertion (3):

Let $M$ be in
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$. Then,

$\displaystyle M\leqslant_{hp}0$ $\displaystyle\Longleftrightarrow\forall x\in
S,i_{x}^{*}M\leqslant_{hp}0$ $\displaystyle\Longleftrightarrow\forall x\in
S,\forall v\text{ valuation }i_{x}^{*}\overline{\rho}_{v}(M)\leqslant_{p}0$
$\displaystyle\Longleftrightarrow\forall v\text{ valuation
}\overline{\rho}_{v}(M)\leqslant_{p}0.$

Indeed the first equivalence follows from Assertion (1)(b); the second
equivalence follows from Proposition 5.1.8 and the fact that the $v$-adic
realization respects the six functors formalism; finally the third equivalence
is a characterization of the perverse t-structure (see [BBDG18, 2.2.12]).

Hence, if the perverse homotopy t-structure induces a t-structure on
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$, it is
the perverse motivic t-structure.

Conversely, assume that the perverse motivic t-structure $t_{pm}$ exists.
Then, its t-non-positive objects are the $t_{hp}$-non-negative objects which
are constructible. Therefore, the $t_{pm}$-positive objects are those
constructible objects $N$ such that for all constructible object $M$ with
$M\leqslant_{hp}0$, we have

$\operatorname{Hom}_{\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)}(M,N)=0.$

But the generators of the $t_{hp}$ are constructible and thus, those objects
are exactly the $t_{hp}$-positive objects which are constructible. ∎

######  Lemma 5.2.2.

Let $S$ be a regular connected scheme and $R$ be a regular ring and let
$\eta\colon\operatorname{Spec}(K)\rightarrow S$ be the generic point of $S$.
Then, the functor

$\eta^{*}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)\rightarrow\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(K,R)$

is conservative and t-exact with respect to the ordinary t-structure.

###### Proof.

The functor $\eta^{*}$ is t-exact.

Assume that $f\colon C\rightarrow D$ is a map in
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$ such that
$\eta^{*}(f)$ is invertible. Then, for any integer $n$, the map
$H^{n}(\eta^{*}(f))$ is invertible. Since the functor $\eta^{*}$ is t-exact,
the map $\eta^{*}H^{n}(f)$ is invertible.

Let $\xi$ be a geometric point with image $\eta$. By [GR71, V.8.2], the
canonical map $G_{K}\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$ is
surjective. Thus, the map

$\mathrm{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi),R)\rightarrow\mathrm{Rep}^{A}(G_{K},R)$

induced by $\eta^{*}$ is conservative. Therefore, for any integer $n$, the map
$H^{n}(f)$ is invertible. Thus, the map $f$ is invertible. ∎

######  Remark 5.2.3.

Let $S$ be a regular scheme and let $R$ be a regular good ring. Assume that
the residue characteristic exponents of $S$ are invertible in $R$. Then, the
perverse homotopy t-structure of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$ induces a t-structure on
$\mathcal{DM}^{smA}_{\operatorname{\acute{e}t},c}(S,R)$. If $T$ is a connected
component of $S$, denote by $i_{T}\colon T\rightarrow S$ the clopen immersion.
The heart of this t-structure is then the full subcategory made of those
objects of the form

$\bigoplus\limits_{T\in\pi_{0}(S)}(i_{T})_{*}\rho_{!}L_{T}[\delta(T)]$

where $L_{T}$ is an object of $\operatorname{Loc}_{T}(R)$ for any connected
component $T$ of $S$.

######  Proposition 5.2.4.

Let $S$ be a regular scheme and let $R$ be a regular good ring. Then, the
Artin motive $\mathbbm{1}_{S}$ is in degree $\delta(S)$ with respect to the
perverse homotopy t-structure.

###### Proof.

Since $S$ is regular, the function $\delta=-\operatorname{codim}_{S}$ is a
dimension function on $S$ (for which $\delta(S)=0$). Without loss of
generality, we may work with this dimension function. We already know that
$\mathbbm{1}_{S}\leqslant_{hp}0$ since it is a generator of the t-structure.

Let $x$ be a point of $S$. Then, by the absolute purity property [CD16,
5.2.7], we have

$\omega^{0}i_{x}^{!}\mathbbm{1}_{S}=\omega^{0}(\mathbbm{1}_{x}(\delta(x)))[2\delta(x)].$

If $\delta(x)=0$, the point $x$ is a generic point of $S$. Therefore, we have

$\omega^{0}i_{x}^{!}\mathbbm{1}_{S}=\mathbbm{1}_{x}$

and therefore, the Artin motive $\omega^{0}i_{x}^{!}\mathbbm{1}_{S}$ in degree
$\delta(x)=0$.

Assume that $\delta(x)<0$. Using Theorem 3.1.4, the Artin motive
$\omega^{0}(\mathbbm{1}_{x}(\delta(x)))$ is in degree $1$ for the ordinary
t-structure. Thus, the Artin motive $\omega^{0}i_{x}^{!}\mathbbm{1}_{S}$ is in
degree $1-2\delta(x)\geqslant-\delta(x)$. ∎

######  Definition 5.2.5.

Let $S$ be a regular scheme and let $R$ be a regular good ring. Assume that
the residue characteristic exponents of $S$ are invertible in $R$. The
_abelian category of perverse smooth Artin motives_
$\mathrm{M}^{smA}_{\mathrm{perv}}(S,R)$ is the heart of the perverse homotopy
t-structure on the stable category
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$.

###  5.3. Induced t-structure on constructible motives with rational
coefficients and weak Lefschetz properties

######  Proposition 5.3.1.

Let $S$ be an excellent scheme and let $K$ be a number field.

  1. (1)

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

  2. (2)

Let $i\colon F\rightarrow S$ be a closed immersion and let $j\colon
U\rightarrow S$ be the open complement. The sequence

$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(F,K)\overset{i_{*}}{\rightarrow}\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(S,K)\overset{j^{*}}{\rightarrow}\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(U,K)$

satisfies the axioms of the gluing formalism (see [BBDG18, 1.4.3] and
Proposition 5.1.4) and the perverse homotopy t-structure on
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(S,K)$ is obtained by gluing
the t-structures of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(U,K)$ and
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(F,K)$, (see [BBDG18, 1.4.9]).

  3. (3)

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

$\rho_{v}\colon\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,K)\rightarrow\mathcal{D}^{A}(S,K_{v})$

is t-exact and conservative when the right hand side is endowed with the
perverse homotopy t-structure.

###### Proof.

We can assume that $S$ is reduced. The first part of (2) is a direct
consequence of Proposition 5.1.4 and Proposition 1.4.7. To prove the rest of
the proposition, we need to adapt the ideas of [BBDG18, 2.2.10] to our
setting. This technique is very similar to [Rui22, 4.7.1].

We first need a few definitions:

  * •

If $\mathcal{S}$ is a stratification of $S$, we say that a locally closed
subscheme $X$ of $S$ is an $\mathcal{S}$-subscheme of $S$ if $X$ is a union of
strata of $\mathcal{S}$.

  * •

A pair $(\mathcal{S},\mathcal{L})$ is _admissible_ if

    * –

$\mathcal{S}$ is a stratification of $S$ with regular strata everywhere of the
same dimension.

    * –

for every stratum $T$ of $\mathcal{S}$, $\mathcal{L}(T)$ is a finite set of
isomorphism classes of objects of $\mathrm{M}^{smA}_{\mathrm{perv}}(S,K)$ (see
Definition 5.2.5).

  * •

If $(\mathcal{S},\mathcal{L})$ is an admissible pair and $X$ is an
$\mathcal{S}$-subscheme of $S$, the category
$\mathcal{DM}_{\mathcal{S},\mathcal{L}}(X,K)$ of _$(\mathcal{S},\mathcal{L})$
-constructible Artin motives_ over $X$ is the full subcategory of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(X,K)$ made of those objects $M$
such that for any stratum $T$ of $\mathcal{S}$ contained in $X$, the
restriction of $M$ to $T$ is a smooth Artin motive and its perverse cohomology
sheaves are successive extensions of objects whose isomorphism classes lie in
$\mathcal{L}(T)$.

  * •

A pair $(\mathcal{S}^{\prime},\mathcal{L}^{\prime})$ refines a pair
$(\mathcal{S},\mathcal{L})$ if every stratum $S$ of $\mathcal{S}$ is a union
of strata of $\mathcal{S}^{\prime}$ and any perverse smooth Artin motive $M$
over a stratum $T$ of $\mathcal{S}$ whose isomorphism class lies in
$\mathcal{L}(T)$ is
$(\mathcal{S}^{\prime},\mathcal{L}^{\prime})$-constructible.

If $(\mathcal{S},\mathcal{L})$ is an admissible pair and if $X$ is an
$\mathcal{S}$-subscheme of $S$, the category
$\mathcal{DM}_{\mathcal{S},\mathcal{L}}(X,K)$ is a stable subcategory of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,K)$ (i.e. it is closed under
finite (co)limits).

If $(\mathcal{S},\mathcal{L})$ is an admissible pair and $i\colon U\rightarrow
V$ is an immersion between $\mathcal{S}$-subschemes of $S$, the functors
$i_{!}$ and $i^{*}$ preserve $(\mathcal{S},\mathcal{L})$-constructible
objects.

Now, we say that an admissible pair $(\mathcal{S},\mathcal{L})$ is
_superadmissible_ if letting $i\colon T\rightarrow S$ be the immersion of a
stratum $T$ of $\mathcal{S}$ into $S$ and letting $M$ be a perverse smooth
Artin motive over $T$ whose isomorphism class lies in $\mathcal{L}(T)$, the
complex $\omega^{0}i_{*}M$ is $(\mathcal{S},\mathcal{L})$-constructible.

We now claim that an admissible pair $(\mathcal{S},\mathcal{L})$ can always be
refined into a superadmissible one.

Assume that this is true when we replace $S$ with the union of the strata of
dimension $n$ or more. Let $i:T\rightarrow S$ and
$i^{\prime}:T^{\prime}\rightarrow S$ be immersions of strata of $\mathcal{S}$.
Let $M$ be a perverse smooth Artin motive on $T$ whose isomorphism class lies
in $\mathcal{L}(T)$. If $T$ and $T^{\prime}$ are of dimension at least $n$,
then, the perverse cohomology sheaves of $(i^{\prime})^{*}\omega^{0}i_{*}M$
are obtained as successive extensions of objects of whose isomorphism classes
lie in $\mathcal{L}(T^{\prime})$ by induction.

If $T=T^{\prime}$, then, by Proposition 1.4.4, we have

$(i^{\prime})^{*}\omega^{0}i_{*}M=\omega^{0}i^{*}\omega^{0}i_{*}M=\omega^{0}i^{*}i_{*}M=M$

whose isomorphism class lies in $\mathcal{L}(T)$.

Otherwise, if $T$ is of dimension $n-1$, we can always replace $T$ with an
open subset, so that the closure of $T$ is disjoint from $T^{\prime}$. Then,
by Proposition 1.4.4, we have

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

Assume now that $T$ is of dimension $n$ or more and that $T^{\prime}$ is of
dimension $n-1$. We can replace $T^{\prime}$ with an open subset so that for
any $M$ in $\mathcal{L}(T)$, the motive $(i^{\prime})^{*}\omega^{0}i_{*}M$ is
smooth Artin by Propositions 1.4.7 and 2.3.1. We can also add all the
isomorphism classes of the
${}^{hp}H^{n}\left((i^{\prime})^{*}\omega^{0}i_{*}M\right)$ to
$\mathcal{L}(T^{\prime})$ as there are finitely many of them.

Hence, any admissible pair can be refined into a superadmissible one.

If $(\mathcal{S},\mathcal{L})$ is a superadmissible pair and if $i\colon
U\rightarrow V$ is an immersion between $\mathcal{S}$-subschemes of $S$, we
claim that the functors $\omega^{0}i_{*}$ and $\omega^{0}i^{!}$ preserve
$(\mathcal{S},\mathcal{L})$-constructibility. The proof is the same as
[BBDG18, 2.1.13] using as above the commutation of $\omega^{0}$ with the six
functors described in Proposition 1.4.4.

Hence, if $(\mathcal{S},\mathcal{L})$ is a superadmissible pair, we can define
a t-structure on $(\mathcal{S},\mathcal{L})$-constructible objects by gluing
the perverse homotopy t-structures on the strata. By Proposition 5.2.1, the
positive (resp. negative) objects of this t-structure are the positive (resp.
negative) objects of the perverse homotopy t-structure that are
$(\mathcal{S},\mathcal{L})$-constructible.

Thus, the perverse homotopy t-structure induces a t-structure on
$(\mathcal{S},\mathcal{L})$-constructible objects. Therefore, the subcategory
of $(\mathcal{S},\mathcal{L})$-constructible objects is stable under the
truncation functors of the perverse homotopy t-structure.

Any object $M$ of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,K)$ is
$(\mathcal{S},\mathcal{L})$-constructible for some superadmissible pair
$(\mathcal{S},\mathcal{L})$. Indeed, by Proposition 2.3.1, there is an
admissible pair $(\mathcal{S},\mathcal{L})$ such that $M$ is
$(\mathcal{S},\mathcal{L})$-constructible. Such a pair is can always be
refined into one that is superadmissible.

Hence, $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,K)$ is stable under the
truncation functors of the perverse homotopy t-structure. Therefore, the
perverse homotopy t-structure induces a t-structure on
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,K)$.

Now, the second part (2) follows from Proposition 5.1.4. Finally the third
assertion follows from the construction of the perverse homotopy t-structure
on constructible objects that we used to prove the first assertion, from the
same construction in the $\ell$-adic setting [Rui22, 4.7.1] and from the fact
that the functor $\omega^{0}$ commutes with the $\ell$-adic realization
functor by [Rui22, 4.6.3]. ∎

######  Remark 5.3.2.

Another possible proof of the first assertion is to show that the perverse
homotopy t-structure can be constructed on constructible Artin motives by
using Vaish’s formalism of punctual gluing [Vai19, 3].

Using the same proof as Proposition 4.2.4, replacing Proposition 4.1.6 with
Proposition 5.1.7 yields

######  Corollary 5.3.3.

Let $S$ be an excellent scheme and let $R$ be a ring. Then, the perverse
homotopy t-structure on the stable category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(S,R)$ induces a
t-structure on the stable subcategory
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},\mathbb{Q}-c}^{A}(S,R)$.

We now prove a weak version of the Affine Lefschetz Theorem in our setting. We
begin with the case of schemes of dimension at most $1$.

######  Proposition 5.3.4.

Let $S$ be an excellent scheme, let $f\colon U\rightarrow S$ be a quasi-finite
morphism and let $K$ be a number field. Assume that $\dim(S)\leqslant 1$,
then, the functor

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

is $t_{hp}$-exact.

###### Proof.

Corollary 5.1.3 implies that when $g$ is a finite morphism, the functor
$g_{!}$ is perverse homotopy t-exact . When the scheme $S$ is $0$-dimensional,
the morphism $f$ is finite and the result holds.

Assume that the scheme $S$ is $1$-dimensional. Using Zariski’s Main Theorem
[Gro67, 18.12.13], we have $f=g\circ j$ with $g$ finite and $j\colon
U\rightarrow X$ an open immersion. Therefore it suffices to show that the
functor

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

is $t_{hp}$-exact.

We already know by Proposition 5.1.2 that $j_{!}$ is right t-exact.

Let $M$ be a constructible Artin motive over $U$. Assume that $M$
$t_{hp}$-non-negative. We want to show that the Artin motive $j_{!}(M)$ is
$t_{hp}$-non-negative.

Let $x$ be a point of $X$. Using Proposition 5.2.1, we can replace $X$ with a
neighborhood of $x$. Since the scheme $X$ is $1$-dimensional, we can therefore
assume that the complement of $U$ is the closed subscheme $\\{x\\}$ and
replace $U$ with any neighborhood of the generic points. In particular, we can
assume that $U$ is nil-regular and that $M$ is smooth Artin using the
continuity property of Proposition 1.2.8.

Let $i\colon\\{x\\}\rightarrow X$ be the canonical immersion. Applying the
localization triangle (0.0.1) to the Artin motive $\omega^{0}j_{*}(M)$ yields
an exact triangle

$j_{!}(M)\rightarrow\omega^{0}j_{*}(M)\rightarrow
i_{*}i^{*}\omega^{0}j_{*}(M).$

By Proposition 5.1.2, the motive $\omega^{0}j_{*}(M)$ is $t_{hp}$-non-
negative. Furthermore, Proposition 1.4.4 ensures that