# Étale motives of geometric origin

Raphaël Ruimy, Swann Tubach

###### Abstract

Over qcqs finite-dimensional schemes, we prove that étale motives of geometric
origin can be characterised by a constructibility property which is purely
categorical, giving a full answer to the question “Do all constructible étale
motives come from geometry?” which dates back to Cisinski and Déglise’s work.
We also show that they afford the continuity property and satisfy h-descent
and Milnor excision.

###### Contents

  1. 1 Introduction.
  2. 2 First descent properties.
  3. 3 Continuity.
  4. 4 Constructible motives are of geometric origin.
  5. 5 Milnor excision and h-descent.

##  1 Introduction.

Consider a scheme $X$ and a derived111We define our $\infty$-categories with
derived coefficients so that we may use derived rings of the form
$\Lambda\otimes_{\mathbb{Z}}^{L}\mathbb{Z}/n\mathbb{Z}$. However, all main
theorems are stated for ordinary commutative rings as the extension to derived
rings with heavy boundedness assumptions seems artificial. commutative ring of
coefficients $\Lambda$. We denote by
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ the stable $\infty$-category of
étale motives defined as the Tate-stabilisation of the category of
$\mathbb{A}^{1}$-local étale hypersheaves of $\Lambda$-modules over the
category of smooth $S$-schemes. This version of étale motives without
transfers was studied by Ayoub in [Ayo14]; it is equivalent over Noetherian
schemes to the category of h-motives studied by Cisinski and Déglise in
[CD16]. It is endowed with the six functors formalism by [AGV22, Theorem
4.6.1]. Denote by $\Lambda_{X}$ the unit object of
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$.

In this paper we explore some properties of étale motives of geometric origin:

######  Definition.

Let $X$ be a scheme and let $\Lambda$ be a commutative ring. Let $M$ be an
object of $\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$. We say that the
motive $M$ is _of geometric origin_ (or simply _geometric_) if it belongs to
the thick subcategory
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$ of
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ generated by the
$f_{\sharp}(\Lambda_{Y})(n)$ for $f\colon Y\to X$ smooth.

An ind-regular ring is a filtered colimit of regular rings. Our main goal is
to prove that motives of geometric origin form an étale hypersheaf and to give
a purely categorical characterisation of geometric étale motives:

######  Main Theorem.

(4.1 and 2.5) Let $\Lambda$ be a commutative ind-regular ring.

  1. 1.

Let $X$ be a qcqs finite-dimensional scheme. An étale motive $M$ over $X$ is
of geometric origin if and only if it is _constructible_ , that is for any
open affine $U\subseteq X$, there is a finite stratification $U_{i}\subseteq
U$ made of constructible locally closed subschemes such that each $M|_{U_{i}}$
is dualisable.

  2. 2.

The functor $\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ is an
étale hypersheaf over qcqs finite-dimensional schemes.

This question dates back to [CD16], where Cisinski and Déglise considered
étale-locally geometric222Beware that in _loc. cit._ , they call
_constructible_ what we chose to call _of geometric origin_ in this paper. We
chose this terminology because they are the motives that can be built from
objects coming from geometry. In addition, objects that are called _locally
constructible_ by Cisinski and Déglise are more similar to classical
constructible sheaves. In any case, the purpose of this paper is to show that
all those notions coincide. motives and showed that over Noetherian schemes,
the latter coincides with constructible motives. This result does not hold for
a general ring of coefficients (see 4.5).

To prove the above theorem, we need to generalise to non-Noetherian schemes
the continuity properties proved in loc. cit.

######  Theorem.

(Continuity; 3.5) Let $\Lambda$ be a commutative ring. Consider a qcqs finite-
dimensional scheme $X$ that is the limit of a projective system of qcqs
finite-dimensional schemes $(X_{i})$ with affine transition maps. Then, the
natural map

$\mathrm{colim}_{i}\hskip
2.84526pt\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{i},\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$

is an equivalence.

Using continuity, the most critical case of the main theorem is the case where
$X=\mathrm{Spec}(k)$ with $k$ a field and where $\Lambda$ is regular. In that
particular case, the result can be deduced from the case of rational
coefficients which was proved in [CD16] combined with results of [Rui22] on
Artin motives. The latter are consequences of results of [BG23] which imply
that étale sheaves of the form $f_{\sharp}\Lambda_{L}$ for $L/k$ a finite
separable extension, generate the category of étale-locally perfect constant
sheaves as a thick subcategory of itself.

We end the paper with some non-trivial descent properties of geometric étale
motives. Namely we prove that they satisfy h-descent in the sense of [Stacks,
Tag 0ETQ] generalizing [CD16, Corollary 5.5.7.] to the non-Noetherian setting
and we prove that they satisfy Milnor excision in the sense of [EHIK21a],
giving a generalisation of the analogous of [EHIK21b, Theorem 5.6] for
geometric motives:

######  Theorem.

(5.2 and 5.3) Let $\Lambda$ be a commutative ring (_resp._ an ind-regular
commutative ring). The functor
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ satisfies Milnor
excision (_resp._ h-descent) over qcqs finite-dimensional schemes.

######  Remark.

In an upcoming paper, we will define a category of geometric Nori motives with
coefficients in an arbitrary commutative ring $\Lambda$ over qcqs finite-
dimensional schemes and show that it satisfies arc-descent. We therefore
expect $\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ to satisfy
arc-descent.

##  2 First descent properties.

Until the end of the proof of the main theorem, we will use the following
notation:

######  Definition 2.1.

Let $X$ be a scheme and let $\Lambda$ be a derived333As mentioned above, we
make the definition here for a derived ring so that we may use it for torsion
rings of the form $\Lambda\otimes^{L}_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}$ with
$\Lambda$ a commutative ring. ring. We denote by
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda)$ the subcategory of
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ made of _constructible motives_
, that is those motives such that for any open affine $U\subseteq X$, there is
a finite stratification $U_{i}\subseteq U$ made of constructible locally
closed subschemes such that each $M|_{U_{i}}$ is dualisable.

######  Remark 2.2.

If $X$ is Noetherian, then if $K$ is an object of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda)$, we can find a finite
stratification $(X_{i})$ of $X$ such that each $K_{\mid X_{i}}$ is dualisable.

######  Remark 2.3.

The functors of type $f^{*}$, $f_{\sharp}$ and $\otimes$ readily preserve
geometric objects. The functor $i_{*}$, for $i$ a closed immersion, preserves
geometric objects by using the localisation triangle.

On the other hand, the functors of type $f^{*}$ and $\otimes$ preserve
constructible objects. The functors $j_{\sharp}$ and $i_{*}$, for $j$ an open
immersion and $i$ a closed immersion, also preserve constructible objects.

We first prove descent properties for geometric objects and for constructible
objects.

######  Lemma 2.4.

Let $\Lambda$ be a commutative ring. The functor
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ is a Nisnevich
hypersheaf over qcqs schemes.

###### Proof.

It suffices to show that
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ it is a subsheaf
of $\mathrm{DM}^{\mathrm{\acute{e}t}}(-,\Lambda)$ (a subsheaf of a hypersheaf
is itself a hypersheaf for qcqs sites). Assume that $(U,V)$ is a Nisnevich
cover of $X$ with $U$ open and $V\to X$ étale and let $M$ be a motive over
$X$. Then if $M|_{U}$ and $M|_{V}$ are geometric, so is $M|_{U\times_{X}V}$,
so that by Nisnevich excision, the motive $M$ itself is geometric. ∎

######  Lemma 2.5.

Let $\Lambda$ be a commutative ring. The functor
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(-,\Lambda)$ is an étale hypersheaf over
qcqs schemes.

###### Proof.

It suffices to show that $\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(-,\Lambda)$ is
a subsheaf of $\mathrm{DM}^{\mathrm{\acute{e}t}}(-,\Lambda)$. The proof is
very similar to the proof of [HRS23, Lemma 4.5], so we only give a sketch.

Take a qcqs scheme $X$ and an étale-locally constructible motive $M$ over $X$.
We can assume $X$ to be affine and to have a surjective étale map $f\colon
Y\to X$ such that $f^{*}(M)$ is constructible. We then reduce to $f$ being
finite étale using [Stacks, Tag 03S0] and then to the case where $f$ is a
torsor under some finite group $G$: writing $X$ as a limit of Noetherian
schemes, the map $f$ is the pullback of some étale map defined over a
Noetherian scheme which we can then replace by a torsor under some finite
group.

Finally, we can construct a $G$-equivariant stratification of $Y$ from the
stratification $\mathcal{Y}$ that witnesses the constructibility of
$f^{*}(M)$. This yields a stratification of $X$ whose pullback to $Y$ is
$\mathcal{Y}$. Hence, replacing $X$ with one of its strata, we can assume that
$f^{*}(M)$ is dualisable; as being dualisable is étale-local, the motive $M$
is also dualisable which finishes the proof. ∎

##  3 Continuity.

We now prove the continuity property for geometric motives. We first need some
compatibilities with rationalisation:

######  Proposition 3.1.

Let $\Lambda$ be a commutative ring, let $f\colon Y\to X$ be a morphism of
qcqs finite-dimensional schemes, let $M$ and $N$ in
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ be motives over $X$ with $M$
constructible or geometric and let $P$ be a motive over $Y$. Then, the natural
transformations below are equivalences:

  1. 1.

$\mathrm{Map}(M,N)\otimes\mathbb{Q}\to\mathrm{Map}(M,N\otimes\mathbb{Q})$.

  2. 2.

$\underline{\mathrm{Hom}}(M,N)\otimes\mathbb{Q}\to\underline{\mathrm{Hom}}(M,N\otimes\mathbb{Q}).$

  3. 3.

$f_{*}(P)\otimes\mathbb{Q}\to f_{*}(P\otimes\mathbb{Q})$.

  4. 4.

$i^{!}(N)\otimes\mathbb{Q}\to i^{!}(N\otimes\mathbb{Q})$ (for $i$ a closed
immersion).

###### Proof.

We begin with the case where $M$ is geometric. In this proof, we say that $M$
is rationalisable if

$\mathrm{Map}(M,N)\otimes\mathbb{Q}\to\mathrm{Map}(M,N\otimes\mathbb{Q})$

is an equivalence. Rationalisable objects are closed under the functors of
type $g_{\sharp}$. Therefore, to prove (1), we can assume that
$M=\Lambda_{X}$. Note first that if $K$ is a complex of étale sheaves of
abelian groups, the canonical map:

$\mathrm{H}^{i}_{\mathrm{\acute{e}t}}(X,K)\otimes\mathbb{Q}\to\mathrm{H}^{i}_{\mathrm{\acute{e}t}}(X,K\otimes\mathbb{Q})$

is an isomorphism for any integer $i$. Indeed, [CM21, Corollary 3.29] gives a
bound to the étale cohomological dimension with rational coefficients over
qcqs finite-dimensional schemes, and the above assertion then follows from
[CD16, Lemma 1.1.10]. By the same argument as [CD16, Corollary 5.4.8] this
gives (1).

Geometric objects generate $\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ as a
localizing subcategory of itself. Hence to show that the maps of (2), (3) and
(4) are equivalences, it suffices to show that they become equivalences after
applying $\mathrm{Map}(C,-)$ for $C$ geometric. Since the functors of type
$f^{*}$, $i_{*}$ and $\otimes$ preserve geometric objects, this follows by
adjunction from the equivalence of (1).

We now tackle the case where $M$ is constructible. The third equivalence
implies that the functors of type $g^{*}$ preserve rationalisable objects.
Hence, being rationalisable is Zariski-local: if we have a finite Zariski
cover $\\{U_{i}\to X\\}$ and a motive $M$ over $X$ such that the $M|_{U_{i}}$
are rationalisable, so are the restrictions of $M$ over the intersections of
the $U_{i}$; as we can write $M$ as a finite colimit with terms of the form
$(V\to X)_{\sharp}(M|_{V})$ with $V$ an intersection of the $U_{i}$, the
motive $M$ is also rationalisable. Hence, we can assume $X$ to be affine.

Note now that (4) implies that rationalisable objects are preserved by the
functor $i_{*}$ for $i$ a closed immersion. By dévissage and localisation, we
are therefore reduced to the case where $M$ is dualisable. In that case,
replacing $N$ by $N\otimes M^{\vee}$ we can further reduce to the case where
$M=\Lambda_{X}$ which we already proved.

Assertion (2) in the constructible setting can be proved in the same way as in
the geometric setting. ∎

For $\Lambda$ a derived ring, denote by
$\mathrm{Sh}(X_{\mathrm{\acute{e}t}},\Lambda)$ the $\infty$-category of étale
hypersheaves of $\Lambda$-modules. The above proposition gives a control on
the rationalised part of our categories, and the following theorem will be the
control on the torsion part.

######  Theorem 3.2 (Bachmann).

Let $n\in\mathbb{N}^{*}$ be an integer, let $\Lambda$ be a derived
$\mathbb{Z}/n\mathbb{Z}$-algebra and let $X$ be a qcqs scheme such that $n$ is
invertible on $X$. The natural functor

$\rho_{!}\colon\mathrm{Sh}(X_{\mathrm{\acute{e}t}},\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$

is an equivalence. We will denote by
$\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{\acute{e}t}},\Lambda)$ the inverse
image of $\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda)$ under this
equivalence: as $\rho_{!}$ is symmetric monoidal it has the same description
as 2.1, we call those _constructible étale sheaves_.

###### Proof.

Thanks to [BH21, Theorem 3.1] the above functor is an equivalence if
$\Lambda=\mathbb{Z}/n\mathbb{Z}$, and then this form of the theorem follows by
taking $\Lambda$-modules on both sides. ∎

######  Lemma 3.3 (Martini-Wolf [MW22, Corollary 7.4.12]).

Let $\Lambda$ be a derived ring and let $X$ be a qcqs scheme. An object $K$ of
$\mathrm{Sh}(X_{\mathrm{\acute{e}t}},\Lambda)$ is dualisable if and only if
there exists an étale covering $f\colon U\to X$ such that $f^{*}K$ is the
constant sheaf associated to a perfect complex.

######  Proposition 3.4.

Let $\Lambda$ be a commutative ring, let $R$ be a perfect derived
$\Lambda$-algebra and let $X$ be a qcqs scheme. Assume that $X=\lim_{i\in
I}X_{i}$ is a limit of qcqs schemes with affine transitions. Then the natural
functor

$\mathrm{colim}_{i}\mathrm{D}_{\mathrm{cons}}((X_{i})_{\mathrm{\acute{e}t}},R)\to\mathrm{D}_{\mathrm{cons}}(X_{\mathrm{\acute{e}t}},R)$

is an equivalence, where the colimit is taken in $\mathrm{Cat}_{\infty}$.

###### Proof.

This is an adaptation of [BM21, Proposition 5.10]. Up to writing each $X_{i}$
as a limit of Noetherian schemes with affine transition, we can assume the
$X_{i}$ to be Noetherian. We first show that the functor is fully faithful.
Let $i_{0}\in I$. It is enough to prove that for any two complexes $K$ and $L$
in $\mathrm{D}_{\mathrm{cons}}((X_{i_{0}})_{\mathrm{\acute{e}t}},R)$ the map

$\mathrm{colim}_{i\geqslant
i_{0}}\mathrm{Map}_{\mathrm{D}_{\mathrm{cons}}((X_{i})_{\mathrm{\acute{e}t}},R)}(K_{\mid
X_{i}},L_{\mid
X_{i}})\to\mathrm{Map}_{\mathrm{D}_{\mathrm{cons}}(X,\Lambda)}(K_{\mid
X},L_{\mathrm{\mid X}})$

is an equivalence. By dévissage and localisation, we may assume that $K$ is
dualisable and then reduce to the case where $K=R$. By adjunction, we thus
have to show that the map

$\mathrm{colim}_{i\geqslant
i_{0}}\mathrm{Map}_{\mathrm{Mod}_{\Lambda}}(\Lambda,\mathrm{R}\Gamma(X_{i},L))\to\mathrm{Map}_{\mathrm{Mod}_{\Lambda}}(\Lambda,\mathrm{R}\Gamma(X,L))$

is an equivalence. Now, as perfect $R$-modules are perfect $\Lambda$-modules
because $R$ is perfect over $\Lambda$, the object $L$, seen as an object of
$\mathrm{Sh}((X_{0})_{\mathrm{\acute{e}t}},\Lambda)$ is constructible, in
particular bounded so that up to a dévissage, it is concentrated in one
degree. The result then follows from the commutation of étale cohomology with
colimits over the $(X_{i})$.

For the essential surjectivity, by dévissage it is enough to prove that any
dualisable object lies in the image. By 3.3, if $K$ is a dualisable object of
$\mathrm{Sh}(X_{\mathrm{\acute{e}t}},R)$ there exists an étale covering
$f\colon U\to X$ and a perfect complex of $R$-modules $P$ such that $f^{*}(K)$
is the constant sheaf with value $P$. By Zariski descent we may even assume
that $X$ is affine and up to cover the source of $f$ by an affine Zariski
cover, we may also assume that the source of $f$ is affine. Then by spreading
out ([EGA IV, Théorème 8.10.5, Proposition 17.7.8]) the morphism $f$ is the
pullback of some étale $f^{i}$ with target $X_{i}$. Let $(f_{n})$ and
$(f^{i}_{n})$ be the Čech nerves of $f$ and $f^{i}$. As the image of
$(f_{n}^{i})_{\sharp}P$ in $\mathrm{D}_{\mathrm{cons}}(X,R)$ is
$(f_{n})_{\sharp}P$, fully faithfulness ensures that the simplicial object
$((f_{n})_{\sharp}P)_{n\in\Delta}$ is the image of a simplicial object
$((f_{n}^{i})_{\sharp}P)_{n\in\Delta}$ in
$\mathrm{D}_{\mathrm{cons}}(X_{i},R)$. By étale descent, we have that
$K=\mathrm{colim}_{n}(f_{n})_{\sharp}P$ in $\mathrm{D}_{\mathrm{cons}}(X,R)$.
Moreover if we denote by $L=\mathrm{colim}_{n}(f_{n}^{i})_{\sharp}P$ in
$\mathrm{Sh}((X_{i})_{\mathrm{\acute{e}t}},R)$, we have that $(f^{i})^{*}L=P$
hence $L$ is dualisable. As the pullback functor
$p_{i}^{*}\colon\mathrm{Sh}((X_{i})_{\mathrm{\acute{e}t}},R)\to\mathrm{Sh}(X_{\mathrm{\acute{e}t}},R)$
commutes with colimits, we obtain that $K$ is the image of $L$, which is a
constructible object over $X_{i}$, finishing the proof.

∎

######  Theorem 3.5.

(Continuity) Let $\Lambda$ be a commutative ring. Consider a qcqs finite-
dimensional scheme $X$ that is the limit of a projective system of qcqs
finite-dimensional schemes $(X_{i})$ with affine transition maps. Then, the
natural map

$\mathrm{colim}\hskip
2.84526pt\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{i},\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$

is an equivalence.

###### Proof.

We can again assume that each $X_{i}$ is Noetherian. Fully faithfulness can be
checked after tensorisation of the mapping spectra by $\mathbb{Q}$ and by
$\mathbb{Z}/p\mathbb{Z}$. By 3.1, the $-\otimes\mathbb{Q}$ passes inside the
mapping spectra, and then our objects become compact by [EHIK21b, Lemma 5.1
(i)] and this follows from continuity of the presentable category as in
[AGV22, Lemma 2.5.11]. Modding out by $p$ passes in the mapping spectra;
hence, noting that $\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$ is a
perfect $\Lambda$-algebra, the result will then follow from the rigidity
theorem in the form of 3.2 and 3.4, once we know that geometric objects are
constructible with torsion coefficients. This last assertion is true over
$\mathbb{Z}/p\mathbb{Z}$ by [CD16, Theorem 6.3.26]; as the functor
$-\otimes_{\mathbb{Z}}\Lambda$ preserves constructible étale sheaves and as
the generators of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z})$
are of the form $M\otimes_{\mathbb{Z}}\Lambda$ for some $M$ in
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\mathbb{Z}/p\mathbb{Z})$,
the result is true with
$\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$-coefficients.

To prove the essential surjectivity, first using that the Čech nerve of a
covering coming from a finite open covering is finite, we can reduce to $X$
affine using Zariski descent. Then Zariski descent again (but this time on the
functor that sends $g\colon Y\to X$ smooth to $g_{\sharp}g^{*}\Lambda_{X}$)
implies that it suffices to reach $f_{\sharp}(\Lambda_{Y})(n)$ with $Y$ and
$X$ affine and $f\colon Y\to X$ smooth. The usual spreading out properties of
[EGA IV, Théorème 8.10.5, Proposition 17.7.8] then give the result. ∎

######  Corollary 3.6.

Let $\Lambda$ be a commutative ring and let $X$ be a finite-dimensional qcqs
scheme.

  1. 1.

Let $f\colon Y\to X$ be a separated finite type morphism. Then the functor
$f_{!}$ restricts to a functor

$f_{!}\colon\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(Y,\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$

between geometric motives.

  2. 2.

The $\infty$-category
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$ is the thick
subcategory of $\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ generated by the
$p_{*}(\Lambda_{Y})(n)$ for $p\colon Y\to X$ projective and $n\in\mathbb{Z}$.

  3. 3.

The $\infty$-category $\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ is the
localising subcategory of itself generated by the $p_{*}(\Lambda_{Y})(n)$ for
$p\colon Y\to X$ projective and $n\in\mathbb{Z}$.

###### Proof.

For the first point, assume first that $f=p$ is proper. Let $M$ be an object
of $\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(Y,\Lambda)$, and write
$X=\lim X_{i}$ as a limit of schemes that are of finite type over the
integers. By [EGA IV, Théorème 8.10.5.(xii)] one can find a $p_{i_{0}}\colon
Y_{i_{0}}\to X_{i_{0}}$ proper such that the pullback to $X$ is $p$. In
particular

$Y=\lim_{i\geqslant i_{0}}Y_{i_{0}}\times_{X_{i_{0}}}X_{i}$

and there is some $i$ such that $M$ is the pull back of a motive $N$ over
$Y_{i}$ by continuity. Then by [CD19, Proposition 4.2.11] the object
$(p_{i})_{*}M_{i}$ is geometric in
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X_{i},\Lambda)$. Thus, its pullback to $X$
is geometric, but by the proper base change theorem this is $p_{*}M$. Finally,
if $f$ is separated of finite type, choosing a Nagata compactification of $f$
yields the result.

Now for the second point, write $X=\lim_{i}X_{i}$ as a limit of schemes that
are of finite type over the integers. Let $M$ be an object of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$, that comes by
continuity above from an object $M_{i}$ of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{i},\Lambda)$ for some
index $i$. Thanks to [Ayo07, Lemme 2.2.23] we know that the result is true for
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{i},\Lambda)$, so that
there exists a finite diagram

$F\colon J\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{i},\Lambda)$

such that $F(j)$ is of the form $p^{j}_{*}\Lambda(n_{j})[m_{j}]$ for $p^{j}$
projective and $m_{j}$ and $n_{j}$ integers, and such that $M_{i}$ is a direct
factor of $\mathrm{colim}_{J}F$. Thus, the object $M$ is a direct factor of
$(\mathrm{colim}_{J}F)_{\mid X}$, which by proper base change is the colimit
of the objects $q^{j}_{*}\Lambda(n_{j})[m_{j}]$, with $q^{j}$ the base change
of $p^{j}$ to $X$. This finishes the proof, as the third point readily follows
from the second one. ∎

As a consequence of continuity, we see that geometric motives are
constructible.

######  Corollary 3.7.

Let $\Lambda$ be a commutative ring. Geometric motives with coefficients
$\Lambda$ are constructible.

###### Proof.

Let $X$ a qcqs scheme. Write $X$ as a limit of Noetherian finite-dimensional
schemes $X_{i}$ with affine transition maps. Any geometric motive with
coefficients $\mathbb{Z}$, over $X_{i}$ is constructible using [CD16, Theorem
6.3.26] then this is also true with coefficients $\Lambda$ by tensoring by
$\Lambda$ over $\mathbb{Z}$. If $M$ is a geometric object, it is the pullback
of some $M_{i}$ over some $X_{i}$. As the motive $M_{i}$ is constructible, so
is $M$. ∎

We now prove the converse when $X$ is $\Lambda$-finite444See the beginning of
[EHIK21b, Section 5] for the precise definition. This hypothesis implies that
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$ is compactly generated and any
finite dimensional scheme is $\Lambda$-finite if $\Lambda$ is a
$\mathbb{Q}$-algebra. by linking both notions to compactness:

######  Proposition 3.8.

Let $\Lambda$ be a commutative ring. Let $X$ be a qcqs $\Lambda$-finite scheme
and let $M$ be an object of $\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$.
Then, the following are equivalent:

  1. 1.

$M$ is of geometric origin.

  2. 2.

$M$ is constructible.

  3. 3.

The functor $\underline{\mathrm{Hom}}(M,-)$ is compatible with colimits.

  4. 4.

$M$ is a compact object of $\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda)$.

###### Proof.

The equivalence between geometric objects and compact objects is [EHIK21b,
Lemma 5.1 (i)]. On the other hand, the formula

$\mathrm{Map}(f_{\sharp}(\Lambda_{Y})(n)\otimes
M,-)=\mathrm{Map}(f_{\sharp}(\Lambda_{Y})(n),\underline{\mathrm{Hom}}(M,-))$

for $f\colon Y\to X$ smooth and $n$ an integer, combined with the fact that
$f_{\sharp}(\Lambda_{Y})(n)$ is compact implies that conditions (3) and (4)
are equivalent.

We proved in 3.7 that motives of geometric origin are constructible. We now
prove the converse. Let $M$ be constructible over $X$. We can assume that $X$
is affine by Zariski hyperdescent of geometric motives. By dévissage and
localisation, we can further assume that $M$ is dualisable. In that case
$\underline{\mathrm{Hom}}(M,-)$ is compatible with colimits which finishes the
proof. ∎

From this, we deduce continuity for constructible motives. We begin with a
general lemma:

######  Lemma 3.9.

Let $\Lambda$ be a commutative ring and let $M$ be an object of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda)$ with $X$ a qcqs finite-
dimensional scheme. Then, there is a geometric motive $P$ as well as a map
$P\to M\oplus M[1]$ whose cone $C$ is such that $C/n\simeq C\oplus C[1]$ for
some integer $n$.

###### Proof.

Take $M$ a constructible motive. 3.1 implies that
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda\otimes\mathbb{Q})$
is the idempotent-completion in
$\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\Lambda\otimes\mathbb{Q})$ of the
rationalisation of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$. The
characterisation of the idempotent completion [BG23, Recollection 2.13] as
well as 3.1 yields that $(M\oplus M[1])\otimes\mathbb{Q}$ belongs to the
rationalisation of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$. This gives a
geometric motive $P$ and a map $P\to M\oplus M[1]$ which becomes an
isomorphism after tensoring with $\mathbb{Q}$. Let $C$ be the cone of this
map, then $C\otimes\mathbb{Q}=0$. Using 3.1 once again, we see that the
multiplication by $n$ seen as a map $C\to C$ vanishes for $n$ large enough.
This together with the exact triangle $C\xrightarrow{\times n}C\to C/n$
implies that $C/n\simeq C\oplus C[1]$. ∎

######  Corollary 3.10.

Let $\Lambda$ be a commutative ring. Consider a qcqs finite-dimensional scheme
$X$ that is the limit of a projective system of qcqs finite-dimensional
schemes $(X_{i})$ with affine transition maps. Then, the natural map

$\mathrm{colim}_{i}\hskip
2.84526pt\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X_{i},\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda)$

is an equivalence.

###### Proof.

We first prove that the functor above is fully faithful. This can be checked
by tensoring the mapping spectra with $\mathbb{Q}$ and
$\mathbb{Z}/p\mathbb{Z}$, and then it follows from 3.1, 3.8 and 3.4.

We now prove the essential surjectivity of the functor. Let $M$ be a
constructible motive. We want to prove that $M$ is in the essential image of
the functor

$\mathrm{colim}\hskip
2.84526pt\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X_{i},\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda).$

Write the exact triangle $P\to M\oplus M[1]\to C$ of the above lemma. 3.5
implies that $P$ belongs to the essential image. As constructible motives are
idempotent-complete, we can assume that $M$ is of the form $C/n$. But then,
this follows from continuity with torsion coefficients which is a combination
of 3.2 and 3.4. ∎

##  4 Constructible motives are of geometric origin.

######  Theorem 4.1.

Let $\Lambda$ be a commutative ind-regular ring and let $X$ be a qcqs finite-
dimensional scheme. Then

$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)=\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda).$

######  Lemma 4.2.

Let $\Lambda$ be a commutative ring. Let $k$ be a field of characteristic
exponent $p$. Then, the Artin motive functor

$\rho_{!}\colon\mathrm{D}(k_{\mathrm{\acute{e}t}},\Lambda[1/p])\to\mathrm{DM}^{\mathrm{\acute{e}t}}(k,\Lambda[1/p])\xleftarrow{\sim}\mathrm{DM}^{\mathrm{\acute{e}t}}(k,\Lambda)$

is a fully faithful monoidal functor. Furthermore any torsion motive lies in
its essential image.

###### Proof.

If the result is known for $\Lambda=\mathbb{Z}$, then the result for an
arbitrary ring of coefficients $\Lambda$ follows from tensoring both sides
with $\mathrm{Mod}_{\Lambda}$: this operation preserves full faithfulness
according to [Hai22, Lemma 2.14]. The result then follows from [Rui22,
Corollary 1.2.3.4, Corollary 2.2.2.7].∎

######  Lemma 4.3.

Let $\Lambda=\mathrm{colim}\Lambda_{i}$ be a filtered colimit of rings and let
$X$ be a Noetherian scheme. Then the natural functor

$\mathrm{colim}_{i}\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda_{i})\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(X,\Lambda)$

is an equivalence.

###### Proof.

This proof was communicated to the authors by Denis-Charles Cisinski. Fully
faithfulness can be checked after tensorisation by $\mathbb{Q}$ and
$\mathbb{Z}/p\mathbb{Z}$. The rationalised statement follows from 3.8 and the
fact that
$\Lambda\mapsto\mathrm{Mod}_{\Lambda}(\mathrm{DM}^{\mathrm{\acute{e}t}}(X,\mathbb{Z}))$
commutes with filtered colimits ([HA, Theorem 4.8.5.11]). The mod $p$
statement can be proven as follows: a dévissage reduces to showing the
statement for dualisable objects, and then for maps out of the constant sheaf,
and even from maps out of the constant sheaf $\mathbb{Z}$, and it is now a
consequence of commutativity of étale cohomology with uniformly bounded below
complexes as in [HRS23, Lemma 5.3].

Now for the essential surjectivity first using 3.9, as it is clear that
geometric objects are in the image, we reduce to reaching an object of the
form $C/n$ which is of the form $\rho_{!}K$ with $K$ in
$\mathrm{D}_{\mathrm{cons}}(X,\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z})$.
Then, a dévissage reduces to a dualisable object for which by 3.3 there exists
an étale covering $f\colon U\to X$ such that $f^{*}(K)$ is constant with value
a complex $P$ in
$\mathrm{Perf}_{\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}}$. As in the
proof of 3.4 we see that $K=\mathrm{colim}_{n\in\Delta}(f_{n})_{\sharp}P$ with
$(f_{n})$ the Čech nerve of $f$. As $P$ is of the form
$Q\otimes_{\Lambda_{i}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}}(\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z})$
(this follows from the fact that the objects of $\mathrm{Perf}_{(-)}$ are the
compact objects of $\mathrm{Mod}_{(-)}$ which commutes with colimts), we see
that
$K=(\mathrm{colim}_{n\in\Delta}(f_{n})_{\sharp}Q)\otimes_{\Lambda_{i}\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}}\Lambda\otimes_{\mathbb{Z}}\mathbb{Z}/n\mathbb{Z}$
is the image of a dualisable object. ∎

######  Lemma 4.4.

Let $\Lambda$ be a commutative ind-regular ring. The theorem holds when $X$ is
the spectrum of a field $k$.

###### Proof.

Let $M$ be a constructible motive. We have to show that $M$ is geometric. 3.9
yields a geometric motive $P$ and a map $P\to M\oplus M[1]$ whose cone $C$ is
such that $C\otimes_{\mathbb{Z}}\mathbb{Q}=0$.

Furthermore, as $C$ is dualisable, we have

$\underline{\mathrm{Hom}}(C,\Lambda_{k})\otimes_{\mathbb{Z}}\mathbb{Q}=\underline{\mathrm{Hom}}(C,\Lambda_{k}\otimes_{\mathbb{Z}}\mathbb{Q})=\underline{\mathrm{Hom}}(C\otimes\mathbb{Q},\Lambda_{k}\otimes_{\mathbb{Z}}\mathbb{Q})=0.$

Hence, the dual of $C$ is also a torsion motive. It suffices to show that any
such $C$ is geometric.

Assume first that $\Lambda$ is a regular ring. Writing $C=\rho_{!}D$, then $D$
is a dualisable object of $\mathrm{D}(k_{\mathrm{\acute{e}t}},\Lambda[1/p])$.
In particular, it belongs to the thick stable subcategory generated by the
$f_{\sharp}\underline{\Lambda[1/p]}$ for $f$ finite étale according to [Rui22,
Proposition 2.1.2.15]. Thus, its image $C$ through $\rho_{!}$ is geometric as
$\rho_{!}$ is compatible with the functors $f_{\sharp}$ and $f^{*}$ for $f$
finite étale using [CD16, §4.4.2].

Now we let $\Lambda$ be an ind-regular commutative ring, we write it as
$\Lambda=\mathrm{colim}_{i}\Lambda_{i}$ with each $\Lambda_{i}$ regular. By
4.3 any object of $\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(k,\Lambda)$ comes
from a object in $\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(k,\Lambda_{i})$ for
some $i$, hence is geometric by the above paragraph. ∎

###### Proof.

(of 4.1) Using 3.5 and 3.10, we can assume $X$ to be Noetherian. This is now
an adaptation of the arguments given in [CD16, Lemma 6.3.25]. Let $M$ be a
constructible motive on $X$. By Noetherian induction, using the localisation
property and the stability of geometric motives under pushforwards by closed
immersions it suffices to show that there is an open immersion $j\colon U\to
X$ such that $j^{*}(M)$ is geometric, i.e. we can replace $X$ with any non-
empty open subscheme. Therefore, we can assume $X$ to be irreducible. Let
$\eta$ be the generic point of $X$. Consider the commutative diagram:

${\underset{U\text{ open, }\eta\in
U}{\mathrm{colim}}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(U,\Lambda)}$${\underset{U\text{
open, }\eta\in
U}{\mathrm{colim}}\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(U,\Lambda)}$${\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(\eta,\Lambda)}$${\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(\eta,\Lambda).}$$\scriptstyle{\alpha}$$\scriptstyle{u}$$\scriptstyle{v}$$\scriptstyle{\beta}$

Then, 4.4 above implies that $\beta$ is an equivalence and 3.5 implies that
$\alpha$ is an equivalence. On the other hand $u$ is fully faithful and $v$ is
an equivalence by 3.10. Hence, they are both equivalences. The fact that $u$
is an equivalence yields the result. ∎

######  Example 4.5.

This result fails without the assumption that $\Lambda$ is regular: take
$\Lambda=\mathbb{F}_{2}[x]/(x^{2}-1)$. Then, by rigidity, we have

$\mathrm{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec}(\mathbb{R}),\Lambda)\underset{\rho_{!}}{\xleftarrow{\sim}}\mathrm{Sh}(\mathbb{R}_{\mathrm{\acute{e}t}},\Lambda)\simeq\mathrm{D}(\Lambda[C_{2}])$

where $C_{2}$ is the cyclic group with $2$ elements. Taking dualisable
objects, we get an equivalence between
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(\mathrm{Spec}(\mathbb{R}),\Lambda)$ and
the subcategory $\mathrm{D}_{\Lambda\text{-perf}}(\Lambda[C_{2}])$ of
$\mathrm{D}(\Lambda[C_{2}])$ made of those complexes with a perfect underlying
complex of $\Lambda$-modules.

If $f\colon X\to\mathbb{R}$ is smooth, this equivalence sends
$M_{\mathbb{R}}(X)(n)$ to the complex $f_{\sharp}(\Lambda)(n)$. The latter is
isomorphic to
$(f_{\sharp}(\mathbb{F}_{2})(n))\otimes_{\mathbb{F}_{2}}\Lambda$. As by [BG23,
Corollary 4.5], $f_{\sharp}(\mathbb{F}_{2})(n)$ belongs to the thick
subcategory generated by $\mathbb{F}_{2}$ and $\mathbb{F}_{2}[C_{2}]$, the
complex $f_{\sharp}(\Lambda)(n)$ therefore belongs to the thick subcategory
$\mathrm{D}_{\mathrm{perm}}(\Lambda[C_{2}])$ generated by $\Lambda$ and
$\Lambda[C_{2}]$. Hence, through this equivalence
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(\mathrm{Spec}(\mathbb{R}),\Lambda)$
corresponds to $\mathrm{D}_{\mathrm{perm}}(\Lambda[C_{2}])$.

Consider the complex $\Lambda_{x}$ of $\Lambda[C_{2}]$-modules whose
underlying complex is $\Lambda$ placed in degree $0$ and with the action of
the generator of $C_{2}$ given by multiplication by $x$. The complex
$\Lambda_{x}$ belongs to $\mathrm{D}_{\Lambda\text{-perf}}(\Lambda[C_{2}])$
but by [BG22, Example 3.15] it does not belong to
$\mathrm{D}_{\mathrm{perm}}(\Lambda[C_{2}])$. Hence, the inclusion
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(\mathrm{Spec}(\mathbb{R}),\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{c}(\mathrm{Spec}(\mathbb{R}),\Lambda)$
is strict.

In fact, the result fails even for a characteristic $0$ non-regular ring: take
$\Lambda^{\prime}=\mathbb{Z}[x]/(x^{2}-1)$, and consider the
$\Lambda^{\prime}[C_{2}]$-module $\Lambda^{\prime}_{x}$, defined in the same
way. It can be seen as an étale motive through the Artin motive functor
$\rho_{!}\colon\mathrm{Sh}(\mathbb{R}_{\mathrm{\acute{e}t}},\Lambda^{\prime})\to\mathrm{DM}^{\mathrm{\acute{e}t}}(\mathrm{Spec}(\mathbb{R}),\Lambda^{\prime})$.
This object is dualisable, and if it were geometric then its image under the
functor $-\otimes_{\Lambda^{\prime}}\Lambda$, which is $\Lambda_{x}$ would
also be geometric, which is not.

##  5 Milnor excision and h-descent.

######  Proposition 5.1.

Let $\Lambda$ be a commutative ring. The functor
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ satisfies cdh-
descent over qcqs finite-dimensional schemes. Over qcqs schemes of finite
valuative dimension, it also satisfies cdh hyperdescent.

###### Proof.

Since the presentable $\infty$-category
$\mathrm{DM}^{\mathrm{\acute{e}t}}(-,\Lambda)$ satisfies Nisnevich descent,
the localisation property and proper base change, it satisfies cdh-descent as
the proof of [CD19, Theorem 3.3.10] applies in this level of generality. Thus
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ satisfies non
effective cdh descent. The effectivity of descent follows from cdh-excision: a
descent data of geometric motives gives us a motive as a finite limit of
proper pushforwards, which is itself geometric by 3.6 (1). For hyperdescent,
this follows from the fact that the cdh topos is hypercomplete over qcqs
schemes of finite valuative dimension by [EHIK21a, Corollary 2.4.16]. ∎

We now prove that geometric étale motives satisfy h-descent in the sense of
Rydh.

######  Theorem 5.2.

Let $\Lambda$ be a commutative ind-regular ring. The functor
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ satisfies
h-descent in the sense of [Stacks, Tag 0ETQ] over qcqs finite-dimensional
schemes.

###### Proof.

By [BS17, Theorem 2.9], h-descent is equivalent to cdh-descent combined with
fppf-descent. Also, fppf-descent is the same as finite flat descent together
with Nisnevich descent, see [GK15, Definition 2.5(15)]. Therefore it suffices
to prove that $\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ has
descent for the finite flat topology. Let $f\colon T\to S$ be a finite
morphism.

By writing $S=\lim_{i}S_{i}$ where the $S_{i}$ are affine $\mathbb{Z}$-schemes
of finite type, we can find an index $i_{0}$ such that $f$ is the pullback of
a finite map $f_{i_{0}}\colon T_{i_{0}}\to S_{i_{0}}$ using [EGA IV, Théorème
8.10.5]. By Noetherian induction, we can find a finite stratification
$\\{S_{i_{0}}^{1},\ldots,S_{i_{0}}^{p}\\}$ of $S_{i_{0}}$ such that
$S_{i_{0}}^{k}$ is open in $\bigcup_{k^{\prime}\geqslant
k}S_{i_{0}}^{k^{\prime}}$ for each $k$ and $f_{i_{0}}$ is the composition of a
finite étale map and a purely inseparable map over each $S_{i_{0}}^{k}$.
Hence, pulling back this stratification to $S$, we get a stratification
$\\{S^{1},\ldots,S^{p}\\}$ of $S$ with the same properties.

First we prove non-effective descent. For this, it suffices to prove that for
any objects $M$ and $N$ of
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(S,\Lambda)$, the morphism

$\mathrm{Map}_{\mathrm{DM}^{\mathrm{\acute{e}t}}}(M,N)\to\lim_{n\in\Delta}\mathrm{Map}_{\mathrm{DM}^{\mathrm{\acute{e}t}}}(M,(f_{n})_{*}f_{n}^{*}N)$

is an equivalence, where for any non-negative integer $n$, we let $f_{n}\colon
T_{n}\to S$ be the projection given by the Čech nerve of $f$. Assume first
that we have a closed immersion $i\colon Y\to S$ with open complement $j\colon
U\to S$ such that the above map is an equivalence for $(j^{*}M,j^{*}N)$ and
$(i^{*}M,i^{*}N)$. Then using that limit of exact triangles are exact
triangles we have a morphism of exact triangles of spectra:

${{\mathrm{Map}(M,j_{\sharp}j^{*}N)}}$${{\mathrm{Map}(M,N)}}$${{\mathrm{Map}(M,i_{*}i^{*}N)}}$${{\lim\limits_{n\in\Delta}\mathrm{Map}(M,(f_{n})_{*}f_{n}^{*}j_{\sharp}j^{*}N)}}$${{\lim\limits_{n\in\Delta}\mathrm{Map}(M,(f_{n})_{*}f_{n}^{*}N)}}$${{\lim\limits_{n\in\Delta}\mathrm{Map}(M,(f_{n})_{*}f_{n}^{*}i_{*}i^{*}N)}}$

But we have canonical identifications

$(f_{n})_{*}f_{n}^{*}j_{\sharp}j^{*}N\simeq
j_{\sharp}(f_{n})_{*}f_{n}^{*}j^{*}N\text{ and
}(f_{n})_{*}f_{n}^{*}i_{*}i^{*}N\simeq i_{*}(f_{n})_{*}f_{n}^{*}i^{*}N,$

thus the left and right vertical maps of the above diagram are equivalences by
assumption, so that the middle one also. Therefore, by induction on the number
$p$ of strata of $S$, this proves the full faithfulness by induction, given
that we have descent along finite étale maps by 2.5 and that localisation
ensures that $\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ is
an equivalence along purely inseparable maps.

For the effectivity of descent, we also prove the result by induction. Keeping
the same notations as before, we have the following morphism of cofiber
sequences of $\infty$-categories:

${{\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(Y,\Lambda)}}$${{\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)}}$${{\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(U,\Lambda)}}$${{\lim\limits_{n\in\Delta}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(Y_{n},\Lambda)}}$${{\lim\limits_{n\in\Delta}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{n},\Lambda)}}$${{\lim\limits_{n\in\Delta}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(U_{n},\Lambda)}}$$\scriptstyle{\sim}$$\scriptstyle{\tilde{i_{*}}}$$\scriptstyle{\tilde{j^{*}}}$$\scriptstyle{j^{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{\sim}$

in which the left vertical map is an equivalence by induction, and the right
vertical map is an equivalence by étale descent. The functors $\tilde{i_{*}}$
and $\tilde{j_{\sharp}}$ have left adjoint $\tilde{i^{*}}$ and
$\tilde{j_{\sharp}}$ that are computed term wise. In particular as we have a
cofiber sequence of stable $\infty$-categories, in
$\lim_{\Delta}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{n},\Lambda)$
we have an exact sequence

$\tilde{j_{\sharp}}\tilde{j^{*}}\to\mathrm{Id}\to\tilde{i_{*}}\tilde{i^{*}},$

which is compatible with the same exact sequence in
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$. Thus, if
$(M_{n})$ is some descent data in
$\lim_{\Delta}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{n},\Lambda)$,
there are objects $N_{U}$ and $N_{Y}$ in
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(U,\Lambda)$ and
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(Y,\Lambda)$ such that they
correspond to the restrictions of $(M_{n})$ to $(U_{n})$ and $(Y_{n})$. Thus
in
$\lim_{\Delta}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X_{n},\Lambda)$
there is a map $\tilde{i_{*}}N_{Y}\to\tilde{j_{\sharp}}N_{U}[1]$ of fiber
$(M_{n})$, which by full faithfulness gives a similar map in
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)$, whose fiber has
image $(M_{n})$. ∎

We finish the paper by proving Milnor excision.

######  Proposition 5.3.

Let $\Lambda$ be a commutative ring. The functor
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$ satisfies Milnor
excision ([EHIK21a, Definition 3.2.2]) over qcqs finite-dimensional schemes.

###### Proof.

First, using [EHIK21a, Lemma 3.2.6] any Milnor square (i.e. of the form below,
bicartesian, with $f$ affine and $i$ a closed immersion) of qcqs finite-
dimensional schemes

${W}$${Y}$${Z}$${X}$$\scriptstyle{g}$$\scriptstyle{k}$$\scriptstyle{f}$$\scriptstyle{i}$

can be approximated by Milnor squares of finitely presented
$\mathbb{Z}$-schemes (and therefore of Noetherian schemes) such that $f$ is of
finite type. By continuity (3.5), and as filtered colimits are compatible with
finite limits in $\mathrm{Cat}_{\infty}$, we can therefore assume our schemes
to be Noetherian and finite-dimensional and $f$ to be of finite type.

Because pushforwards preserve geometric motives over quasi-excellent schemes
by [CD16, Theorem 6.2.13], the functor

$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(X,\Lambda)\to\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(Y,\Lambda)\times_{\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(W,\Lambda)}\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(Z,\Lambda)$
(5.3.1)

admits a right adjoint. By localization, the family of functors
$(f^{*},i^{*})$ is conservative thus the above functor is conservative. To
show that it is an equivalence, it suffices to show that its right adjoint is
fully faithful. Thus take geometric motives $M_{Y}$, $M_{Z}$ and $M_{W}$ over
$Y$, $Z$ and $W$ such that we have equivalences $k^{*}(M_{Y})\simeq
M_{W}\simeq g^{*}(M_{Z})$. Let $M$ be the fiber of the map $f_{*}(M_{Y})\oplus
i_{*}(M_{Z})\to f_{*}k_{*}(E_{W})$. It is the image under the right adjoint of
5.3.1 of the object $(M_{Y},M_{Z},W)$. To finish the proof, it suffices to
show that the natural maps $f^{*}(M)\to M_{Y}$ and $i^{*}(M)\to M_{Z}$ are
equivalences. Assume first that the ring $\Lambda$ is flat over the integers.
Using 3.1, we can therefore assume $\Lambda$ to be a $\mathbb{Q}$-algebra or
to be a $\mathbb{Z}/p\mathbb{Z}$-algebra for some prime number $p$. The first
case follows from [EHIK21b, Theorem 5.6, Lemma 2.1] while the second case
follows from [BM21, Theorem 5.14] and [EHIK21a, Corollary 3.2.11]. Now for a
ring $\Lambda$ of finite type over the integers that we write as $R/I$ with
$R$ a polynomial ring, we proceed as follows: as $\Lambda$ is perfect over
$R$, the three motives $M_{Y}$, $M_{Z}$ and $M_{W}$ map to geometric objects
of $\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,R)$, together with
descent data. In particular, the maps $f^{*}(M)\to M_{Y}$ and $i^{*}(M)\to
M_{Z}$ are equivalences in
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,R)$ hence in
$\mathrm{DM}^{\mathrm{\acute{e}t}}_{\mathrm{gm}}(-,\Lambda)$. Thus the functor
5.3.1 is an equivalence whenever $\Lambda$ is of finite type over the
integers. Finally for a general ring, we can write it as a colimit of ring of
finite type over the integers and with 4.3 this is just a consequence of the
commutation of filtered colimits with finite limits. ∎

## Acknowledgments.

The authors would like to thank Denis-Charles Cisinski very warmly for his
interest in this work and his help on some technical points that have improved
greatly the quality of the paper, in particular for mentioning the existence
of 3.3 and for indications regarding the proof of 4.3. We thank Martin
Gallauer for answering a question on [BG22, Example 3.15]. We are grateful to
Sophie Morel and Frédéric Déglise for their support and for useful comments.
Finally, we would like to thank Fabrizio Andreatta, Federico Binda, Vova
Sosnilo and Alberto Vezzani for some useful conversations.

We were funded by the ANR HQDIAG, RR was funded by the _Prin 2022: The
arithmetic of motives and L-functions_ and ST had his PhD fundings given by
the É.N.S de Lyon.

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Raphaël Ruimy<EMAIL_ADDRESS>
Dipartimento di Matematica "F. Enriques" - Università degli Studi di Milano,

via Cesare Saldini 50, I-20133 Milano, Italy

Swann Tubach<EMAIL_ADDRESS>
E.N.S Lyon, UMPA,

46 Allée d’Italie, 69364 Lyon Cedex 07, France