$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}$,
$\operatorname{\mathcal{DM}}^{\operatorname{coh}}_{\operatorname{\acute{e}t},c}$,
$\operatorname{\mathcal{DM}}^{n}_{\operatorname{\acute{e}t},c}$ or
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}$.

######  Proposition 1.2.8.

Let $R$ be a good ring. Consider a scheme $X$ which is the limit of a
projective system of schemes with affine transition maps $(X_{i})_{i\in I}$.

Then, in the $2$-category of stable monoidal categories, we have

$\operatorname{colim}\mathscr{D}_{c}(X_{i},R)=\mathscr{D}_{c}(X,R).$

###### Proof.

The proof is the same as [CD16, 6.3.9], replacing [Gro66, 8.10.5, 17.7.8] with
[Leh19b, 1.24]. ∎

###  1.3. Torsion étale motives and torsion étale sheaves

####  1.3.1. The Six Functors and the Rigidity Theorem

######  Definition 1.3.1.

Let $S$ be a scheme, let $R$ be a ring and let $p$ be a prime number. We
define stable categories as follows.

  1. (1)

The stable category of _torsion étale motives_
$\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$ as the
subcategory of the stable category
$\mathcal{DM}_{\operatorname{\acute{e}t}}(S,R)$ made of those objects $M$ such
that the motive $M\otimes_{\mathbb{Z}}\mathbb{Q}$ vanishes.

  2. (2)

The stable category of _$p^{\infty}$ -torsion étale motives_
$\mathcal{DM}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S,R)$ as
the subcategory of the stable category
$\mathcal{DM}_{\operatorname{\acute{e}t}}(S,R)$ made of those objects $M$ such
that the motive $M\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]$ vanishes.

  3. (3)

The stable category of _torsion étale sheaves_
$\operatorname{\mathcalboondox{Sh}}_{\mathrm{tors}}(S_{\operatorname{\acute{e}t}},R)$
as the subcategory of the stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ made of
those sheaves $M$ such that the sheaf $M\otimes_{\mathbb{Z}}\mathbb{Q}$
vanishes.

  4. (4)

The stable category of _$p^{\infty}$ -torsion étale sheaves_
$\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(S_{\operatorname{\acute{e}t}},R)$
as the subcategory of the stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ made of
those sheaves $M$ such that the sheaf $M\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]$
vanishes.

######  Proposition 1.3.2.

Let $R$ be a ring and let $p$ be a prime number. Then, the six functors of
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(-,R)$ induce functors
over $\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(-,R)$ and
$\mathcal{DM}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(-,R)$.

###### Proof.

The six functors commute with tensor product with $\mathbb{Q}$ and with
$\mathbb{Z}[1/p]$ by Proposition 1.2.3 which implies the result. ∎

Scholze has shown in [Sch22, 7.15] that the category
$\operatorname{\mathcalboondox{Sh}}_{\mathrm{tors}}(-_{\operatorname{\acute{e}t}},\mathbb{Z})$
is endowed with the six functors formalism. The proof applies to an arbitrary
noetherian ring $R$. We deduce in the same fashion that they induce functors
over
$\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(-_{\operatorname{\acute{e}t}},R)$.

The following result is an analog of the rigidity theorem [CD16, 5.5.4] for
torsion motives.

######  Proposition 1.3.3.

Let $S$ be a scheme and let $R$ be a good ring. Then,

  1. (1)

The functor

$\prod_{p\text{
prime}}\mathcal{DM}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S,R)\rightarrow\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$

which sends an object $(M_{p})_{p\text{ prime}}$ to the object
$\bigoplus\limits_{p\text{ prime}}M_{p}$ is an equivalence.

  2. (2)

Let $p$ be a prime number. Let $j_{p}\colon S[1/p]\rightarrow S$ be the open
immersion. Then, the functor

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

is an equivalence.

  3. (3)

Let $p$ be a prime number. Let $S$ be a scheme over which $p$ is invertible.
Then, the map $\rho_{!}$ of Section 1.2.4 induces an equivalence

$\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\mathcal{DM}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S,R)$

  4. (4)

All those equivalences commute with the six functors formalism.

###### Proof.

Proof of Assertion (1): For any prime number $p$, let $\mathbb{Z}(p^{\infty})$
be Prüfer $p$-group. If $A$ is an abelian group, denote by $\underline{A}$ the
image through the functor

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

of the constant sheaf with value $A$. We have an exact triangle

$\mathbbm{1}_{S}\rightarrow\mathbbm{1}_{S}\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]\rightarrow\underline{\mathbb{Z}(p^{\infty})},$

therefore, if $M$ is a $p^{\infty}$-torsion étale motive, we have

$M=M\otimes_{\mathbb{Z}}\underline{\mathbb{Z}(p^{\infty})}[-1].$

Furthermore, if $p$ and $q$ are distinct prime numbers, the canonical map

$\mathbb{Z}(q^{\infty})\rightarrow\mathbb{Z}(q^{\infty})\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]$

is an isomorphism.

For any prime number $p$, let $M_{p}$ and $N_{p}$ be $p^{\infty}$-torsion
étale motives. Notice that if $p$ is a prime number, we have

$\displaystyle\bigoplus\limits_{\begin{subarray}{c}q\text{ prime}\\\ q\neq
p\end{subarray}}N_{q}$
$\displaystyle=\bigoplus\limits_{\begin{subarray}{c}q\text{ prime}\\\ q\neq
p\end{subarray}}N_{q}\otimes_{\mathbb{Z}}\mathbb{Z}(q^{\infty})[-1]$
$\displaystyle=\bigoplus\limits_{\begin{subarray}{c}q\text{ prime}\\\ q\neq
p\end{subarray}}N_{q}\otimes_{\mathbb{Z}}(\mathbb{Z}(q^{\infty})\otimes_{\mathbb{Z}}\mathbb{Z}[1/p])[-1]$
$\displaystyle=\left(\bigoplus\limits_{\begin{subarray}{c}q\text{ prime}\\\
q\neq p\end{subarray}}N_{q}\right)\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]$

Therefore, we get

$\operatorname{Map}\left(M_{p},\bigoplus\limits_{\begin{subarray}{c}q\text{
prime}\\\ q\neq p\end{subarray}}N_{q}\right)=0.$

Hence, the map

$\prod_{p\text{
prime}}\operatorname{Map}(M_{p},N_{p})\rightarrow\operatorname{Map}\left(\bigoplus\limits_{p\text{
prime}}M_{p},\bigoplus\limits_{p\text{ prime}}N_{p}\right)$

is an equivalence. Therefore, the functor

$\prod_{p\text{
prime}}\mathcal{DM}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S,R)\rightarrow\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$

is fully faithful.

We now prove that this functor is essentially surjective. We have an exact
triangle

$\mathbbm{1}_{S}\rightarrow\mathbbm{1}_{S}\otimes_{\mathbb{Z}}\mathbb{Q}\rightarrow\underline{\mathbb{Q}/\mathbb{Z}}.$

Let now $M$ be a torsion étale motive. We get that

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

Furthermore, we have

$\underline{\mathbb{Q}/\mathbb{Z}}=\bigoplus\limits_{p\text{
prime}}\underline{\mathbb{Z}(p^{\infty})}.$

and since the functor $\rho_{!}$ commutes with colimits, we therefore get that

$M=\bigoplus\limits_{p\text{
prime}}\left(M\otimes_{\mathbb{Z}}\underline{\mathbb{Z}(p^{\infty})}[-1]\right).$

As

$\left(M\otimes_{\mathbb{Z}}\underline{\mathbb{Z}(p^{\infty})}[-1]\right)\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]=0,$

the motive $M\otimes_{\mathbb{Z}}\underline{\mathbb{Z}(p^{\infty})}[-1]$ is of
$p^{\infty}$-torsion and therefore, the functor

$\prod_{p\text{
prime}}\mathcal{DM}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S,R)\rightarrow\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$

is essentially surjective which finishes the proof of the first assertion.

Proof of Assertion (2):

The $2$-morphism

$j_{p}^{*}(j_{p})_{*}\rightarrow\operatorname{Id}$

is an equivalence. Since by Proposition 1.3.2, both functors $j_{p}^{*}$ and
$(j_{p})_{*}$ induce functors on $p^{\infty}$-torsion motives, the functor

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

is essentially surjective.

To show that the functor $j_{p}^{*}$ is fully faithful when restricted to the
category of $p^{\infty}$-torsion étale motives, it suffices to show that the
natural map $\operatorname{Id}\rightarrow(j_{p})_{*}j_{p}^{*}$ is an
equivalence when restricted to the category of $p^{\infty}$-torsion étale
motives. Notice that the functor:

$-\otimes_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}\colon\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},p^{\infty}-\mathrm{tors}}(S,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R/pR)$

is conservative by Lemma 1.2.5 and commutes with the functors $(j_{p})_{*}$
and $j_{p}^{*}$ by [CD16, 5.4.5]. Therefore the full faithfulness of
$j_{p}^{*}$ follows from the same result with coefficients in $R/pR$.

Since $R/pR$ is a $\mathbb{Z}/p\mathbb{Z}$-algebra, we have by [CD16, 5.4] a
premotivic adjunction

$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,\mathbb{Z}/p\mathbb{Z})\leftrightarrows\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R/pR).$

Therefore, the result follows from [CD16, A.3.4]

Proof of Assertion (3):

Let $N$ be a $p^{\infty}$-torsion étale motive over $S$. Then, as before, we
have

$N=N\otimes_{\mathbb{Z}}\underline{\mathbb{Z}(p^{\infty})}[-1]=\operatorname{colim}N\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1].$

If $n$ is a non-negative integer, let

$N_{n}=N\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1].$

The étale motive $N_{n}$ lies in
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R/p^{n}R)$ and lies
therefore in the essential image of the functor $\rho_{!}$ by the rigidity
theorem [CD16, 5.5.4]. Let $M_{n}$ be a $p^{\infty}$-torsion étale sheaf such
that

$\rho_{!}M_{n}=N_{n}.$

The rigidity theorem also ensures that the map

$\operatorname{Id}\otimes_{\mathbb{Z}}(\times p)\colon N_{n}\rightarrow
N_{n+1}$

is the image through $\rho_{!}$ of a map

$\phi_{n}\colon M_{n}\rightarrow M_{n+1}.$

Letting $M$ be the colimit of the $M_{n}$ with transition maps the $\phi_{n}$,
we have

$\rho_{!}M=N$

since the functor $\rho_{!}$ preserves colimits.

Hence, the functor $\rho_{!}$ is essentially surjective.

Recall now that the map $\rho_{!}$ has a right adjoint functor $\rho^{!}$
defined in Section 1.2.4. To finish the proof it suffices to show that the
natural map

$\operatorname{Id}\rightarrow\rho^{!}\rho_{!}$

is an equivalence. Since the functor
$-\otimes_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$ is conservative and commutes
with the functors $\rho^{!}$ and $\rho_{!}$ by Proposition 1.2.3, it suffices
to show the same result with coefficients $\mathbb{Z}/p\mathbb{Z}$ which
follows from the rigidity theorem [CD16, 5.5.4].

Proof of Assertion (4):

The fourth assertion follows from the fact that all those functors can be
promoted into $\operatorname{Sm}$-premotivic functors (as defined in [CD16,
A.1.7]) and from [CD16, A.1.17]. ∎

Putting those results together, we get the following result.

######  Corollary 1.3.4.

Keep the notations of Proposition 1.3.3. The functor

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

which sends an object $(M_{p})_{p\text{ prime}}$ to the object
$\bigoplus\limits_{p\text{ prime}}j_{p}^{*}(\rho_{!}M_{p})$ is an equivalence.
This equivalence commutes with the six functors.

####  1.3.2. The ordinary t-structure

######  Proposition 1.3.5.

Let $S$ be a scheme and let $R$ be a ring. The ordinary t-structure on
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ is
generated in the sense of Proposition 1.1.6 by the étale sheaves of the form
$R_{S}(X)$ with $X$ étale over $S$.

###### Proof.

The proposition follows from Proposition 1.1.7. ∎

######  Proposition 1.3.6.

Let $S$ be a scheme and $R$ be a ring. Then, the ordinary t-structure of the
stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ induces
a t-structure on the stable subcategory
$\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(S_{\operatorname{\acute{e}t}},R)$.

###### Proof.

Let $\tau_{\geqslant 0}$ denote the truncation with respect to the ordinary
t-structure of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$. Let $M$
be an étale sheaf of $p^{\infty}$-torsion. The functor
$\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]$ is t-exact with respect to the ordinary
t-structure as $\mathbb{Z}[1/p]$ is flat over $\mathbb{Z}$. Therefore, we have

$\tau_{\geqslant 0}(M)\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]=\tau_{\geqslant
0}(M\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]).$

Furthermore, the sheaf $\tau_{\geqslant
0}(M\otimes_{\mathbb{Z}}\mathbb{Z}[1/p])$ vanishes and therefore, the sheaf
$\tau_{\geqslant 0}(M)$ is of $p^{\infty}$-torsion. We conclude by using
[BBDG18, 1.3.19]. ∎

A product of t-categories is endowed with a t-structure by taking the product
of the subcategories of t-non-positive objects as the subcategory of t-non-
positive objects and the product of the subcategories of t-non-negative
objects as the subcategory of t-non-negative objects.

This allows us to define an ordinary t-structure on $p^{\infty}$-torsion étale
motives.

######  Definition 1.3.7.

Let $S$ be a scheme and let $R$ be a good ring. The _ordinary t-structure_
over the stable category
$\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$ is the only
t-structure such that the functor

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

of Corollary 1.3.4 is a t-exact equivalence when the left hand side is endowed
with the product of the ordinary t-structures.

######  Remark 1.3.8.

Let $S$ be a scheme and let $R$ be a good ring. If all the residue
characteristics of $S$ are invertible in $R$, then, the functor $\rho_{!}$ of
Section 1.2.4 induces a t-exact equivalence

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

when both sides are endowed with their ordinary t-structures.

######  Proposition 1.3.9.

Let $f\colon T\rightarrow S$ be a morphism of schemes and let $R$ be a good
ring. The functor $f^{*}$ induces a t-exact functor

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

when both sides are endowed with their ordinary t-structures.

###### Proof.

Since the functor of Corollary 1.3.4 commutes with the six functors, it
suffices to show that if $p$ is a prime number and is invertible on $S$, the
functor

$f^{*}\colon\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(T_{\operatorname{\acute{e}t}},R)$

is t-exact when both sides are endowed with their ordinary t-structures which
follows from the t-exactness of the functor

$f^{*}\colon\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcalboondox{Sh}}(T_{\operatorname{\acute{e}t}},R).$

∎

####  1.3.3. Strengthening of the continuity property

The ordinary t-structure gives us a setting in which we can generalize the
continuity property of [CD16, 6.3.7].

######  Proposition 1.3.10.

Let $\mathscr{D}$ be one of the fibered categories
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}$ or
$\operatorname{\mathcalboondox{Sh}}(-_{\operatorname{\acute{e}t}})$ and let
$R$ be a good ring. Consider a scheme $X$ which is the limit of a projective
system of schemes with affine transition maps $(X_{i})_{i\in I}$.

Let $(M_{i})_{i\in I}$ and $(N_{i})_{i\in I}$ be two cartesian sections of the
fibered category $\mathscr{D}(-,R)$ over the diagram of schemes $(X_{i})_{i\in
I}$ and denote by $M$ and $N$ the respective pullback of $M_{i}$ and $N_{i}$
along the map $X\rightarrow X_{i}$. If each $M_{i}$ is constructible and each
$N_{i}\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}$ is bounded below with respect
to the ordinary t-struture on $\mathscr{D}_{\mathrm{tors}}(X_{i},R)$, then the
canonical maps

$\operatorname{colim}_{i\in
I}\operatorname{Hom}_{\mathscr{D}(X_{i},R)}(M_{i},N_{i})\rightarrow\operatorname{Hom}_{\mathscr{D}(X,R)}(M,N)$
$\operatorname{colim}_{i\in
I}\operatorname{Map}_{\mathscr{D}(X_{i},R)}(M_{i},N_{i})\rightarrow\operatorname{Map}_{\mathscr{D}(X,R)}(M,N)$

are equivalences.

###### Proof.

Following the proof of [CD16, 6.3.7], we can assume that the ring $R$ is a
$\mathbb{Q}$-algebra or a $\mathbb{Z}/n\mathbb{Z}$-algebra with $n$ a power of
some prime number $p$.

If $R$ is a $\mathbb{Q}$-algebra, the result is [CD16, 5.2.5] when
$\mathscr{D}=\mathcal{DM}_{\operatorname{\acute{e}t}}$ and the result follows
from [AGV73, VI.8.5.7] when
$\mathscr{D}=\operatorname{\mathcalboondox{Sh}}(-_{\operatorname{\acute{e}t}})$.

If $R$ is a $\mathbb{Z}/n\mathbb{Z}$-algebra, we can assume that $p$ (and
therefore $n$) is invertible on $S$ by virtue of [CD16, A.3.4]. Applying the
Rigidity Theorem [CD16, 5.5.4], we can assume that
$\mathscr{D}=\operatorname{\mathcalboondox{Sh}}(-_{\operatorname{\acute{e}t}})$.
In this case, the result follows from [AGV73, IX.2.7.3]. ∎

####  1.3.4. The perverse t-structure

We now define a perverse t-structure on the category of torsion étale motives
over base schemes endowed with a dimension function following [Gab08]. First
recall the definition of dimension functions (see for instance [BD17, 1.1.1]).

######  Definition 1.3.11.

Let $S$ be a scheme. A _dimension function_ on $S$ is a function

$\delta\colon S\rightarrow\mathbb{Z}$

such that for any immediate specialization $y$ of a point $x$ (i.e a
specialization such that $\operatorname{codim}_{\overline{\\{x\\}}}(y)=1$), we
have $\delta(x)=\delta(y)+1$.

Two dimension functions on a scheme differ by a Zariski-locally constant
function. Moreover, if a scheme $S$ is universally catenary and integral, the
formula

$\delta(x)=\dim(S)-\operatorname{codim}_{S}(x)$

defines a dimension function on $S$. Finally, any $S$-scheme of finite type
$f:X\rightarrow S$ inherits a canonical dimension function by setting

$\delta(x)=\delta(f(x))+\mathrm{tr.deg}\hskip 2.84526ptk(x)/k(f(x)).$

Let $R$ be a ring. The auto-dual perversity $p_{1/2}\colon Z\mapsto-\delta(Z)$
induces two t-structures $[p_{1/2}]$ and $[p_{1/2}^{+}]$ on
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$. In this
paper, we only consider the t-structure $[p_{1/2}]$.

######  Definition 1.3.12.

([Gab08, 2]) Let $S$ be a scheme endowed with a dimension function $\delta$
and let $R$ be a ring. If $x$ is a point of $S$, denote by
$i_{x}\colon\\{x\\}\rightarrow S$ the inclusion. We define

  * •

the full subcategory ${}^{p}\operatorname{\mathcalboondox{Sh}}^{\leqslant
0}(S_{\operatorname{\acute{e}t}},R)$ as the subcategory made of those étale
sheaves $M$ such that for any point $x$ of $S$, we have

$i_{x}^{*}M\leqslant p_{1/2}(\overline{\\{x\\}})=-\delta(x)$

with respect to the ordinary t-structure of
$\operatorname{\mathcalboondox{Sh}}(k(x)_{\operatorname{\acute{e}t}},R)$.

  * •

the full subcategory ${}^{p}\operatorname{\mathcalboondox{Sh}}^{\geqslant
0}(S_{\operatorname{\acute{e}t}},R)$ as the subcategory made of those bounded
below étale sheaves $M$ such that for any point $x$ of $S$, we have

$i_{x}^{!}M\geqslant p_{1/2}(\overline{\\{x\\}})=-\delta(x)$

with respect to the ordinary t-structure of
$\operatorname{\mathcalboondox{Sh}}(k(x)_{\operatorname{\acute{e}t}},R)$.

[Gab08, 6] implies that this defines a t-structure on
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$. Which
we call the _perverse t-structure_.

######  Proposition 1.3.13.

Let $S$ be a scheme endowed with a dimension function $\delta$ and let $R$ be
a ring. The perverse t-structure on
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ is
generated in the sense of Proposition 1.1.6 by the étale sheaves of the form
$f_{!}\underline{R}[\delta(X)]$ with $f\colon X\rightarrow S$ quasi-finite.

###### Proof.

When $S$ is the spectrum of a field, the result follows from Proposition
1.3.5.

Denote by $t_{0}$ the t-structure generated by the étale sheaves of the form
$f_{!}\underline{R}[\delta(X)]$ with $f\colon X\rightarrow S$ quasi-finite. It
suffices to show that a sheaf $M$ is perverse t-non-negative if and only if it
is $t_{0}$-non-negative.

Hence, it suffices to show that a sheaf $M$ is $t_{0}$-non-negative if and
only if it is bounded below and for any point $x$ of $S$, the étale sheaf
$i_{x}^{!}M$ is $t_{0}$-non-negative.

To prove the "if" part, notice that if $f$ is a quasi-finite morphism, the
adjunction $(f_{!},f^{!})$ is a $t_{0}$-adjunction. Indeed, the full
subcategory of those étale sheaves $N$ such that $f_{!}N$ is $t_{0}$-non-
positive is closed under extensions and small colimits and contains the set of
generators of $t_{0}$. Thus, this subcategory contains all $t_{0}$-non-
positive objects. Hence, the functors $f_{!}$ is right $t_{0}$-exact and
therefore, the functor $f^{!}$ is left-$t_{0}$-exact. Hence, if $M$ is
$t_{0}$-non-negative and if $x$ is a point of $S$, the étale sheaf
$i_{x}^{!}M$ is $t_{0}$-non-negative which yields

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

with respect to the ordinary t-structure on
$\operatorname{\mathcalboondox{Sh}}(k(x)_{\operatorname{\acute{e}t}},R)$.

Furthermore, if $M$ is $t_{0}$-non-negative, if $c$ is a lower bound for the
function $\delta$ on $S$, the complex $\operatorname{Map}(R_{S}(X),M)$ is
$(c-1)$-connected for any étale $S$-scheme $X$. Hence, the étale sheaf sheaf
$M$ is bounded below and it is therefore perverse t-non-negative.

We now prove the converse. Let $M$ be a bounded below étale sheaf such that
for all point $x$ of $S$, the étale sheaf $i_{x}^{!}M$ is $t_{0}$-non-
negative. Let $f\colon X\rightarrow S$ be quasi-finite. For any point $y$ of
$X$, let $\mathrm{Z}(y)$ be the set of open neighborhoods of $y$ in the
closure $\overline{\\{y\\}}$ of $\\{y\\}$. If $U$ is an element of
$\mathrm{Z}(y)$, denote by $f_{U}\colon U\rightarrow S$ the inclusion. Then,
by [BD17, 3.1.5], we have the $\delta$-niveau spectral sequence

$E^{1}_{p,q}\Rightarrow\pi_{-p-q}\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}(f_{!}\underline{R},M)$

with

$E^{1}_{p,q}=\bigoplus_{y\in X,\hskip
2.84526pt\delta(y)=p}\operatorname{colim}\limits_{U\in\mathrm{Z}(y)}\pi_{-p-q}\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}((f_{U})_{!}\underline{R},M)$

If $y$ is a point of $X$ and if $x=f(y)$, let
$\iota_{y}\colon\\{y\\}\rightarrow X$ be the inclusion and let
$f_{y}\colon\\{y\\}\rightarrow\\{x\\}$ be the morphism induced by $f$. Then,
since the sheaf $M$ is bounded below, Proposition 1.3.10 yields

$\displaystyle\operatorname{colim}\limits_{U\in\mathrm{Z}(y)}\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}((f_{U})_{!}\underline{R},M)$
$\displaystyle=\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}((f\circ\iota_{y})_{!}\underline{R},M)$
$\displaystyle=\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}((i_{x})_{!}(f_{y})_{!}\underline{R},M)$
$\displaystyle=\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(k(x)_{\operatorname{\acute{e}t}},R)}((f_{y})_{!}\underline{R},i_{x}^{!}M).$

Thus, we get

$E^{1}_{p,q}=\bigoplus_{y\in X,\hskip
2.84526pt\delta(y)=p}\pi_{-p-q}\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(k(f(y))_{\operatorname{\acute{e}t}},R)}((f_{y})_{!}\underline{R},i_{f(y)}^{!}M).$

Since for any point $x$ of $S$, the sheaf $i_{x}^{!}M$ is $t_{0}$-non-
negative, for any point $y$ of $X$, the complex
$\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(k(f(y))_{\operatorname{\acute{e}t}},R)}((f_{y})_{!}\underline{R},i_{f(y)}^{!}M)$
is $(-\delta(y)-1)$-connected and thus it is $(-\delta(X)-1)$-connected.

Thus, if $n>\delta(X),$ the $R$-module
$\pi_{-n}\operatorname{Map}_{\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)}(f_{!}\underline{R},M)$
vanishes. Hence, the étale sheaf $M$ is $t_{0}$-non-negative. ∎

######  Proposition 1.3.14.

Let $S$ be a scheme endowed with a dimension function and let $R$ be a ring.
Then, the perverse t-structure of the stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ induces
a t-structure on the stable subcategory
$\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(S_{\operatorname{\acute{e}t}},R)$.

###### Proof.

To apply the same proof as in Proposition 1.3.6, we only need to show that the
functor $-\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]$ is perverse t-exact. This is
true because if $x$ is a point of $S$, denoting by
$i_{x}\colon\\{x\\}\rightarrow S$ the inclusion, the functors $i_{x}^{*}$ and
$i_{x}^{!}$ commute with the functor $-\otimes_{\mathbb{Z}}\mathbb{Z}[1/p]$. ∎

######  Definition 1.3.15.

Let $S$ be a scheme endowed with a dimension function and let $R$ be a ring.
The _perverse t-structure_ over the stable category
$\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$ is the only
t-structure such that the functor

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

of Corollary 1.3.4 is a t-exact equivalence when the left hand side is endowed
with the product of the perverse t-structures.

######  Remark 1.3.16.

Let $S$ be a scheme endowed with a dimension function and let $R$ be a ring.
If all the residue characteristics of $S$ are invertible in $R$, then, the
functor $\rho_{!}$ of Section 1.2.4 induces a t-exact equivalence

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

when both sides are endowed with their perverse t-structures.

The definition of the perverse t-structure and the fact that the equivalence
$\rho_{!}$ of Section 1.2.4 commutes with the six functors yield the following
characterization of the perverse t-structure.

######  Proposition 1.3.17.

Let $S$ be a scheme endowed with a dimension function and let $R$ be a ring.
If $x$ is a point of $S$, denote by $i_{x}\colon\\{x\\}\rightarrow S$ the
inclusion. Then,

  * •

The subcategory ${}^{p}\mathcal{DM}^{\leqslant
0}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$ is made of those torsion
motives $M$ such that for any point $x$ of $S$, we have

$i_{x}^{*}M\leqslant-\delta(x)$

with respect to the ordinary t-structure of
$\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(k(x),R)$.

  * •

The subcategory ${}^{p}\mathcal{DM}^{\geqslant
0}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$ is made of those bounded
below torsion motives $M$ such that for any point $x$ of $S$, we have

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

with respect to the ordinary t-structure of
$\mathcal{DM_{\operatorname{\acute{e}t},\mathrm{tors}}}(k(x),R)$.

######  Proposition 1.3.18.

(Affine Lefschetz Theorem for torsion motives) Let $f\colon T\rightarrow S$ be
an affine morphism of schemes and let $R$ be a ring. Assume that the scheme
$S$ is quasi-excellent and endowed with a dimension function. Then, the
functor $f_{!}$ induces a left t-exact functor

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

when both sides are endowed with their perverse t-structures.

###### Proof.

Since the functor of Corollary 1.3.4 commutes with the six functors, it
suffices to show that if $p$ is a prime number and in invertible on$S$, the
functor

$f_{!}\colon\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(T_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcalboondox{Sh}}_{p^{\infty}-\mathrm{tors}}(S_{\operatorname{\acute{e}t}},R)$

is left t-exact when both sides are endowed with their perverse t-structures.

Let $M$ be a $p^{\infty}$-torsion étale sheaf which is perverse t-non-
negative. We have

$M=\operatorname{colim}_{n}M\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1].$

Let $n$ be a positive integer. Since we have an exact triangle

$M\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1]\rightarrow
M\overset{\times p^{n}}{\rightarrow}M,$

the sheaf $M\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1]$ is perverse
t-non-negative.

Therefore, by Lemma 1.3.19, the sheaf
$f_{!}(M\otimes_{\mathbb{Z}}\mathbb{Z}/p^{n}\mathbb{Z}[-1])$ is perverse
t-non-negative. Since the functor $f_{!}$ is a left adjoint, it is compatible
with colimits and therefore, using Proposition 1.2.7, the sheaf $f_{!}(M)$ is
perverse t-non-negative. ∎

######  Lemma 1.3.19.

Let $f\colon T\rightarrow S$ be an affine morphism of schemes scheme, let $p$
be a prime number which is invertible on $S$ and let $n$ be a positive
integer. Then, the functor

$f_{!}\colon\operatorname{\mathcalboondox{Sh}}(T_{\operatorname{\acute{e}t}},\mathbb{Z}/p^{n}\mathbb{Z})\rightarrow\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},\mathbb{Z}/p^{n}\mathbb{Z})$

is left t-exact.

###### Proof.

By definition of the perverse t-structure, we can assume that the scheme $S$
is the spectrum of a strictly henselian ring. In that case, since the étale
sheaves of the form $g_{!}\underline{\mathbb{Z}/p^{n}\mathbb{Z}}[\delta(X)]$
with $g\colon X\rightarrow S$ quasi-finite are compact, the perverse
t-structure is compatible with filtered colimits.

Let $M$ be an étale sheave over $T_{\operatorname{\acute{e}t}}$ which is
t-non-negative we want to prove that the sheaf $f_{!}M$ is . The sheaf $M$ is
a filtering colimit of constructible sheaves. Therefore, we can assume that
the sheaf $M$ is constructible. The result is then exactly the Affine
Lefschetz Theorem [LYF14, XV.1.1.2]. ∎

###  1.4. Reminders and complements on Artin motives

####  1.4.1. Generators and stability

We now state and generalize some results from [AZ12] and [Leh19b] that will be
useful in our discussion. The following two results were stated in [AZ12] for
schemes over a perfect field and extended over any schemes in [Leh19b] where
they assume that $R=\mathbb{Q}$. However, their proofs apply if $R$ is any
commutative ring.

######  Proposition 1.4.1.

([Leh19b, 1.28]) Let $S$ be a scheme and let $R$ be a ring. The category
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$ (resp.
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(S,R)$) is the
localizing (resp. thick) subcategory of
$\mathcal{DM}_{\operatorname{\acute{e}t}}(S,R)$ generated by any of the
following families objects.

  1. (1)

The family of the $M_{S}^{\operatorname{BM}}(X)$, with $X$ quasi-finite over
$S$.

  2. (2)

The family of the $M_{S}^{\operatorname{BM}}(X)=h_{S}(X)$, with $X$ finite
over $S$.

  3. (3)

The family of the $M_{S}^{\operatorname{BM}}(X)=M_{S}(X)$, with $X$ étale over
$S$.

######  Proposition 1.4.2.

([Leh19b, 1.17]) Let $R$ be a ring. The categories
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},(c)}(-,R)$ are closed under the
following operations.

  1. (1)

The functor $f^{*}$, where $f$ is any morphism.

  2. (2)

The functor $f_{!}$, where $f$ is a quasi-finite morphism.

  3. (3)

Tensor product.

Furthermore, the fibered category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},(c)}(-,R)$ satisfies
the localization property (0.0.1).

If $j\colon U\rightarrow S$ is an open immersion, the motive
$j_{*}\mathbbm{1}_{U}$ need not be Artin, the fibered category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(-,R)$ does not
satisfy the localization property (0.0.2).

Finally, the description of generators yields the following statement.

######  Proposition 1.4.3.

Let $S$ be a scheme and let $R$ be a ring. Then, the functor $\rho_{!}$ of
Section 1.2.4 has its essential image contained in
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R)$. We still
denote by

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

the induced functor.

###### Proof.

The functor $\rho_{!}$ of Section 1.2.4 is colimit preserving. Therefore, the
result follows from Proposition 1.1.7 and from the fact that if $X$ is an
étale $S$-scheme, we have

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

∎

####  1.4.2. The Artin truncation functor

Let $S$ be a scheme and let $R$ be a ring. The adjunction theorem [Lur09,
5.5.2.9] ensures the existence of a right adjoint exact functor to the
inclusion of Artin motives into cohomological motives:

$\omega^{0}\colon\operatorname{\mathcal{DM}}^{\operatorname{coh}}_{\operatorname{\acute{e}t}}(S,R)\rightarrow\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(S,R).$

This functor was first introduced in [AZ12, 2.2] and was further studied in
[Leh19b] and [Vai16]. We have a $2$-morphism
$\delta\colon\omega^{0}\rightarrow\mathrm{id}$ given by the counit of the
adjunction. We recall the basic properties of $\omega^{0}$ and its
compatibilities with the six functors formalism from [AZ12, 2.15] and [Leh19b,
3.3,3.7]:

######  Proposition 1.4.4.

Let $R$ be a ring.

  1. (1)

The $2$-morphism $\delta$ induces an equivalence
$\omega^{0}\rightarrow\mathrm{id}$ when restricted to Artin motives. Moreover,
we have $\delta(\omega^{0})=\omega^{0}(\delta)$ and this $2$-morphism is an
equivalence $\omega^{0}\circ\omega^{0}\rightarrow\omega^{0}$.

  2. (2)

Let $f$ be a morphism of schemes.

    1. (a)

The $2$-morphism
$\delta(f^{*}\omega^{0})\colon\omega^{0}f^{*}\omega^{0}\rightarrow
f^{*}\omega^{0}$ is invertible.

    2. (b)

We have a $2$-morphism $\alpha_{f}\colon
f^{*}\omega^{0}\rightarrow\omega^{0}f^{*}$ such that

      1. (i)

we have $\delta(f^{*})\circ\alpha_{f}=f^{*}(\delta)$111A more rigorous
statement would be that $f^{*}(\delta)$ is a composition of $\delta(f^{*})$
and $\alpha_{f}$ (in the $\infty$-categorical sense). and
$\alpha_{f}\circ\delta(f^{*}\omega^{0})=\omega^{0}f^{*}(\delta)$.

      2. (ii)

the $2$-morphism $\alpha_{f}$ is invertible when $f$ is étale.

      3. (iii)

letting $i\colon F\rightarrow S$ be a closed immersion, letting $j\colon
U\rightarrow S$ be the complementary open immersion and letting $M$ be an
object of $\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)$ such
that $j^{*}(M)$ belongs to
$\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t}}(U,R)$, the map
$\alpha_{i}(M)$ is invertible.

  3. (3)

Let $f$ be a morphism of finite type.

    1. (a)

The $2$-morphism
$\omega^{0}f_{*}(\delta)\colon\omega^{0}f_{*}\omega^{0}\rightarrow\omega^{0}f_{*}$
is invertible.

    2. (b)

We have a $2$-morphism $\beta_{f}\colon\omega^{0}f_{*}\rightarrow
f_{*}\omega^{0}$ such that:

      1. (i)

we have $f_{*}(\delta)\circ\beta_{f}=\delta(f_{*})$ and
$\beta_{f}\circ\omega^{0}f_{*}(\delta)=\delta(f_{*}\omega^{0})$.

      2. (ii)

the $2$-morphism $\beta_{f}$ is invertible when the morphism $f$ is finite.

  4. (4)

Let $f$ be a quasi-finite morphism.

    1. (a)

The maps $\delta(f_{!}\omega^{0}):\omega^{0}f_{!}\omega^{0}\rightarrow
f_{!}\omega^{0}$ and
$\omega^{0}f^{!}(\delta):\omega^{0}f^{!}\omega^{0}\rightarrow\omega^{0}f^{!}$
are invertible.

    2. (b)

We have $2$-morphisms $\eta_{f}:f_{!}\omega^{0}\rightarrow\omega^{0}f_{!}$ and
$\gamma_{f}:\omega^{0}f^{!}\rightarrow f^{!}\omega^{0}$ such that

      1. (i)

we have $\delta(f_{!})\circ\eta_{f}=f_{!}(\delta)$,
$\eta_{f}\circ\delta(f_{!}\omega^{0})=\omega^{0}f_{!}(\delta)$,
$f^{!}(\delta)\circ\gamma_{f}=\delta(f^{!})$ and
$\gamma_{f}\circ\omega^{0}f^{!}(\delta)=\delta(f^{!}\omega^{0})$.

      2. (ii)

the $2$-morphism $\eta_{f}$ is invertible when $f$ is finite and coincides
with $\beta_{f}^{-1}$ modulo the natural isomorphism $f_{!}\simeq f_{*}$.

      3. (iii)

the $2$-morphism $\gamma_{f}$ is invertible when $f$ is étale and coincides
with $\alpha_{f}^{-1}$ modulo the natural isomorphism $f^{*}\simeq f^{!}$.

  5. (5)

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

    1. (a)

The map $\alpha_{f}$ is an equivalence when $f$ is smooth.

    2. (b)

Assume that $S$ is regular. Let $X$ be a smooth and proper $S$-scheme and
$\pi_{0}(X/S)$ be its Stein factorization over $S$. Then, the canonical map
$h_{S}(\pi_{0}(X/S))\rightarrow\omega^{0}h_{S}(X)$ is an equivalence.

###### Proof.

This is [Leh19b, 3.3,3.7]. ∎

As a consequence, we get an analog of the localization triangle (0.0.2) for
Artin motives:

######  Corollary 1.4.5.

Let $i\colon Z\rightarrow S$ be a closed immersion, let $j\colon U\rightarrow
S$ be the complementary open immersion and let $R$ be a ring. Then, we have an
exact triangle

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

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

###### Proof.

Apply the functor $\omega^{0}$ to the exact localization triangle (0.0.2) and
use the fact that the functor $\eta_{i}$ of Proposition 1.4.4 above an
equivalence to get the result. ∎

We will see later that this triangle need not induce a triangle in
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(S,R)$ in
general.

In [AZ12], the authors proved that if $S$ is a quasi-projective scheme over a
field of characteristic $0$ and if $R=\mathbb{Q}$, then, the functor
$\omega^{0}_{S}$ preserves constructible objects (see [AZ12, 2.15 (vii)]). In
[Leh19b, 3.7], Pepin Lehalleur extended the result to schemes allowing
resolutions of singularities by alterations. In the proof, Pepin Lehalleur
uses this hypothesis only to ensure that the localization triangle (0.0.2)
belongs to
$\operatorname{\mathcal{DM}}^{\operatorname{coh}}_{\operatorname{\acute{e}t},c}(-,R)$.
Using the appendix, we see that this assumption is in fact unnecessary and we
deduce the following statement.

######  Proposition 1.4.6.

Let $S$ be a quasi-excellent scheme and $R$ a $\mathbb{Q}$-algebra. Then, the
functor $\omega^{0}_{S}$ preserves constructible objects.

Using the Artin truncation functor $\omega^{0}$, we prove that there is a
trace of the six functors formalism on constructible Artin motives with
rational coefficients ; this is not true with integral coefficient. Namely, if
$k$ is a field, and $i$ is the inclusion of a point in $\mathbb{P}^{1}_{k}$,
the motive $\omega^{0}i^{!}\mathbbm{1}_{\mathbb{P}^{1}_{k}}$ is not
constructible (see Corollary 3.1.8 below).

######  Proposition 1.4.7.

Let $R$ be a $\mathbb{Q}$-algebra. Then, the six functors from
$\mathcal{DM}_{\operatorname{\acute{e}t}}(-,R)$ induce

  1. (1)

a functor $f^{*}$ for any morphism $f$,

  2. (2)

a functor $\omega^{0}f_{*}$ for any separated morphism of finite type $f$
whose target is quasi-excellent,

  3. (3)

a functor $f_{!}$ for any quasi-finite morphism $f$,

  4. (4)

a functor $\omega^{0}f^{!}$ for any quasi-finite morphism $f$ with a quasi-
excellent source,

  5. (5)

functors $\otimes$ and $\underline{\operatorname{Hom}}$

on $\operatorname{\mathcal{DM}}^{A}_{\operatorname{\acute{e}t},c}(-,R)$.
Moreover, the fibered category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(S,R)$ satisfies
the localization property (0.0.1) and if $i\colon Z\rightarrow S$ is a closed
immersion and if $j\colon U\rightarrow S$ is the complementary open immersion,
we have an exact triangle

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

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

###### Proof.

We already proved the proposition in the case of left adjoint functors in
Proposition 1.4.2. The case of right adjoint functors follows from the fact
that
$\operatorname{\mathcal{DM}}^{\operatorname{coh}}_{\operatorname{\acute{e}t},c}(-,R)$
is endowed with the six functors formalism and from Proposition 1.4.6. ∎

Finally, the functor $\omega^{0}$ is compatible with change of coefficients.

######  Proposition 1.4.8.

Let $\sigma\colon R\rightarrow R^{\prime}$ be a ring morphism of the form
$\sigma_{A}\colon R\rightarrow R\otimes_{\mathbb{Z}}A$ or
$\sigma_{n}:R\rightarrow R/nR$ where $A$ is a localization of $\mathbb{Z}$ and
$n$ is an integer. Assume that the ring $R$ is flat over $\mathbb{Z}$. Then,
we have invertible $2$-morphisms

$u\colon\sigma_{*}\omega^{0}\rightarrow\omega^{0}\sigma_{*}\text{ and
}v\colon\sigma^{*}\omega^{0}\rightarrow\omega^{0}\sigma^{*}$

such that the diagrams

${\sigma_{*}\omega^{0}}$${\omega^{0}\sigma_{*}}$${\sigma_{*}}$$\scriptstyle{u}$$\scriptstyle{\sigma_{*}(\delta)}$$\scriptstyle{\delta(\sigma_{*})}$
and
${\sigma^{*}\omega^{0}}$${\omega^{0}\sigma^{*}}$${\sigma^{*}}$$\scriptstyle{u}$$\scriptstyle{\sigma^{*}(\delta)}$$\scriptstyle{\delta(\sigma^{*})}$

are commutative.

###### Proof.

Let us first show that any object of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R^{\prime})$ lies in
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$. Let $X$ be an étale
$S$-scheme.

In the case where $R^{\prime}=R\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}$,
since $R$ is flat over $\mathbb{Z}$, we have an exact sequence

$0\rightarrow R\overset{\times n}{\rightarrow}R\rightarrow
R\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}\rightarrow 0.$

Since the six functors formalism commute with change of coefficients by
Proposition 1.2.3, this implies that $M_{S}(X,R^{\prime})$ is the cone of the
map

$M_{S}(X,R)\overset{\times n}{\longrightarrow}M_{S}(X,R).$

Therefore, the set of generators of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R^{\prime})$ belongs to
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$.

In the case where $R^{\prime}=R\otimes_{\mathbb{Z}}A$, write
$A=\Sigma^{-1}\mathbb{Z}$ where $\Sigma$ is a multiplicative subset of
$\mathbb{Z}$. Then, the motive $M_{S}(X,R^{\prime})$ is the colimit of the
inductive system given by the multiplication by $s$ maps
$M_{S}(X,R)\rightarrow M_{S}(X,R)$ for $s\in\Sigma$. Therefore, the set of
generators of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R^{\prime})$
lies in $\mathcal{DM}^{A}(S,R)$.

Therefore, the functor $\sigma_{*}$ preserves Artin motives. The existence of
a $2$-morphisms $u$ such that $\sigma_{*}(\delta)=\delta(\sigma_{*})\circ u$
is therefore a consequence of the definition of $\omega^{0}$. Likewise, since
the functor $\sigma^{*}$ preserves Artin motives, we get a $2$-morphism $v$
such that $\sigma^{*}(\delta)=\delta(\sigma^{*})\circ v$. We now show that $u$
and $v$ are invertible.

To show that the $2$-morphism $u$ is invertible, it suffices to show that for
any object $M$ of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$ and any
object $N$ of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R^{\prime})$,
the map

$\operatorname{Map}(M,\sigma_{*}\omega^{0}N)\rightarrow\operatorname{Map}(M,\sigma_{*}N)$

induced by $\sigma_{*}(\delta)$ is invertible. This amounts to the fact that
the map

$\operatorname{Map}(\sigma^{*}M,\omega^{0}N)\rightarrow\operatorname{Map}(\sigma^{*}M,N)$

induced by $\delta$ is invertible which is true by definition of the functor
$\omega^{0}$.

To show that the $2$-morphism $v$ is invertible, it suffices to show that for
any object $M$ of
$\mathcal{DM}^{A}_{\operatorname{\acute{e}t},c}(S,R^{\prime})$ and any object
$N$ of $\mathcal{DM}^{A}_{\operatorname{\acute{e}t}}(S,R)$, the map

$\operatorname{Map}(M,\sigma^{*}\omega^{0}N)\rightarrow\operatorname{Map}(M,\sigma^{*}N)$

induced by $\sigma^{*}(\delta)$ is invertible. We can assume that
$M=M_{S}(X,R^{\prime})$ for $X$ an étale $S$-scheme. We have
$M=\sigma^{*}M_{S}(X,R)$. But the diagram

${\operatorname{Map}(\sigma^{*}M_{S}(X,R),\sigma^{*}\omega^{0}N)}$${\operatorname{Map}(\sigma^{*}M_{S}(X,R),\sigma^{*}N)}$${\sigma^{*}\operatorname{Map}(M_{S}(X,R),\omega^{0}N)}$${\sigma^{*}\operatorname{Map}(M_{S}(X,R),N)}$$\scriptstyle{\sigma^{*}(\delta)\circ-}$$\scriptstyle{\sigma^{*}(\delta\circ-)}$

is commutative, its vertical arrows are invertible by [CD16, 5.4.5, 5.4.11]
and its lower horizontal arrow is invertible by definition of $\omega^{0}$.
This finishes the proof. ∎

####  1.4.3. Torsion motives are Artin motives

######  Proposition 1.4.9.

Let $S$ be a scheme and let $R$ be a ring. The stable category of torsion
étale motive $\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$ is
the subcategory of the stable category
$\mathcal{DM}_{\operatorname{\acute{e}t}}^{A}(S,R)$ made of those Artin
motives $M$ such that the motive $M\otimes_{\mathbb{Z}}\mathbb{Q}$ vanishes.

###### Proof.

If an Artin motive $M$ is such that the motive
$M\otimes_{\mathbb{Z}}\mathbb{Q}$ vanishes, it is by definition a torsion
motive. Therefore, it suffices to show that torsion motives are Artin motives.

Let $M$ be a torsion motive. Using Proposition 1.3.3, we can decompose $M$ as

$M=\bigoplus\limits_{p\text{ prime}}M_{p}$

where $M_{p}$ is of $p^{\infty}$-torsion. Therefore, we can assume that $M$ is
of $p^{\infty}$-torsion for some prime number $p$. Proposition 1.3.3 gives a
$p^{\infty}$-torsion étale sheaf $N$ over the open subscheme $S[1/p]$ of $S$
such that

$M=(j_{p})_{!}\rho_{!}N$

where $j_{p}\colon S[1/p]\rightarrow S$ is the open immersion.

By Proposition 1.4.3, the motive $\rho_{!}N$ is an Artin motive. Therefore, by
Proposition 1.4.2, the motive $M$ is an Artin motive. ∎

######  Corollary 1.4.10.

Let $S$ be a scheme and let $R$ be a ring. The functor
$\delta\colon\omega^{0}\rightarrow\operatorname{Id}$ induces an equivalence
over $\mathcal{DM}_{\operatorname{\acute{e}t},\mathrm{tors}}(S,R)$.

###  1.5. Artin representations and lisse étale sheaves

####  1.5.1. Finite permutation resolutions

In this paragraph, we recall a result of Balmer and Gallauer (see [BG23]). If
$G$ is a finite group, $R$ is a ring and $H$ is a subgroup of $G$, the
$R$-module $R[G/H]$ is endowed with a left $R[G]$-module structure letting $G$
act on $G/H$ by left multiplication.

######  Proposition 1.5.1.

Let $G$ be a finite group, let $R$ be a regular ring and let $M$ be a
representation of $G$ with coefficients in $R$. Then, the complex $M\oplus
M[1]$ of $R[G]$-modules is quasi-isomorphic as a complex of $R[G]$-modules to
a complex of the form

$\cdots\rightarrow 0\rightarrow P_{n}\rightarrow
P_{n-1}\rightarrow\cdots\rightarrow P_{0}\rightarrow 0\rightarrow\cdots$

where each $P_{i}$ is placed in degree $-i$ and is isomorphic to
$R[G/H_{1}]\oplus\cdots\oplus R[G/H_{m}]$ for some subgroups
$H_{1},\ldots,H_{m}$ of $G$.

###### Proof.

[Rui22, 1.3.1]. ∎

####  1.5.2. Discrete representations, Artin representations

Recall the following definitions:

######  Definition 1.5.2.

Let $\pi$ be a topological group and let $R$ be a ring. An $R[\pi]$-module $M$
is _discrete_ if the action of $\pi$ on $M$ is continuous when $M$ is endowed
with the discrete topology. We will denote by $\mathrm{Mod}(\pi,R)$ the
Grothendieck abelian category of discrete $R[\pi]$-modules.

######  Remark 1.5.3.

Keep the same notations. An $R[\pi]$-module is discrete if and only if the
stabilizer of every point is open. Equivalently, this means that $M$ is the
union of all the $M^{U}$ for $U$ an open subgroup of $\pi$.

If the topological group $\pi$ is profinite, this also means that $M$ is the
union of all the $M^{H}$ for $H$ a normal subgroup of $\pi$ of finite index.

######  Example 1.5.4.

Let $k$ be a field of characteristic exponent $p$. If $n$ is a positive
integer coprime to $p$, the absolute Galois group $G_{k}$ of $k$ acts on the
abelian group $\mu_{n}$ of roots of unity in the algebraic closure of $k$. Let
$(T_{n})$, where $n$ runs through the set of positive integers which are
coprime to $p$, be an inductive system of discrete representations of $G_{k}$
such that $T_{n}$ is of $n$-torsion. Set $T=\operatorname{colim}T_{n}$ and let
$m$ be an integer, we can define a discrete representation

$T(m):=\operatorname{colim}T_{n}\otimes\mu_{n}^{\otimes m}$

of $G_{k}$. We call this representation the _$m$ -th discrete Tate twist_ of
$T$ (beware that it depends on the inductive system and not only on $T$ itself
in general). Notice that by definition, $T$ is isomorphic to every $T(m)$ as
an abelian group but not as a $G_{k}$-module in general.

For example, let $X\rightarrow\operatorname{Spec}(k)$ be a morphism of finite
type, let $\overline{k}$ be the algebraic closure of $k$, let
$X_{\overline{k}}$ be the scalar extension of $X$ to $\overline{k}$ and let
$n$ be a non-negative integer. Then, the $m$-th discrete Tate twist of the
discrete Galois representation
$H^{n}_{\operatorname{\acute{e}t}}(X_{\overline{k}},\mathbb{Q}/\mathbb{Z}[1/p])$
will by convention be the discrete representation

$\operatorname{colim}\limits_{(N,p)=1}H^{n}_{\operatorname{\acute{e}t}}(X_{\overline{k}},\mathbb{Z}/N\mathbb{Z})\otimes\mu_{N}^{\otimes
m}.$

######  Definition 1.5.5.

Let $\pi$ be a profinite group and let $R$ be a ring. An _Artin
representation_ of $\pi$ is a discrete $R[\pi]$-module which is of finite
presentation as an $R$-module. We will denote by
$\operatorname{Rep}^{A}(\pi,R)$ the abelian category of Artin representations.

######  Remark 1.5.6.

Keep the same notations. An $R[\pi]$-module is an Artin representation if and
only if it is of finite presentation as an $R$-module and the action of $\pi$
factors through a finite quotient. Indeed, if $M$ is an Artin representation,
the stabilizer of a finite generating family is open and thus contains a
normal subgroup $N$ of finite index. Therefore, the action of $\pi$ factors
through $\pi/N$.

####  1.5.3. Locally constant étale sheaves

######  Definition 1.5.7.

Let $S$ be a scheme and let $R$ be a ring. We define:

  1. (1)

The abelian category $\operatorname{Sh}(S_{\operatorname{\acute{e}t}},R)$ of
_étale sheaves of $R$-modules_ as the heart of the ordinary t-structure on the
stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$.

  2. (2)

The abelian category $\mathrm{Loc}_{S}(R)$ of _locally constant sheaves of
$R$-modules_ as the subcategory of the abelian category
$\operatorname{Sh}(S_{\operatorname{\acute{e}t}},R)$ made of those sheaves
which are étale-locally constant with finitely presented fibers.

  3. (3)

The abelian category $\operatorname{Ind}\mathrm{Loc}(S,R)$ _Ind-locally
constant sheaves of $R$-modules_ as the subcategory of the abelian category
$\operatorname{Sh}(S_{\operatorname{\acute{e}t}},R)$ made of those sheaves
which filtered colimits of locally constant sheaves of $R$-modules.

  4. (4)

The _category of lisse étale sheaves_
$\operatorname{\mathcalboondox{Sh}}_{\mathrm{lisse}}(S,R)$ as the subcategory
of dualizable objects of the stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$.

  5. (5)

The _category of Ind-lisse étale sheaves_
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\mathrm{lisse}}(S,R)$
as the localizing subcategory of the stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$
generated by $\operatorname{\mathcalboondox{Sh}}_{\mathrm{lisse}}(S,R)$.

These categories are related to each other by the following proposition.

######  Proposition 1.5.8.

Let $S$ be a scheme and let $R$ be a regular ring.

  1. (1)

An étale sheaf is lisse if and only if it is locally constant with perfect
fibers.

  2. (2)

The ordinary t-structure of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ induces
a t-structure on the category
$\operatorname{\mathcalboondox{Sh}}_{\mathrm{lisse}}(S,R)$ whose heart is the
abelian category $\mathrm{Loc}_{S}(R)$.

  3. (3)

The ordinary t-structure of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ induces
a t-structure on the category
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\mathrm{lisse}}(S,R)$
whose heart is the abelian category $\operatorname{Ind}\mathrm{Loc}(S,R)$.

###### Proof.

The first assertion follows from [Sta23, 0FPP]. To prove the second assertion,
notice that, as a consequence of the first assertion, lisse sheaves are
bounded with respect to the ordinary t-structure of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$.
Therefore, Lemma 1.1.5 ensures that it suffices to show that

$\operatorname{Loc}_{S}(R)=\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)\cap\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)^{\heartsuit}.$

The fact that locally constant sheaves with finitely presented fibers have
perfect fibers follows from the hypothesis that the ring $R$ is regular, since
in this case, an $R$-modules is perfect if and only if it is finitely
presented by [Sta23, 066Z]. Moreover, an abelian sheaf with perfect fibers is
finitely presented. This finishes the proof of the second assertion.

Any Ind-lisse étale sheaf is a filtered colimit of lisse étale sheaves.
Furthermore, by [Lur17, 1.3.5.21], the ordinary t-structure of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ is
compatible with filtered colimits. Therefore, if $M=\operatorname{colim}_{i\in
I}M_{i}$ is an Ind-lisse étale sheaf, where $I$ is a filtered simplicial set
and each $M_{i}$ is lisse, we have

$\tau_{\geqslant 0}(M)=\tau_{\geqslant
0}(\operatorname{colim}M_{i})=\operatorname{colim}\tau_{\geqslant 0}(M_{i}).$

Therefore, the sheaf $\tau_{\geqslant 0}(M)$ is Ind-lisse. Hence, by [BBDG18,
1.3.19], the ordinary t-structure of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ induces
a t-structure on the category
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\mathrm{lisse}}(S,R)$.
An object $L$ of the heart of this t-structure is a filtered colimit of lisse
sheaves $M_{i}$. But then, we have,

$L=\operatorname{colim}M_{i}=H^{0}(\operatorname{colim}M_{i})=\operatorname{colim}H^{0}(M_{i}).$

Therefore, the sheaf $L$ is Ind-locally constant. Conversely, Ind-locally
constant sheaves lie in the heart of the ordinary t-structure and are Ind-
lisse. This finishes the proof of the third assertion. ∎

The above proposition shows in particular that $\operatorname{Loc}_{S}(R)$ is
a weak Serre subcategory of
$\operatorname{Sh}(S_{\operatorname{\acute{e}t}},R)$. We have the following
characterization of lisse sheaves:

######  Proposition 1.5.9.

Let $S$ be a connected scheme and let $R$ be a ring. Let $\xi$ be a geometric
point of $S$. The fiber functor associated to $\xi$ induces an equivalence of
abelian monoidal categories

$\xi^{*}\colon\mathrm{Loc}_{S}(R)\rightarrow\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi),R)$

where $\pi_{1}^{\mathrm{\operatorname{\acute{e}t}}}(S,\xi)$ is the étale
fundamental group of $S$ with base point $\xi$ of [GR71, V].

###### Proof.

Let $\mathrm{Loc}_{S}$ be the Galois category of locally constant sheaves of
sets on $S_{\operatorname{\acute{e}t}}$. By definition of the étale
fundamental group, the fiber functor

$\xi^{*}\colon\mathrm{Loc}_{S}\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)-\mathrm{Set}$

is an equivalence.

The constant sheaf $R_{S}(S)$ is sent to the set $R$ endowed with the trivial
action of $\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$. Thus, the functor
$\xi^{*}$ induces an equivalence

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

∎

######  Proposition 1.5.10.

Let $f\colon T\rightarrow S$ be a morphism of schemes, let
$\xi^{\prime}\colon\operatorname{Spec}(\Omega)\rightarrow T$ be a geometric
point of $T$, let $\xi=f\circ\xi^{\prime}$ and let $R$ be a ring. Denote by
$\pi_{1}^{\operatorname{\acute{e}t}}(f)\colon\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime})\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$
the group homomorphism induced by $f$.

  1. (1)

The square

${\mathrm{Loc}_{S}(R)}$${\mathrm{Loc}_{T}(R)}$${\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi),R)}$${\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime}),R)}$$\scriptstyle{f^{*}}$$\scriptstyle{\xi^{*}}$$\scriptstyle{(\xi^{\prime})^{*}}$$\scriptstyle{\pi_{1}^{\operatorname{\acute{e}t}}(f)^{*}}$

is commutative.

  2. (2)

Assume that $f$ is finite and étale, then, the map
$\pi_{1}^{\operatorname{\acute{e}t}}(f)$ is injective and its image is a
finite index subgroup. Furthermore, the functor

$f_{*}\colon\operatorname{\mathcalboondox{Sh}}(T,R)\rightarrow\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$

induces a functor $\mathrm{Loc}_{T}(R)\rightarrow\mathrm{Loc}_{S}(R)$ and the
square

${\mathrm{Loc}_{T}(R)}$${\mathrm{Loc}_{S}(R)}$${\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime}),R)}$${\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi),R)}$$\scriptstyle{f_{*}}$$\scriptstyle{(\xi^{\prime})^{*}}$$\scriptstyle{\xi^{*}}$$\scriptstyle{\mathrm{Ind}_{\pi^{\prime}}^{\pi}}$

where $\pi^{\prime}=\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime})$ and
$\pi=\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$, is commutative.

###### Proof.

If $X$ is a scheme, let $\mathrm{Loc}_{X}$ be the Galois category of locally
constant sheaves of sets on $X_{\operatorname{\acute{e}t}}$. By definition of
the map $\pi_{1}^{\mathrm{\operatorname{\acute{e}t}}}(f)$, the square:

${\mathrm{Loc}_{S}}$${\mathrm{Loc}_{T}}$${\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)-\text{Set}}$${\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime})-\text{Set}}$$\scriptstyle{f^{*}}$$\scriptstyle{\xi^{*}}$$\scriptstyle{(\xi^{\prime})^{*}}$$\scriptstyle{\pi_{1}^{\operatorname{\acute{e}t}}(f)^{*}}$

is commutative which implies the first assertion.

The first part of the second assertion is [GR71, V.6.3]. To finish the proof,
notice that on the one hand, the functor $f_{*}$ is left adjoint to the
functor $f^{*}$ and that on the other hand, the functor
$\mathrm{Ind}_{\pi^{\prime}}^{\pi}$ is left adjoint to the functor
$\pi_{1}^{\operatorname{\acute{e}t}}(f)^{*}$ which yields the result. ∎

Since the functor $f^{*}$ (for any morphism of schemes $f$) and the functor
$f_{*}$ (for any finite étale morphism $f$) are compatible with colimits, this
yields the following result.

######  Corollary 1.5.11.

Let $f\colon T\rightarrow S$ be a morphism of schemes, let $\xi^{\prime}$ be a
geometric point of $T$, let $\xi=f\circ\xi^{\prime}$ and let $R$ be a ring.
Denote by
$\pi_{1}^{\operatorname{\acute{e}t}}(f)\colon\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime})\rightarrow\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$
the group homomorphism induced by $f$.

  1. (1)

The functor
$f^{*}\colon\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)\rightarrow\operatorname{\mathcalboondox{Sh}}(T,R)$
induces a functor

$\operatorname{Ind}\mathrm{Loc}_{T}(R)\rightarrow\operatorname{Ind}\mathrm{Loc}_{S}(R)$

which sends an Ind-locally constant sheaf $\operatorname{colim}M_{i}$ to the
Ind-locally constant sheaf
$\operatorname{colim}[(\xi^{\prime})^{*}]^{-1}\left(\pi_{1}^{\operatorname{\acute{e}t}}(f)^{*}(\xi^{*}M_{i})\right)$.

  2. (2)

Assume that $f$ is finite and étale, then, the functor
$f_{*}\colon\operatorname{\mathcalboondox{Sh}}(T,R)\rightarrow\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$
induces a functor

$\operatorname{Ind}\mathrm{Loc}_{T}(R)\rightarrow\operatorname{Ind}\mathrm{Loc}_{S}(R)$

which sends an Ind-locally constant sheaf $\operatorname{colim}M_{i}$ to the
Ind-locally constant sheaf
$\operatorname{colim}[(\xi^{\prime})^{*}]^{-1}\left(\mathrm{Ind}_{\pi^{\prime}}^{\pi}(\xi^{*}M_{i})\right)$
where $\pi^{\prime}=\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime})$ and
$\pi=\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$.

In the case when $S$ is the spectrum of a field $k$, the results of this
paragraph can be formulated in a more satisfying way.

######  Proposition 1.5.12.

Let $k$ be a field and let $R$ be a ring.

  1. (1)

If
$\xi\colon\operatorname{Spec}(\overline{k})\rightarrow\operatorname{Spec}(k)$
is the separable closure of $k$, the fiber functor associated to $\xi$ induces
an equivalence of abelian monoidal categories

$\xi^{*}\colon\operatorname{Sh}(k_{\operatorname{\acute{e}t}},R)\rightarrow\mathrm{Mod}(G_{k},R)$

and an equivalence of monoidal stable categories

$\xi^{*}\colon\operatorname{\mathcalboondox{Sh}}(k_{\operatorname{\acute{e}t}},R)\rightarrow\mathcal{D}(\mathrm{Mod}(G_{k},R)).$

  2. (2)

The functor $\xi^{*}$ induces an equivalence

$\xi^{*}\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(k_{\operatorname{\acute{e}t}},R)\rightarrow\mathcal{D}_{\mathrm{perf}}(\mathrm{Mod}(G_{k},R))$

where $\mathcal{D}_{\mathrm{perf}}(\mathrm{Mod}(G_{k},R))$ denotes the
subcategory of the stable category $\mathcal{D}(\mathrm{Mod}(G_{k},R))$ made
of those complexes which are perfect as complexes of $R$-modules.

  3. (3)

We have

$\operatorname{\mathcalboondox{Sh}}(k_{\operatorname{\acute{e}t}},R)=\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(k_{\operatorname{\acute{e}t}},R)$
$\operatorname{Sh}(k_{\operatorname{\acute{e}t}},R)=\operatorname{Ind}\operatorname{Loc}(k,R).$

  4. (4)

The canonical functor

$\mathcal{D}^{b}(\operatorname{Rep}^{A}(G_{k},R))\rightarrow\mathcal{D}_{\mathrm{perf}}(\mathrm{Mod}(G_{k},R))$

is an equivalence.

###### Proof.

The first assertion follows from [AGV73, VIII.2.1].

The second assertion follows from the first assertion and the description of
lisse sheaves Proposition 1.5.8(1).

Furthermore, any complex of discrete $R$-linear $G_{k}$-module is a (filtered)
colimit of complexes with perfect underlying complex of $R$-modules.
Therefore, we get

$\operatorname{\mathcalboondox{Sh}}(k_{\operatorname{\acute{e}t}},R)=\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(k_{\operatorname{\acute{e}t}},R)$

and since the ordinary t-structure on the stable category
$\operatorname{\mathcalboondox{Sh}}(k_{\operatorname{\acute{e}t}},R)$ is
compatible with filtered colimits, we also get

$\operatorname{Sh}(k_{\operatorname{\acute{e}t}},R)=\operatorname{Ind}\operatorname{Loc}(k,R).$

The last assertion is a consequence of [BGI71, I.4.12]. We give an alternative
proof for the reader’s convenience. Denote by

$\psi\colon\mathcal{D}^{b}(\operatorname{Rep}^{A}(G_{k},R))\rightarrow\mathcal{D}_{\mathrm{perf}}(\mathrm{Mod}(G_{k},R))$

the canonical functor. It suffices to show that the canonical map

$\operatorname{Map}_{\mathcal{D}^{b}(\operatorname{Rep}^{A}(G_{k},R))}(M,N)\rightarrow\operatorname{Map}_{\mathcal{D}(\mathrm{Mod}(G_{k},R)))}(\psi(M),\psi(N))$

is an equivalence for any bounded complexes $M$ and $N$ of Artin
representations of $G_{k}$.

Notice that the objects of $\mathcal{D}^{b}(\operatorname{Rep}^{A}(G_{k},R))$
are dualizable. Therefore, we can assume that $M$ is the $R$-module $R$
endowed with the trivial $G_{k}$-action placed in degree $0$. Furthermore, by
dévissage, we can assume that $N$ is placed in degree $0$.

Therefore, it suffices to show that for all $n$, the map

$\mathrm{Ext}^{n}_{\operatorname{Rep}^{A}(G_{k},R)}(R,N)\rightarrow
H^{n}(G_{k},N)$

is an equivalence.

Now, we have

$H^{n}(G_{k},N)=\operatorname{colim}H^{n}(G_{k}/H,N^{H})$

where $H$ runs through the set of normal subgroups of finite index of $G_{k}$.

Since $N$ is an Artin representation, any normal subgroup $H$ of $G_{k}$
contains a normal subgroup $H^{\prime}$ such that the action of $H^{\prime}$
on $N$ is trivial. Hence, the set of normal subgroups $H$ of finite index of
$G_{k}$ such that $N^{H}=N$ is final in the set of all normal subgroups of
finite index of $G_{k}$.

Therefore, we have $H^{n}(G_{k},N)=\operatorname{colim}H^{n}(G_{k}/H,N)$ where
$H$ runs through the set of normal subgroups of finite index of $G_{k}$ such
that the action of $H$ on $N$ is trivial.

In addition, if $H$ is a normal subgroup of $G_{k}$ which acts trivially on
$N$, the abelian group $H^{n}(G_{k}/H,N)$ coincides with the
$\mathrm{Ext}^{n}_{\operatorname{Rep}^{A}(G_{k}/H,R)}(R,N)$ and can by [Sta23,
06XU] be computed as the abelian group of classes of sequences

$0\rightarrow R\rightarrow M_{1}\rightarrow\cdots\rightarrow M_{n}\rightarrow
N\rightarrow 0$

where each $M_{i}$ is an $R$-linear representation of $G_{k}/H$ and two
sequences $M$ and $M^{\prime}$ are equivalent if there exists a third sequence
$M^{\prime\prime}$ and morphisms of sequences $M\leftarrow
M^{\prime\prime}\rightarrow M^{\prime}$.

Furthermore, using [Sta23, 06XU] again, the abelian group
$\mathrm{Ext}^{n}_{\operatorname{Rep}^{A}(G_{k},R)}(R,N)$ can be computed as
the abelian group of classes of sequences

$0\rightarrow R\rightarrow M_{1}\rightarrow\cdots\rightarrow M_{n}\rightarrow
N\rightarrow 0$

where each $M_{i}$ is an Artin representation of $G_{k}$ and two sequences $M$
and $M^{\prime}$ are equivalent if there exists a third sequence
$M^{\prime\prime}$ and morphisms of sequences $M\leftarrow
M^{\prime\prime}\rightarrow M^{\prime}$.

Any such sequence in fact is a sequence of $R$-linear representation of
$G_{k}/H$ for some normal subgroup $H$ and any equivalence $M\leftarrow
M^{\prime\prime}\rightarrow M^{\prime}$ between two sequences $M$ and
$M^{\prime}$ comes from an equivalence of sequences of $R$-linear
representation of $G_{k}/H$ for some normal subgroup $H$. Therefore, the map

$\operatorname{colim}\mathrm{Ext}^{n}_{\operatorname{Rep}^{A}(G_{k}/H,R)}(R,N)\rightarrow\mathrm{Ext}^{n}_{\operatorname{Rep}^{A}(G_{k},R)}(R,N),$

where $H$ runs through the set of normal subgroups of finite index of $G_{k}$
such that the action of $H$ on $N$ is trivial, is an equivalence. This
finishes the proof. ∎

Propositions 1.5.12 and 1.5.11 yield the following proposition.

######  Proposition 1.5.13.

Let $f\colon\operatorname{Spec}(L)\rightarrow\operatorname{Spec}(K)$ be a
field extension, let
$\xi^{\prime}\colon\operatorname{Spec}(\overline{L})\rightarrow\operatorname{Spec}(L)$
be the separable closure of $L$ and let $\xi=f\circ\xi^{\prime}$. Denote by
$\phi\colon G_{L}\rightarrow G_{K}$ the group homomorphism induced by $f$.

  1. (1)

The square

${\operatorname{Sh}(K_{\operatorname{\acute{e}t}},R)}$${\operatorname{Sh}(L_{\operatorname{\acute{e}t}},R)}$${\mathrm{Mod}(G_{K},R)}$${\mathrm{Mod}(G_{L},R)}$$\scriptstyle{f^{*}}$$\scriptstyle{\xi^{*}}$$\scriptstyle{(\xi^{\prime})^{*}}$$\scriptstyle{\phi^{*}}$

is commutative.

  2. (2)

Assume that $L$ is a finite extension of $K$. Then, the square

$\leavevmode\hbox to169.7pt{\vbox
to53.02pt{\pgfpicture\makeatletter\hbox{\hskip
84.84985pt\lower-26.55817pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-84.84985pt}{-26.45833pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
26.29898pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-21.99344pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\operatorname{Sh}(L_{\operatorname{\acute{e}t}},R)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
26.29898pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
0.0pt\hfil&\hfil\hskip 51.50032pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.19481pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\operatorname{Sh}(K_{\operatorname{\acute{e}t}},R)}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
27.50035pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
30.08856pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.78302pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathrm{Mod}(G_{L},R)}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
30.08856pt\hfil&\hfil\hskip 23.99997pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
0.0pt\hfil&\hfil\hskip 54.76129pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-26.45578pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathrm{Mod}(G_{K},R)}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
30.76132pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-28.26231pt}{11.63892pt}\pgfsys@lineto{25.98822pt}{11.63892pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.1882pt}{11.63892pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-4.02736pt}{15.35278pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{f_{*}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-54.76129pt}{2.50143pt}\pgfsys@lineto{-54.76129pt}{-14.69865pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{-54.76129pt}{-14.89864pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-52.40852pt}{-8.51385pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{(\xi^{\prime})^{*}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{54.08853pt}{2.50143pt}\pgfsys@lineto{54.08853pt}{-14.69865pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{54.08853pt}{-14.89864pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{54.08853pt}{2.50143pt}\pgfsys@lineto{54.08853pt}{-14.69865pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.0}{-1.0}{1.0}{0.0}{54.08853pt}{-14.89864pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{56.4413pt}{-8.51387pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\xi^{*}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-24.47273pt}{-23.95833pt}\pgfsys@lineto{22.72725pt}{-23.95833pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{22.92723pt}{-23.95833pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.48694pt}{-19.14557pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\mathrm{Ind}_{G_{L}}^{G_{K}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $\pi^{\prime}=\pi_{1}^{\operatorname{\acute{e}t}}(T,\xi^{\prime})$ and
$\pi=\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$, is commutative.

####  1.5.4. Generators of lisse and Ind-lisse sheaves

######  Proposition 1.5.14.

Let $S$ be a scheme and let $R$ be a regular ring.

  1. (1)

The thick stable subcategory of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$
generated by the sheaves of the form $R_{S}(X)$ with $X$ finite and étale over
$S$ is $\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$.

  2. (2)

The localizing stable subcategory of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$
generated by the sheaves of the form $R_{S}(X)$ with $X$ finite and étale is
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)$.

###### Proof.

By definition of Ind-lisse sheaves the second assertion follows from the
first.

To prove the first assertion, first note that
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$ is a thick
stable subcategory of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$.
Furthermore, if $X$ is a finite étale $S$-scheme, the sheaf $R_{S}(X)$ is
dualizable with dual itself and is therefore lisse. Hence, the thick stable
subcategory of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$
generated by the sheaves of the form $R_{S}(X)$ with $X$ finite and étale over
$S$ is contained in
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$.

By Proposition 1.5.8, lisse sheaves are bounded with respect to the ordinary
t-structure. Therefore, it suffices by dévissage to prove that any locally
constant sheaf lies in the thick subcategory generated by the sheaves of the
form $R_{S}(X)$ with $X$ finite and étale over $S$ which follows from the
stronger Lemma 1.5.15 below. ∎

######  Lemma 1.5.15.

Let $S$ be a scheme and let $R$ be a regular ring. The category
$\operatorname{Loc}_{S}(R)$ is contained in the smallest subcategory of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ which
contains the sheaves of the form $R_{S}(X)$ with $X$ finite and étale over $S$
and is closed under finite colimits, extensions and retracts.

###### Proof.

We can assume that the scheme $S$ is connected. Let $\xi$ be a geometric point
of $S$ and let $M$ be a locally constant étale sheaf. By Proposition 1.5.9,
the sheaf $M$ corresponds to an Artin representation $N$ of
$\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$ through the fiber functor
associated to $\xi$. Let $G$ be a finite quotient of
$\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$ through which $N$ factors. Then,
by Proposition 1.5.1 the complex $M\oplus M[1]$ of finite $R[G]$-modules is
quasi-isomorphic to a complex of the form

$P=\cdots\rightarrow 0\rightarrow P_{n}\rightarrow
P_{n-1}\rightarrow\cdots\rightarrow P_{0}\rightarrow 0\rightarrow\cdots$

where each $P_{i}$ is placed in degree $-i$ and is isomorphic to
$R[G/H_{1}]\oplus\cdots\oplus R[G/H_{m}]$ for some subgroups
$H_{1},\ldots,H_{m}$ of $G$.

Furthermore, we have an exact functor

$\mathcal{D}^{b}(\operatorname{Rep}(G,R))\rightarrow\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$

and if $H$ is a subgroup of $G$, the finite set $G/H$ is endowed with an
action of $\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi)$ and therefore
corresponds to a finite locally constant sheaf on
$S_{\operatorname{\acute{e}t}}$, and therefore to a finite étale $S$-scheme
$X$. The image of $R[G/H]$ through the functor above is then $R_{S}(X)$.

Thus, the object $M\oplus M[1]$, and therefore the object $M$ itself, belong
to the smallest subcategory which contains the sheaves of the form $R_{S}(X)$
with $X$ finite and étale over $S$ and is closed under finite colimits,
extensions and retracts. ∎

Another consequence of Lemma 1.5.15 is the following statement.

######  Proposition 1.5.16.

Let $S$ be a scheme and let $R$ be a regular ring.

  1. (1)

The smallest subcategory of
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$ which
contains the sheaves of the form $R_{S}(X)$ with $X$ finite and étale over $S$
and is closed under finite colimits, extensions and retracts is the
subcategory
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}^{\leqslant 0}(S,R)$
of t-non-positive objects of the ordinary t-structure on
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$.

  2. (2)

The ordinary t-structure on the stable category
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)$
is generated by the sheaves of the form $R_{S}(X)$ with $X$ finite and étale
over $S$.

###### Proof.

By dévissage, the first assertion follows from Lemma 1.5.15.

Let $M$ be a t-non-positive Ind-lisse étale sheaf. Write $M$ as a filtered
colimit of lisse étale sheaves $M_{i}$. Since the ordinary t-structure is
compatible with filtered colimits, we have

$M=\tau_{\leqslant 0}(M)=\tau_{\leqslant
0}(\operatorname{colim}M_{i})=\operatorname{colim}\tau_{\leqslant 0}(M_{i}).$

Using the first assertion, each sheaf $\tau_{\leqslant 0}(M_{i})$ belongs to
the smallest subcategory of
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)$
which contains the sheaves of the form $R_{S}(X)$ with $X$ finite and étale
over $S$ and is closed under small colimits and extensions. Therefore, so does
the sheaf $M$. This finishes the proof. ∎

######  Proposition 1.5.17.

Let $S$ be a scheme and let $R$ be a regular ring. Then, with the notations of
Section 1.1.5, the canonical functor

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)_{\mathfrak{p}}\rightarrow\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R\otimes_{\mathbb{Z}}\mathbb{Z}_{\mathfrak{p}})$

is an equivalence.

###### Proof.

The proof is the same as the proof of Proposition 1.2.4. ∎

###  1.6. Reminders on the $\ell$-adic theory

In this section, we recall the $\ell$-adic formalism. We also recall the main
definitions and the main results of [Rui22].

Let $S$ be a scheme and $\ell$ be a prime number invertible on $S$. Let
$\Lambda$ be a finite extension of $\mathbb{Q}_{\ell}$ or the ring of integers
of a finite extension of $\mathbb{Q}_{\ell}$. There are several definitions of
the _derived category of constructible $\ell$-adic sheaves_
$\mathcal{D}^{b}_{c}(S,\Lambda)$. The first definition was given in [BBDG18]
for $S$ of finite type over a field. In [Eke07] [CD16, 7.2], [BS15] and
[LYF14, XIII.4], this definition was extended to all quasi-compact quasi-
separated schemes.

The definition that was chosen in [Rui22] is the definition of [BS15]. If $S$
is a scheme, denote by $S_{\mathrm{pro\acute{e}t}}$ the proétale site of $S$
introduced in [BS15, 4]. Any totally disconnected topological space $M$ gives
rise to a constant sheaf on $S_{\mathrm{pro\acute{e}t}}$ with value $M$
denoted by $M_{S}$. The derived category of $\ell$-adic sheaves
$\mathcal{D}(S,\Lambda)$ is defined as the derived category of the category
$\operatorname{Sh}(S,\Lambda)$ of sheaves of $\Lambda_{S}$ modules on the site
$S_{\mathrm{pro\acute{e}t}}$.

####  1.6.1. $\ell$-adic realization

Let $S$ be a scheme, let $R$ be a localization of the ring of integers of a
number field $K$, let $v$ be a non-archemedian valuation on the field $K$ and
let $\ell$ be the prime number such that $v$ extends the $\ell$-adic
valuation. Assuming that the prime number $\ell$ is invertible on $S$,
Cisinski and Déglise defined an $\ell$-adic realization functor, or in this
context more accurately a $v$-adic realization functor

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

This functor preserves constructible objects and respects the six functors
formalism.

To describe this $v$-adic realization functor, we must use the description of
$\ell$-adic complexes given in [CD16, 7.2]. Assume first $R$ is a discrete
valuation ring with valuation $v$ and let $\lambda$ be a uniformizer of $R$.

Recall that an étale motive $M$ is _$\lambda$ -complete_ if the canonical map

$M\rightarrow\lim M/\lambda^{n}$

is an equivalence. Cisinski and Déglise defined the derived category of
$\lambda$-adic sheaves as the full subcategory
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,\hat{R}_{\lambda})$
of the stable category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)$made of those
motives which are $\lambda$-complete.

We can define the subcategory
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)_{\lambda}$
of $\lambda$-complete objects in the stable category
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ in a
similar fashion. The proof of [CD16, 7.2.21] implies that the map $\rho_{!}$
of Section 1.2.4 induces an equivalence

$\operatorname{\mathcalboondox{Sh}}_{\lambda}(S,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,\hat{R}_{\lambda})$

with inverse the functor $\rho^{!}$.

Cisinski and Déglise then define the $\lambda$-adic realization functor

$\rho_{\lambda}^{\mathrm{CD}}\colon\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,\hat{R}_{\lambda})$

as the functor that sends a motive $M$ to the motive $\lim M/\lambda^{n}$.
Finally, they defined the subcategory
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}(S,\hat{R}_{\lambda})$
of _constructible $\lambda$-complete étale motives_ as the thick subcategory
of the stable category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,\hat{R}_{\lambda})$
generated by the essential image through the functor
$\rho_{\lambda}^{\mathrm{CD}}$ of the subcategory of constructible motives.

We now explain the link between the models of [CD16, 7.2] and [BS15] for
$\ell$-adic complexes. On the one hand, we can define the subcategory
$\operatorname{\mathcalboondox{Sh}}_{\lambda,c}(S,R)$ of _constructible
$\lambda$-complete étale sheaves_ as the essential image of the stable
subcategory
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}(S,\hat{R}_{\lambda})$
of the stable category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,\hat{R}_{\lambda})$
through the functor $\rho^{!}$.

On the other hand, let

$\nu:S_{\mathrm{pro\acute{e}t}}\rightarrow S_{\operatorname{\acute{e}t}}$

be the canonical morphism of sites. Then, unwinding the definitions of the
maps described in the proofs of [CD16, 7.2.21] and [BS15, 5.5], the map

$\Psi\colon\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)_{\lambda}\rightarrow\mathcal{D}(S,R_{v})$

which sends the étale sheaf $M$ to the proétale sheaf
$\lim\nu^{*}(M)/\lambda^{n}$ and induces an equivalence

$\operatorname{\mathcalboondox{Sh}}_{\lambda,c}(S,R)\rightarrow\mathcal{D}^{b}_{c}(S,R_{v}).$

We can therefore define a $v$-adic realization functor

$\rho_{v}=\Psi\circ\rho^{!}\circ\rho_{\lambda}^{\mathrm{CD}}\colon\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)\rightarrow\mathcal{D}(S,R_{v}).$

In the case when $v$ is non-negative on $R$, we can define the $v$-adic
realization functor by pre-composing the $v$-adic realization functor

$\rho_{v}\colon\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R_{(v)})\rightarrow\mathcal{D}(S,R_{v})$

with the change of coefficients from $R$ to $R_{(v)}$.

In the case when $v$ is not non-negative $R$, we can only define the
realization functor on the subcategory of constructible motives. Notice that
$R_{v}$ is the completion of $K$ with respect to the valuation $v$. Let

$A=\\{x\in K\mid v(x)\geqslant 0\\}.$

The ring $A$ is a discrete valuation ring. Let $\lambda$ be a uniformizer of
$A$. We define the $v$-adic realization functor $\rho_{v}$ as the composition

$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}(S,R)\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}(S,K)\xrightarrow{\rho_{\lambda}\otimes_{A}K}\mathcal{D}^{b}_{c}(S,R_{v}),$

where, with the notations of Section 1.1.5, Proposition 1.2.4 ensures that the
functors

$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}(S,A)_{(0)}\rightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}(S,K)$

and where [CD16, 7.2.23] ensures that the functor

$\mathcal{D}^{b}_{c}(S,A_{v})_{(0)}\rightarrow\mathcal{D}^{b}_{c}(S,K_{v})$

is an equivalence.

####  1.6.2. Ordinary t-structure

In the rest of this section, we let $S$ be a scheme, we let$\ell$ be a prime
number invertible on $S$ and we let $\Lambda$ be a finite extension of
$\mathbb{Q}_{\ell}$ or the ring of integers of a finite extension of
$\mathbb{Q}_{\ell}$.

The ordinary t-structure of the stable category $\mathcal{D}(S,\Lambda)$
induces a t-structure on the subcategory $\mathcal{D}^{b}_{c}(S,\Lambda)$. The
heart of this t-structure is the abelian category
$\operatorname{Sh}_{c}(S,\Lambda)$ _constructible $\ell$-adic sheaves_.

Following [HRS20], the ordinary t-structure of the stable category
$\mathcal{D}(S,\Lambda)$ also induces a t-structure on the subcategory
$\mathcal{D}_{\operatorname{lisse}}(S,\Lambda)$ of dualizable objects of
stable category $\mathcal{D}(S,\Lambda)$. The heart of this t-structure is the
abelian category $\operatorname{Loc}_{S}(\Lambda)$ of locally (with respect to
the proétale topology) constant sheaves with finitely presented fibers.

We can give a more concrete description of the abelian category
$\operatorname{Loc}_{S}(\Lambda)$. The topology on $\Lambda$ induces a
topology on every $\Lambda$-module of finite presentation. Letting $\pi$ be a
topological group, a _continuous representation_ of $\pi$ with coefficients in
$\Lambda$ is a $\Lambda$-module of finite presentation endowed with a
continuous action of $\pi$. We denote by $\mathrm{Rep}(\pi,\Lambda)$ the
abelian category of continuous representation of $\pi$ with coefficients in
$\Lambda$. Then, if the scheme $S$ is connected and if $\xi$ is a geometric
point of $S$, [Rui22, 1.5.4] implies that the fiber functor $\xi^{*}$
associated to $\xi$ induces an equivalence

$\operatorname{Loc}_{S}(\Lambda)\rightarrow\operatorname{Rep}(\pi_{1}^{\mathrm{pro\acute{e}t}}(S,\xi),\Lambda)$

where $\pi_{1}^{\mathrm{pro\acute{e}t}}(S,\xi)$ is the proétale fundamental
group of Bhatt and Scholze defined in [BS15, 7].

####  1.6.3. Perverse t-structure

Assume that the scheme $S$ is endowed with a dimension function $\delta$. If
$x$ is a point of $S$, denote by $i_{x}\colon\\{x\\}\rightarrow S$ the
inclusion. We define

  * •

the full subcategory ${}^{p\leqslant 0}\mathcal{D}^{b}_{c}(S,\Lambda)$ as the
subcategory made of those $\ell$-adic complexes $M$ such that for any point
$x$ of $S$, we have

$i_{x}^{*}M\leqslant-\delta(x)$

with respect to the ordinary t-structure of
$\mathcal{D}^{b}_{c}(k(x),\Lambda)$.

  * •

the full subcategory ${}^{p\geqslant 0}\mathcal{D}^{b}_{c}(S,\Lambda)$ as the
subcategory made of those bounded below $\ell$-adic complexes $M$ such that
for any point $x$ of $S$, we have

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

with respect to the ordinary t-structure of
$\mathcal{D}^{b}_{c}(k(x),\Lambda)$.

Gabber’s proof in [Gab08] can be used to prove that $\delta$ gives rise to a
perverse t-structure on $\mathcal{D}^{b}_{c}(S,\Lambda)$ which we call the
_perverse t-structure_. The heart of this t-structure is the abelian category
of _perverse sheaves_ denoted by $\mathrm{Perv}(S,\Lambda)$.

We denote with the small letter $p$ the notions related to the perverse
t-structure on $\mathcal{D}^{b}_{c}(S,\Lambda)$. For instance, we denote by
${}^{p}\tau_{\leqslant 0}$ is the left truncation with respect to the perverse
t-structure.

Finally, if the scheme $S$ is regular, the perverse t-structure induces a
t-structure on the stable subcategory
$\mathcal{D}_{\operatorname{lisse}}(S,\Lambda)$ which coincides with the
ordinary t-structure shifted by $\delta(S)$.

####  1.6.4. Artin $\ell$-adic complexes

If $X$ is a scheme, recall that $\Lambda_{X}$ denotes the constant sheaf with
value $\Lambda$ on $X$. The following $\ell$-adic analogs of Artin motives and
smooth Artin motives are defined in [Rui22, 2.1.3].

  * •

The stable category $\mathcal{D}^{A}(S,\Lambda)$ of _Artin $\ell$-adic
complexes_ is the thick stable subcategory of $\mathcal{D}^{b}_{c}(S,\Lambda)$
generated by the complexes of the form $f_{*}\mathbb{Z}_{\ell,X}$ with
$f\colon X\rightarrow S$ finite.

  * •

The stable category $\mathcal{D}^{smA}(S,\Lambda)$ of _smooth Artin
$\ell$-adic complexes_ over $S$ with coefficients $\Lambda$ is the thick
stable subcategory of $\mathcal{D}^{b}_{c}(S,\Lambda)$ generated by the
complexes of the form $f_{*}\Lambda_{X}$ with $f\colon X\rightarrow S$ étale
and finite.

  * •

The stable category $\mathcal{D}^{A}_{\operatorname{Ind}}(S,\Lambda)$ is the
localizing subcategory of $\mathcal{D}(S,\Lambda)$ generated by the complexes
of the form $f_{*}\mathbb{Z}_{\ell,X}$ with $f\colon X\rightarrow S$ finite.

Let $R$ be a localization of the ring of integers of a number field, and let
$v$ be a valuation on the fraction field of $R$ such that $v(\ell)$ is
positive. Notice that the $\ell$-adic realization functor induces functors

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

####  1.6.5. Artin $\ell$-adic sheaves

The ordinary t-structure of the stable category
$\mathcal{D}^{b}_{c}(S,\Lambda)$ induces a structure on the stable subcategory
$\mathcal{D}^{smA}(S,\Lambda)$ and on the stable subcategory
$\mathcal{D}^{A}(S,\Lambda)$. We can describe the heart of those t-structures
which are weak Serre subcategories of $\operatorname{Sh}_{c}(S,\Lambda)$.

Letting $\pi$ be a topological group, the category of representation of _Artin
origin_ of $\pi$, denoted by $\operatorname{Rep}^{A}(\pi,\Lambda)^{*}$, is the
smallest Serre subcategory of $\operatorname{Rep}(\pi,\Lambda)$ which contains
Artin representations. It is made of those objects of
$\operatorname{Rep}(\pi,\Lambda)$ obtained as successive extensions of Artin
representations. The abelian category $\operatorname{Sh}^{smA}(S,\Lambda)$ of
_smooth Artin $\ell$-adic sheaves_ is then defined as the subcategory of the
abelian category $\operatorname{Loc}_{S}(\Lambda)$ made of those sheaves whose
image through the fiber functor associated to any geometric point $\xi$ of $S$
in a representation of Artin origin of the topological group
$\pi_{1}^{\mathrm{pro\acute{e}t}}(S,\xi)$. When the scheme $S$ is connected,
the fiber functor associated to any geometric point $\xi$ of $S$ induces an
equivalence

$\operatorname{Sh}^{smA}(S,\Lambda)\rightarrow\operatorname{Rep}^{A}(\pi_{1}^{\mathrm{pro\acute{e}t}}(S,\xi),\Lambda)^{*}.$

With those definitions, [Rui22, 2.2.1] ensures that the stable category
$\mathcal{D}^{smA}(S,\Lambda)$ is the subcategory of the stable category
$\mathcal{D}^{b}_{c}(S,\Lambda)$ made of those objects whose (ordinary)
cohomology sheaves are smooth Artin and that therefore, the ordinary
t-structure on the stable category $\mathcal{D}^{b}_{c}(S,\Lambda)$ induces a
t-structure on the stable subcategory $\mathcal{D}^{smA}(S,\Lambda)$ whose
heart is the abelian category $\operatorname{Sh}^{smA}(S,\Lambda)$.

Finally, the abelian category $\operatorname{Sh}^{A}(S,\Lambda)$ of _Artin
$\ell$-adic sheaves_ is defined as the subcategory of the abelian category
$\operatorname{Sh}(S,\Lambda)$ made of those objects which are locally
constant on every stratum of some stratification of $S$. [Rui22, 2.4.3]
implies that the stable category $\mathcal{D}^{A}(S,\Lambda)$ is then the
subcategory of the stable category $\mathcal{D}^{b}_{c}(S,\Lambda)$ made of
those objects whose cohomology sheaves are Artin. Therefore, the ordinary
t-structure on the stable category $\mathcal{D}^{b}_{c}(S,\Lambda)$ induces a
t-structure on the stable subcategory $\mathcal{D}^{A}(S,\Lambda)$ whose heart
is the abelian category $\operatorname{Sh}^{A}(S,\Lambda)$.

####  1.6.6. Artin perverse sheaves

First, if $S$ is connected and regular, then, by [Rui22, 3.1.1] the perverse
t-structure on the stable category $\mathcal{D}^{b}_{c}(S,\Lambda)$ induces
t-structures on the stable subcategory $\mathcal{D}^{smA}(S,\Lambda)$ and,
letting $\delta$ be a dimension function on $S$, this t-structure coincides
with the ordinary t-structure shifted by $\delta(S)$.

If $S$ is an excellent scheme of dimension $0$, $1$ or $2$, or a scheme of
dimension $3$ such that all the residue fields of its closed points are real
closed or separably closed, then, by [Rui22, 3.3.4, 3.4.1] the perverse
t-structure on $\mathcal{D}^{b}_{c}(S,\Lambda)$ induces a t-structure on
$\mathcal{D}^{A}(S,\Lambda)$. We denote its heart by
$\mathrm{Perv}^{A}(S,\Lambda)$ and we call it the abelian category of _Artin
perverse sheaves_.

####  1.6.7. Perverse homotopy t-structure

If $S$ excellent, letting $\delta$ be a dimension function on $S$, we define
the _perverse homotopy t-structure_ on
$\mathcal{D}^{A}_{\operatorname{Ind}}(S,\Lambda)$ as the t-structure generated
by the complexes of the form $f_{*}\Lambda_{X}[\delta(X)]$ with $f\colon
X\rightarrow S$ finite.

If $S$ is connected and regular. Then, by [Rui22, 4.3.1], the perverse
homotopy t-structure on the stable category
$\mathcal{D}^{A}_{\operatorname{Ind}}(S,\Lambda)$ both induce t-structures on
the stable subcategory $\mathcal{D}^{smA}(S,\Lambda)$. In addition, this
t-structure coincides with the perverse t-structure. Moreover, [Rui22, 4.5.1]
yields the following statements.

  * •

Let $M$ be an object of $\mathcal{D}^{A}(S,\Lambda)$, then $M$ is perverse
homotopy t-negative if and only if it is perverse t-negative.

  * •

If the perverse t-structure on $\mathcal{D}^{b}_{c}(S,\Lambda)$ induces a
t-structure on $\mathcal{D}^{A}(S,\Lambda)$, then the perverse homotopy
t-structure on $\mathcal{D}^{A}_{\operatorname{Ind}}(S,\Lambda)$ induces a
t-structure on $\mathcal{D}^{A}(S,\Lambda)$. When this is the case, those two
t-structures coincide.

Therefore, when the perverse homotopy t-structure induces a t-structure on
$\mathcal{D}^{A}(S,\Lambda)$, the induced t-structure is the best possible
approximation of the perverse t-structure.

Finally, if $S$ is a scheme of finite type over $\mathbb{F}_{p}$ and if
$\Lambda$ is a finite extension of $\mathbb{Q}_{\ell}$, then [Rui22, 4.7.1]
ensures that the perverse homotopy t-structure on
$\mathcal{D}^{A}_{\operatorname{Ind}}(S,\Lambda)$ induces a t-structure on
$\mathcal{D}^{A}(S,\Lambda)$. We denote its heart by
$\mathrm{Perv}^{A}(S,\Lambda)^{\\#}$ and we call it the abelian category of
_Artin homotopy perverse sheaves_.

###  1.7. Some computations of étale cohomology and étale motivic cohomology

Recall that the étale motivic cohomology groups of a scheme $X$ are the groups

$H^{n,m}_{\mathcal{M},\operatorname{\acute{e}t}}(X,R)=\operatorname{Hom}_{\operatorname{DM}_{\operatorname{\acute{e}t}}(X,R)}(\mathbbm{1}_{X},\mathbbm{1}_{X}[n](m)).$

We will need the following computations. The first one is a slight
generalization of a [Ayo14, 11.1].

######  Lemma 1.7.1.

Let $S$ be a scheme, let $R$ be a ring and let $n<0$ and $m\leqslant 0$ be
integers. Then,

$H^{n,m}_{\mathcal{M},\operatorname{\acute{e}t}}(S,R)=0.$

###### Proof.

By dévissage (see [Ayo14, 11.1, Étape A]), we are reduced to the case where
$R=\mathbb{Z}/p\mathbb{Z}$ or is a $\mathbb{Q}$-algebra.

In the first case, by the rigidity theorem [CD16, 4.1.2], the category
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,\mathbb{Z}/p\mathbb{Z})$
is equivalent to the category
$\mathcal{D}(S[1/p]_{\operatorname{\acute{e}t}},\mathbb{Z}/p\mathbb{Z})$ and
we have

$H^{n,m}_{\mathcal{M},\operatorname{\acute{e}t}}(X,\mathbb{Z}/p\mathbb{Z})=H^{n}_{\operatorname{\acute{e}t}}(X,\mathbb{Z}/p\mathbb{Z}(m))$

which vanishes since the integer $n$ is negative. In the case when the ring
$R$ is a $\mathbb{Q}$-algebra, use Ayoub’s result. ∎

We will also need to compute étale cohomology with invertible on the base
scheme and constant coefficients. First recall:

######  Proposition 1.7.2.

(Deninger) Let $X$ be a geometrically unibranch scheme. Then, we have

$H^{i}_{\operatorname{\acute{e}t}}(X,\mathbb{Q})=\begin{cases}\mathbb{Q}^{\pi_{0}(X)}\text{
if }i=0.\\\ 0\text{ otherwise.}\end{cases}$

###### Proof.

By [Den88, 2.1], this result is true when the scheme $X$ is normal. If $X$ is
a general geometrically unibranch scheme, the normalization map $\nu\colon
X^{\nu}\rightarrow X$ is a universal homeomorphism by [Sta23, 0GIQ] and
therefore, the lemma follows from the topological invariance of the small
étale site [Gro67, 18.1.2]. ∎

If $X$ is a scheme and $\ell$ is a prime number which is invertible on $X$,
write $b_{n}(X,\ell)$ the dimension (over $\mathbb{Q}_{\ell}$) of the $n$-th
$\ell$-adic cohomology group of $X$. Let $T_{n}(X,\ell)$ be the torsion
subgroup of the group
$H^{n}_{\operatorname{\acute{e}t}}(X,\mathbb{Z}_{\ell})$. Finally denote by
$\mathbb{Z}(\ell^{\infty})$ the Prüfer $\ell$-group. Then we have the
following result.

######  Proposition 1.7.3.

Let $X$ be a geometrically unibranch scheme, let $R$ be a ring which is flat
over $\mathbb{Z}$ and in which all the residue characteristics of $X$ are
invertible and let $e(X)$ be the set of residue characteristic exponents of
$X$. Then,

$H^{n}_{\operatorname{\acute{e}t}}(X,R)=\begin{cases}R^{\pi_{0}(X)}&\text{ if
}n=0\\\ 0&\text{ if }n=1\\\
R\otimes_{\mathbb{Z}}\left[\bigoplus\limits_{\ell\notin
e(X)}\left(\mathbb{Z}(\ell^{\infty})^{b_{n-1}(X,\ell)}\oplus
T_{n-1}(X,\ell)\right)\right]&\text{ if }n\geqslant 2\end{cases}$

###### Proof.

As $R$ is flat over $\mathbb{Z}$, we may suppose that $R$ is the subring

$\mathbb{Z}[1/e(X)]=\left\\{\frac{n}{p_{1}^{m_{1}}\cdots p_{r}^{m_{r}}}\mid
p_{i}\in e(X)\right\\}$

of $\mathbb{Q}$. We have an exact sequence

$0\rightarrow\mathbb{Z}[1/e(X)]\rightarrow\mathbb{Q}\rightarrow\bigoplus\limits_{\ell\notin
e(X)}\mathbb{Z}(\ell^{\infty})\rightarrow 0.$

Hence, by Proposition 1.7.2, we get

$H^{n}_{\operatorname{\acute{e}t}}(X,\mathbb{Z}[1/e(X)])=\begin{cases}\mathbb{Z}[1/e(X)]^{\pi_{0}(X)}&\text{
if }n=0\\\ 0&\text{ if }n=1\\\ \bigoplus\limits_{\ell\notin
e(X)}\operatorname{colim}_{N}H^{n-1}_{\operatorname{\acute{e}t}}(X,\mathbb{Z}/\ell^{N}\mathbb{Z})&\text{
if }n\geqslant 2\end{cases}$

This yields the result. ∎

######  Remark 1.7.4.

If a complete theory of abelian motives was available, all the $b_{i}(X,\ell)$
would coincide. Thus, for $n\geqslant 2$, we would have

$H^{n}_{\operatorname{\acute{e}t}}(X,R)=R\otimes_{\mathbb{Z}}\left[\mathbb{Q}/\mathbb{Z}[1/e(X)]^{b_{i}(X)}\oplus\bigoplus\limits_{\ell\notin
e(X)}T_{n-1}(X,\ell)\right]$

Notice that, using Artin’s Theorem [AGV73, XI.4.4] or Deligne’s work on Weil’s
conjectures (see [Del80]) all the $b_{i}(X,\ell)$ coincide if $X$ is defined
over an algebraically closed field of characteristic $0$ or over
$\overline{\mathbb{F}}_{p}$.

Finally, if $X$ is a curve over any separably closed field $k$, using [AGV73,
IX.4.7], we have

$H^{2}_{\operatorname{\acute{e}t}}(X,R)=(R\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[1/p])^{2g}$
$H^{3}_{\operatorname{\acute{e}t}}(X,R)=R\otimes_{\mathbb{Z}}\mathbb{Q}/\mathbb{Z}[1/p]$

and $H^{n}_{\operatorname{\acute{e}t}}(X,R)=0$ for $n\geqslant 4$.

##  2\. Description of smooth Artin étale motives and applications

###  2.1. Equivalence between smooth Artin étale motives and lisse étale
sheaves over a regular basis

In this paragraph, we give an explicit description of smooth Artin étale
motives over a regular basis. This description will not work over a non-
regular basis, even when it is normal (see Remark 2.1.2 below).

######  Theorem 2.1.1.

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 $R$.

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

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

###### Proof.

We can assume the scheme $S$ to be connected. If $V$ is an étale cover of $S$,
the functor $\rho_{!}$ sends the sheaf $R_{S}(V)$ to the motive $M_{S}(V)$.
Using Proposition 1.5.14, the fact that the ring $R$ is regular and the fact
that $\rho_{!}$ commutes with colimits and finite limits, it induces exact
functors

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

Assuming that the second functor is an equivalence, the first functor is also
an equivalence since
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$ is the thick
stable subcategory of
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)$
generated by the sheaves of the form $R_{S}(V)$ when $V$ runs through the set
of étale covers of $S$ and
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)$ is the
thick stable subcategory of
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$ generated
by the motives of the form $M_{S}(V)$ when $V$ runs through the set of étale
covers of $S$.

We now prove that the functor

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

induced by $\rho_{!}$ is an equivalence. Assuming that the functor $\rho_{!}$
is fully faithful, its essential surjectivity follows from the fact that
$\mathcal{DM}^{smA}_{\operatorname{\acute{e}t}}(S,R)$ is generated as a
localizing subcategory of itsef by the essential image of the functor
$\rho_{!}$. Hence, it suffices to prove that the functor above is fully
faithful.

Let $\rho^{!}$ be the right adjoint of $\rho_{!}$. The full faithfulness of
$\rho_{!}$ holds for any good ring. It is equivalent to the fact that the
ordinary transformation $\alpha\colon Id\rightarrow\rho^{!}\rho_{!}$ is an
equivalence. Using Lemmas 1.2.5 and 1.2.3, we may assume that the ring $R$ is
either of $\ell$-torsion where $\ell$ is invertible in $S$, or a
$\mathbb{Q}$-algebra. The case where $R$ is of $\ell$-torsion follows from
[CD16, 4.1.2]. Therefore, we can assume that the ring $R$ is a
$\mathbb{Q}$-algebra.

The family of functors $\operatorname{Map}(R_{S}(W),-)$ when $W$ runs through
the set of étale $S$-schemes is conservative. Therefore, it suffices to show
that for any Ind-lisse étale sheaf $M$ and any étale $S$-scheme $W$, the
canonical map

$\operatorname{Map}(R_{S}(W),M)\rightarrow\operatorname{Map}(R_{S}(W),\rho^{!}\rho_{!}M)$

is an equivalence. This is equivalent to the fact that the canonical map

$\operatorname{Map}(R_{S}(W),M)\rightarrow\operatorname{Map}(M_{S}(W),\rho_{!}M)$

is an equivalence.

Since $R$ is a $\mathbb{Q}$-algebra, by [CD16, 1.1.4, 1.1.9] the sheaf
$R_{S}(W)$ is a compact object of
$\operatorname{\mathcalboondox{Sh}}(S_{\operatorname{\acute{e}t}},R)$ and the
motive $M_{S}(W)$ is a compact object of
$\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}(S,R)$. As the stable
category
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)$
generated as a localizing subcategory of itself by objects of the form
$R_{S}(V)$ where $V$ is finite étale over $S$, it therefore suffices to show
that for any étale finite scheme $V$ over $S$ and any étale scheme $W$ over
$S$, the map

$\rho_{W,V}\colon\operatorname{Map}(R_{S}(W),R_{S}(V))\rightarrow\operatorname{Map}(M_{S}(W),M_{S}(V))$

is an equivalence.

Recall now, that the étale sheaf $R_{S}(V)$ is dualizable with itself as a
dual. As the functor $\rho_{!}$ is monoidal, this implies that the motive
$M_{S}(V)$ is also dualizable with itself as a dual. Hence, the map
$\rho_{W,V}$ is an equivalence if and only if the map $\rho_{W\times_{S}V,S}$
is an equivalence. As

$\rho_{X,S}=\bigoplus\limits_{T\in\pi_{0}(X)}\rho_{T,S},$

it suffices to show that the maps $\rho_{T,S}$ are equivalences for $T$ étale
over $S$ and connected. But if $T$ is étale over $S$, the map $\rho_{T,S}$ is
an equivalence if and only if the canonical map

$\psi_{T}\colon\operatorname{Map}(R_{T}(T),R_{T}(T))\rightarrow\operatorname{Map}(\mathbbm{1}_{T},\mathbbm{1}_{T})$

is an equivalence.

Such a $T$ is connected and geometrically unibranch. Therefore, using
Proposition 1.7.2, we have

$\displaystyle\pi_{n}(\operatorname{Map}(R_{T}(T),R_{T}(T)))$
$\displaystyle=H^{n}_{\operatorname{\acute{e}t}}(T,R)$
$\displaystyle=\begin{cases}R&\text{ if }n=0.\\\ 0&\text{
otherwise}.\end{cases}$

Moreover, as such a $T$ is also regular (and connected), using [Leh19b, B.5],

$\displaystyle\pi_{n}(\operatorname{Map}(\mathbbm{1}_{T},\mathbbm{1}_{T}))$
$\displaystyle=H^{n,0}_{\mathcal{M},\operatorname{\acute{e}t}}(T,R)$
$\displaystyle=\begin{cases}R&\text{ if }n=0.\\\ 0&\text{
otherwise}.\end{cases}$

Thus, it is sufficient to show that the map $\pi_{0}(\psi_{T})$ is non-zero in
order to show that it is an equivalence. But $\pi_{0}(\psi_{T})$ is the
natural map

$\operatorname{Hom}(R_{T}(T),R_{T}(T))\rightarrow\operatorname{Hom}(\mathbbm{1}_{T},\mathbbm{1}_{T})$

and this map sends the identity of $R_{T}(T)$ to the identity of
$\mathbbm{1}_{T}$. ∎

######  Remark 2.1.2.

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

Then, the functor $\theta_{!}=L_{{\mathbb{A}^{1}}}\rho_{\\#}$ (see Sections
1.2.2 and 1.2.4) induces monoidal equivalences

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)\longrightarrow\operatorname{\mathcal{DA}}^{\operatorname{eff},smA}_{\operatorname{\acute{e}t}}(S,R)$
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)\longrightarrow\operatorname{\mathcal{DA}}^{\operatorname{eff},smA}_{\operatorname{\acute{e}t},c}(S,R)$

Indeed, for the same reasons as in the proof of Theorem 2.1.1, we reduce to
showing that the natural maps

$\operatorname{Map}(R_{S}(T),R_{S}(S))\rightarrow\operatorname{Map}(L_{{\mathbb{A}^{1}}}R_{S}(T),L_{{\mathbb{A}^{1}}}R_{S}(S))$

are equivalences for $T$ connected and étale over $S$ and $R$ a
$\mathbb{Q}$-algebra (beware that on the right hand side, $R_{S}(T)$ and
$R_{S}(S)$ are seen as sheaves on $\operatorname{Sm}_{S}$). This follows from
Lemma 2.1.3 below.

Hence, if $S$ is regular, the functor

$\Sigma^{\infty}\colon\operatorname{\mathcal{DA}}^{\operatorname{eff},smA}_{\operatorname{\acute{e}t}}(S,R)\rightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$

is an equivalence. This is not true in general if $S$ is normal but non-
regular since [Leh19b, B.7] provides an example where étale cohomology in
degree $n$ differs from motivic cohomology in degree $(n,0)$ for some integer
$n$. In particular Theorem 2.1.1 is false when the scheme $S$ is not regular.

######  Lemma 2.1.3.

Let $S$ be a scheme, let $X$ be a geometrically unibranch smooth $S$-scheme
and let $R$ be a regular good ring, where all residue characteristics of $X$
are invertible. Then, the étale sheaf $R_{S}(X)$ on $\operatorname{Sm}_{S}$ is
${\mathbb{A}^{1}}$-local.

###### Proof.

The functor $L_{{\mathbb{A}^{1}}}$ commutes with the functors
$-\otimes_{\mathbb{Z}}\mathbb{Q}$ and
$-\otimes_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$ by Proposition 1.2.3. Using
[AGV73, XV.2.2] and Proposition 1.7.2, The lemma follows Lemma 1.2.5 applied
to the canonical map $R_{S}(X)\rightarrow L_{{\mathbb{A}^{1}}}R_{S}(X)$. ∎

We now state some consequences of Theorem 2.1.1.

######  Corollary 2.1.4.

Let $R$ be a regular good ring and let $S$ be a connected regular scheme.
Assume that any residue characteristic exponent of $S$ is invertible in $R$.
Then, the canonical map

$H^{n}_{\operatorname{\acute{e}t}}(S,R)\rightarrow
H^{n,0}_{\mathcal{M},\operatorname{\acute{e}t}}(S,R)$

is an isomorphism.

Combining Theorem 2.1.1 with Propositions 1.5.8 and 1.5.9 yields the following
statement.

######  Corollary 2.1.5.

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 $R$.

  1. (1)

The category
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$ is
endowed with a non-degenerate t-structure such that the functor

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

induced by the functor $\rho_{!}$ of Proposition 1.4.3 is t-exact when the
left hand side is endowed with the ordinary t-structure. We still call this
t-structure the _ordinary t-structure_. Its heart is equivalent to the abelian
category $\operatorname{Ind}\operatorname{Loc}(S,R)$.

  2. (2)

The category
$\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)$ is
endowed with a non-degenerate t-structure such that the functor

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

induced by the functor $\rho_{!}$ of Proposition 1.4.3 is t-exact when the
left hand side is endowed with the ordinary t-structure. We still call this
t-structure the _ordinary t-structure_. Its heart is equivalent to the abelian
category $\operatorname{Loc}_{S}(R)$. If $S$ is connected and $\xi$ is a
geometric point of $S$, it is also equivalent to the abelian category
$\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi),R)$.

If $n$ is an integer, we denote by $H^{n}$ the n-th cohomology functor with
respect to the ordinary t-structure on smooth Artin motives.

######  Corollary 2.1.6.

Let $R$ be a regular good ring and let $S$ be a connected regular scheme of
characteristic exponent $p$. Then, the functor $\rho_{!}$ of Proposition 1.4.3
induces monoidal equivalences

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R[1/p])\longrightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)$
$\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R[1/p])\longrightarrow\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R).$

######  Corollary 2.1.7.

Let $k$ be a field of characteristic exponent $p$. Let $R$ be a regular good
ring. Then, the functor $\rho_{!}$ induces monoidal equivalences

$\mathcal{D}(\mathrm{Mod}(G_{k},R[1/p]))\simeq\operatorname{\mathcalboondox{Sh}}(k_{\operatorname{\acute{e}t}},R[1/p])\longrightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t}}^{A}(k,R)$
$\mathcal{D}^{b}(\operatorname{Rep}^{A}(G_{k},R[1/p]))\simeq\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(k,R[1/p])\longrightarrow\operatorname{\mathcal{DM}}_{\operatorname{\acute{e}t},c}^{A}(k,R).$

###### Proof.

Combine Corollary 2.1.6 with Proposition 1.5.12. ∎

######  Remark 2.1.8.

It is not true in general that the canonical functor

$\mathcal{D}^{b}(\operatorname{Rep}^{A}(\pi_{1}^{\operatorname{\acute{e}t}}(S,\xi),R))\rightarrow\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)$

is an equivalence. Indeed, letting $k$ be a separably closed field of
characteristic $0$, the étale fundamental group of $\mathbb{P}^{1}_{k}$ is
trivial. Therefore, the abelian category of its $\mathbb{Z}$-linear Artin
representations is the category of finitely generated abelian groups, which
for any geometric point $\xi$ yields

$\mathrm{Ext}^{3}_{\operatorname{Rep}^{A}\left(\pi_{1}^{\operatorname{\acute{e}t}}(\mathbb{P}^{1}_{k},\xi),\mathbb{Z}\right)}(\mathbb{Z},\mathbb{Z})=0.$

On the other hand, we have

$H_{\operatorname{\acute{e}t}}^{3}(\mathbb{P}^{1}_{k},\mathbb{Z})=\mathrm{Ext}^{3}_{\operatorname{Sh}_{\operatorname{\acute{e}t}}(\mathbb{P}^{1}_{k},\mathbb{Z})}(\mathbb{Z},\mathbb{Z})$

and this group is isomorphic to $\mathbb{Q}/\mathbb{Z}$ by Remark 1.7.4. Thus,
the above functor cannot be an equivalence.

We end this section with a description of the $\ell$-adic realization functor
on smooth Artin motives over a regular scheme.

######  Proposition 2.1.9.

Let $S$ be a regular scheme and let $R$ be a localization of the ring of
integers of a number field $K$, let $v$ be a non-archemedian valuation on the
field $K$ and let $\ell$ be the prime number such that $v$ extends the
$\ell$-adic valuation. Assume that $\ell$ is invertible on $S$ and that all
the residue characteristic exponents of $S$ are invertible in $R$.

  1. (1)

If $v$ is non-negative on $R$, denote by $\lambda$ a uniformizer of $R_{(v)}$.
Let $\Xi$ be the functor

$\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)\rightarrow\operatorname{\mathcalboondox{Sh}}_{\lambda}(S,R_{(v)})$

that sends an étale sheaf $M$ to the $\lambda$-complete sheaf
$\lim(M\otimes_{R}R_{(v)})/\lambda^{n}$.

Then, with the notations of Section 1.6.1, the square

${\operatorname{\mathcalboondox{Sh}}_{\operatorname{Ind}\operatorname{lisse}}(S,R)}$${\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t}}(S,R)}$${\operatorname{\mathcalboondox{Sh}}_{\lambda}(S,R_{(v)})}$${\mathcal{D}(S,R_{v})}$$\scriptstyle{\Xi}$$\scriptstyle{\rho_{!}}$$\scriptstyle{\rho_{v}}$$\scriptstyle{\Psi}$

is commutative.

Furthermore, when restricted to the subcategory of constructible smooth Artin
étale motives, the $v$-adic realization functor is t-exact with respect to the
ordinary t-structures and induces the exact functor

$-\otimes_{R}R_{v}\colon\operatorname{Loc}_{S}(R)\rightarrow\operatorname{Sh}^{smA}(S,R_{v})$

between the hearts.

  2. (2)

If $v$ is not non-negative on $R$, Let $A$ be the subring of $K$ on which the
valuation $v$ is non-negative and let $\lambda$ be a uniformizer of $A$. Then,
with the notations of Section 1.6.1, the square

${\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)}$${\operatorname{\mathcal{DM}}^{smA}_{\operatorname{\acute{e}t},c}(S,R)}$${\operatorname{\mathcalboondox{Sh}}_{\lambda,c}(S,A)_{(0)}}$${\mathcal{D}^{b}_{c}(S,R_{v})}$$\scriptstyle{\Xi^{\prime}}$$\scriptstyle{\rho_{!}}$$\scriptstyle{\rho_{v}}$$\scriptstyle{\Psi^{\prime}}$

is commutative using the notations of Section 1.1.5, if $\Psi^{\prime}$
denotes the functor $\Psi_{(0)}$ and $\Xi^{\prime}$ denotes the functor
$\Xi_{(0)}\circ(-\otimes_{R}K)$.

Furthermore, the $v$-adic realization functor is t-exact with respect to the
ordinary t-structures and induces the exact functor

$-\otimes_{R}R_{v}\colon\operatorname{Loc}_{S}(R)\rightarrow\operatorname{Sh}^{smA}(S,R_{v})$

between the hearts.

###### Proof.

The second assertion follows from the first assertion. We now prove the first
assertion. Recall that with the notations of Section 1.6.1, we have

$\rho_{v}=\Psi\circ\rho^{!}\circ\rho_{\lambda}^{\mathrm{CD}}.$

But as $\rho^{!}$ is a right adjoint functor, it commutes with limits which
yields

$\rho^{!}\circ\rho_{\lambda}^{\mathrm{CD}}\circ\rho_{!}=\Xi.$

To show that the $\ell$-adic realization functor is t-exact when restricted to
the subcategory of constructible smooth Artin étale motives, it suffices to
show that the functor

$\Xi\circ\Psi\colon\operatorname{\mathcalboondox{Sh}}_{\operatorname{lisse}}(S,R)\rightarrow\mathcal{D}^{b}_{c}(S,R_{v})$

is t-exact. Let $M$ be a locally constant étale sheaf with coefficients in
$R$. We have

$\Xi(\Psi(M))=\lim\nu^{*}(M)/\lambda^{n}.$

It suffices to show that $\lim\nu^{*}(M)/\lambda^{n}$ lies in the heart of the
ordinary t-structure on $\mathcal{D}^{b}_{c}(S,R_{v})$. Now, by definition of
the ordinary t-structure (see [BBDG18, 2.2.13]) if $i$ is an integer, we have

$H^{i}(\lim\nu^{*}(M)/\lambda^{n})=\lim
H^{i}(\nu^{*}(M/\lambda^{n}))=\lim\nu^{*}\mathrm{Tor}_{i}^{R}(M,R/\lambda^{n}R).$

Now, the exact sequence

$0\rightarrow R\overset{\times\lambda^{n}}{\rightarrow}R\rightarrow
R/\lambda^{n}R\rightarrow 0$

yields

$\mathrm{Tor}_{i}^{R}(M,R/\lambda^{n}R)=\begin{cases}M/\lambda^{n}M&\text{if
}i=0\\\ M[\lambda^{n}]&\text{if }i=-1\\\ 0&\text{otherwise}\end{cases}$

where $M[\lambda^{n}]$ denotes the $\lambda^{n}$-torsion of $M$. As the
transition map $M[\lambda^{n+1}]\rightarrow M[\lambda^{n}]$ is the
multiplication by $\lambda$, we get $\lim M[\lambda^{n}]=0$. The result
follows. ∎

###  2.2. The conservativity conjecture for Artin motives

The conservativity conjecture, proved for Artin motives with rational
coefficients in [Leh19b], asserts that the $\ell$-adic realization functor
[CD16, 7.2.11] is conservative. With integral coefficients, we need to include
realization functors with respect to every $\ell$. Indeed, if $M$ is an Artin
motive of $p$-torsion and $\ell\neq p$, the $\ell$-adic realization of $M$
vanishes.

Furthermore, the stable category of constructible $\ell$-adic complexes has a
non-pathological behavior only when $\ell$ is invertible on the base scheme.
Therefore, we introduce the following functor.

######  Definition 2.2.1.

Let $S$ be a scheme, let $R$ be a localization of the ring of integers of a
number field $K$, let $v$ be a valuation on $K$ and let $\ell$ be the prime
number such that $v$ extends the $\ell$-adic valuation.

The _reduced $v$-adic realization functor_ $\bar{\rho}_{v}$ is the composition

$\mathcal{DM}_{\operatorname{\acute{e}t}}(S,R)\overset{j_{\ell}^{*}}{\rightarrow}\mathcal{DM}_{\operatorname{\acute{e}t}}(S[1/\ell],R)\overset{\rho_{v}}{\rightarrow}\mathcal{D}(S[1/\ell],R_{v}),$

where $j_{\ell}$ denotes the open immersion $S[1/\ell]\rightarrow S$ and where
the functor $\rho_{v}$ is the $v$-adic realization functor defined in Section
1.6.1.

We still denote by $\bar{\rho}_{v}$ the restriction of this functor to the
stable subcategory of constructible Artin motives.

A consequence of Theorem 2.1.1 is the conservativity conjecture in the case of
Artin motives.

######  Proposition 2.2.2.

Let $S$ be a scheme, let $R$ be a localization of the ring of integers of a
number field $K$. If $v$ is a valuation on $K$, denote by $R_{v}$ the
completion of $R$ with respect to $v$ and by $K_{v}$ the completion of $K$
with respect to $v$.

  1. (1)

If there is a prime number $\ell$ which is invertible on $S$, let $v$ be a
valuation which extends the $\ell$-adic valuation then, the functor

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

is conservative.

  2. (2)

If $\ell$ and $\ell^{\prime}$ are two distinct prime numbers, let $v$ be a
valuation which extends the $\ell$-adic valuation and let $v^{\prime}$ be a
valuation which extends the $\ell^{\prime}$-adic valuation. Then, when working
with coefficients in $K$, the family
$(\bar{\rho}_{v},\bar{\rho}_{v^{\prime}})$ is conservative.

  3. (3)

The family

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

where $v$ runs through the set of non-archimedian valuations on $K$ which are
non-negative on $R$ and $\ell(v)$ denotes the prime number such that $v$
extends the $\ell$-adic valuation is conservative.

###### Proof.

If $x$ is a point of $S$, denote by $i_{x}\colon\\{x\\}\rightarrow S$ the
inclusion. The family of functors $(i_{x}^{*})_{x\in S}$ is conservative by
[CD19, 4.3.17]. In addition, the $v$-adic realization commutes with the six
functors by [CD16, 7.2.16]. Hence, to prove the proposition, we may assume
that $S$ is the spectrum of a field $k$. Let $p$ be the characteristic
exponent of $k$.

We prove the first assertion. Since $\ell\neq p$, we have

$\rho_{v}=\bar{\rho}_{v}.$

Denote by $(BG_{k})_{\mathrm{pro\acute{e}t}}$ the site of profinite continuous
$G_{k}$-sets with covers given by continous surjections (see [BS15, 4.1.10]).
Using Proposition 2.1.9, the diagram

$\leavevmode\hbox to285.53pt{\vbox
to58.81pt{\pgfpicture\makeatletter\hbox{\hskip
142.76538pt\lower-30.39088pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-142.76538pt}{-28.41939pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
40.18846pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.88292pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${{\mathcal{D}^{b}(\mathrm{Rep}^{A}(G_{k},K))}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
40.18846pt\hfil&\hfil\hskip 52.34566pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-24.04015pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${{\operatorname{\mathcalboondox{Sh}}_{\mathrm{lisse}}(k,K)}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
28.34569pt\hfil&\hfil\hskip 52.284pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-23.97849pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${{\mathcal{DM}^{A}_{\mathrm{\acute{e}t}}(k,K)}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
28.28403pt\hfil\cr\vskip 18.00005pt\cr\hfil\hskip
39.54616pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-35.24062pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${{\mathcal{D}^{b}(\mathrm{Rep}(G_{k},K_{v}))}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
39.54616pt\hfil&\hfil\hskip 74.2929pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{