# Étale cohomology of algebraic varieties over Stein compacta

Olivier Benoist  Département de mathématiques et applications, École normale
supérieure, CNRS, 45 rue d’Ulm, 75230 Paris Cedex 05, France
<EMAIL_ADDRESS>
###### Abstract.

We prove a comparison theorem between the étale cohomology of algebraic
varieties over Stein compacta and the singular cohomology of their
analytifications. We deduce that the field of meromorphic functions in a
neighborhood of a connected Stein compact subset of a normal complex space of
dimension $n$ has cohomological dimension $n$. As an application of
$\mathrm{Gal}(\mathbb{C}/\mathbb{R})$-equivariant variants of these results,
we obtain a quantitative version of Hilbert’s 17th problem on compact subsets
of real-analytic spaces.

## Introduction

###  0.1. Hilbert’s 17th problem

Hilbert discovered in [Hil88] that real polynomials
$f\in\mathbb{R}[x_{1},\dots,x_{n}]$ that are positive semidefinite (i.e., that
only take nonnegative values on $\mathbb{R}^{n}$) may not always be written as
sums of squares of polynomials. He however conjectured, in his celebrated
$17$th problem, that this issue may be resolved by allowing denominators: the
polynomial $f$ should always be a sum of squares of rational functions. This
problem was solved positively by E. Artin [Art27, Satz 4] in 1927. Forty years
later, Pfister [Pfi67, Theorem 1] obtained a quantitative improvement of
Artin’s theorem bounding the number of squares required. It states that a
positive semidefinite $f\in\mathbb{R}[x_{1},\dots,x_{n}]$ is a sum of $2^{n}$
squares in $\mathbb{R}(x_{1},\dots,x_{n})$.

###  0.2. The real-analytic variant

As an application of the main results of this article, we prove a quantitative
theorem à la Pfister in real-analytic geometry.

######  Theorem 0.1 (Theorem 6.5).

Let $M$ be a normal real-analytic variety of pure dimension $n$. Let $K\subset
M$ be compact. Any nonnegative real-analytic function on $M$ is a sum of
$2^{n}$ squares of real-analytic meromorphic functions in a neighborhood of
$K$.

Much less is known on Hilbert’s 17th problem in the real-analytic setting,
first considered in [BR75] (see [ABF22] for an up-to-date survey), than in the
algebraic case. A nonnegative real-analytic function on a normal and pure-
dimensional real-analytic variety $M$ is a sum of squares of real-analytic
meromorphic functions in dimension $\leq 2$ (see [Jaw82, Corollary 2] and
[ADCR03, Theorem 1]), under a compactness hypothesis (see [Rui85, Theorem 1]
and [Jaw86, Theorem 1]) or, when $M$ is a manifold, under a discreteness
hypothesis on the zeros of $f$ [BKS81, Theorem 1]. However, whether a
nonnegative real-analytic function on $\mathbb{R}^{n}$ is a sum of squares of
real-analytic meromorphic functions is still an open problem for $n\geq 3$.

In addition, quantitative results are known in dimension $\leq 2$. A
nonnegative real-analytic function on a normal real-analytic surface $M$ is a
sum of $3$ squares of real-analytic functions if $M$ is a manifold [Jaw82,
Corollary 2], and a sum of $5$ squares of real-analytic meromorphic functions
in general (see [ABFR05, Theorem 1.2] or the more general [Fer21, Theorem
1.3]). No bounds on the required number of squares were known if $n\geq 3$
(except in the local case, for which see [Ben20, Theorem 0.2]). Theorem 0.1
rectifies this situation, under a compactness hypothesis.

Although the qualitative content of Theorem 0.1 is not new (see [Jaw86,
Theorem 1]), the quantitative bounds it provides (more precisely, that these
bounds only depend on the dimension of $M$) do imply new cases of (the
qualitative version of) the real-analytic Hilbert’s 17th problem. Indeed, work
of Acquistapace, Broglia, Fernando and Ruiz [ABFR10, Proposition 1.8] readily
implies the following.

######  Corollary 0.2.

Let $f$ be a nonnegative real-analytic function on a real-analytic manifold.
If the zero set of $f$ is a disjoint union of compact sets, then $f$ is a
finite sum of squares of real-analytic meromorphic functions.

The best results to date in this vein handled the case where the zero locus of
$f$ is the union of a compact set and of a discrete set (see [Jaw92, Theorem
2] or [ABFR10, Corollary 1.10]), or concerned infinite sums of squares
[ABFR10, Corollary 1.9]. We also refer to [ABF14, Theorem 1.1] for
quantitative results in this direction.

###  0.3. From real-analytic to complex-analytic geometry

In real algebraic geometry, it is very important to consider not only the sets
of real points of real algebraic varieties, but also their sets of complex
points, endowed with the action of $G:=\mathrm{Gal}(\mathbb{C}/\mathbb{R})$ by
complex conjugation. For the same reason, it is crucial for our proof of
Theorem 0.1 to work in the setting of $G$-equivariant complex-analytic
geometry (see §5.3 for our conventions) instead of real-analytic geometry.

This point of view also leads to a more general theorem on sums of squares,
from which Theorem 0.1 is easily derived using the successive works of Cartan
[Car57b], Grauert [Gra58] and Tognoli [Tog67] on complexifications of real-
analytic spaces. Recall that a compact subset of a complex space is said to be
Stein if it admits a basis of Stein open neighborhoods.

######  Theorem 0.3 (Theorem 6.4).

Let $S$ be a reduced $G$-equivariant Stein space of dimension $n$. Let
$K\subset S$ be a $G$-invariant Stein compact subset. Any $G$-invariant
holomorphic function on $S$ which is nonnegative on $S^{G}$ is a sum of
$2^{n}$ squares of $G$-invariant meromorphic functions in a neighborhood of
$K$.

We believe that this $G$-equivariant variation on the analytic Hilbert’s 17th
problem is novel, and that the new statements it comprises (for instance when
$S=\mathbb{C}^{n}$ and $K\subset S$ is the closed unit ball) are of interest.

###  0.4. Cohomological dimension of fields of meromorphic functions

It has been known since Voevodsky’s proof of the Milnor conjectures [Voe03]
that quadratic forms over a field are largely governed by the Galois
cohomology of the field. As a consequence of these results, the validity of
Theorem 0.3 is controlled by the vanishing of a single Galois cohomology class
of degree $n+1$ (see [Ben20, Proposition 2.1]). Using this point of view,
Theorem 0.3 follows from an upper bound for the cohomological dimension of
fields of $G$-invariant meromorphic functions on $G$-equivariant Stein
compacta (Corollary 5.8). In this introduction, let us only state a
non-$G$-equivariant version of this theorem.

######  Theorem 0.4 (Corollary 5.11 and Remark 5.12 (ii)).

Let $K$ be a connected Stein compact subset of a normal Stein space $S$ of
dimension $n$. The field $\mathcal{M}(K)$ of germs of meromorphic functions in
a neighborhood of $K$ has cohomological dimension $n$.

In dimension $1$, Theorem 0.4 is attributed to M. Artin by Guralnick ([Gur88,
Proposition 3.7], see also [BW21, Proposition A.6]), with no compactness
hypothesis. It is new in dimension $\geq 2$. Theorem 0.4 is an algebraic
statement concerning a field of analytic origin, and its proof uses a mixture
of analytic and algebraic tools. As such, it is deeper than its algebraic
counterpart [Ser94, II, Proposition 11] bounding the cohomological dimension
of function fields of algebraic varieties.

We also obtain bounds on the étale cohomological dimension of affine varieties
over (possibly $G$-equivariant) Stein compacta (see Theorems 5.6 and 5.9). We
refer to the work of Bhatt and Mathew [BM21, Theorem 7.3 and Remark 7.4] for
related bounds in non-archimedean analytic geometry.

###  0.5. Étale and singular cohomology

By a complex-analytic incarnation due to Hamm [Ham83, Satz 1] of the weak
Lefschetz theorem, a Stein space of dimension $n$ has the homotopy type of a
CW complex of dimension $\leq n$. Its singular cohomology therefore vanishes
in degree $>n$. Our strategy of proof of Theorem 0.4 is to transfer this
topological information on the singular cohomology of a Stein compactum to
algebraic information on the étale cohomology of its ring of holomorphic
functions, and eventually on the Galois cohomology of its field of meromorphic
functions.

In algebraic geometry, such a transfer tool exists in the shape of M. Artin’s
comparison theorem [SGA43, XVI, Théorème 4.1] between the étale cohomology of
a complex algebraic variety $X$ and the singular cohomology of its
analytification $X^{\mathrm{an}}$. In order to implement the strategy
described above, we prove the following analogue of Artin’s comparison theorem
in analytic geometry.

######  Theorem 0.5 (Theorem 5.1).

Let $S$ be a Stein space. Let $X$ be an $\mathcal{O}(S)$-scheme of finite type
and let $\mathbb{L}$ be a torsion étale abelian sheaf on $X$. Assume that $X$
is proper over $\mathcal{O}(S)$ or that $\mathbb{L}$ is constructible. If one
lets $U$ run over all Stein open neighborhoods of a Stein compact subset $K$
of $S$, the change of topology morphisms

$\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(U)},\mathbb{L}_{\mathcal{O}(U)})\to\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}((X_{\mathcal{O}(U)})^{\mathrm{an}},\mathbb{L}^{\mathrm{an}})$

are isomorphisms for $k\geq 0$.

Theorem 0.5 is our main result. The bulk of this article is dedicated to its
proof. For applications to sums of squares, we really need a $G$-equivariant
extension of Theorem 0.5 (see Theorem 5.5), which we deduce from Theorem 0.5
by means of the Hochschild–Serre spectral sequence.

###  0.6. Proof of the comparison theorem

Two proofs of Artin’s comparison theorem appear in [SGA43]. The first [SGA43,
XI, §4] (which only works for smooth varieties and locally constant
coefficients) compares the étale and the classical topology by means of a
Leray spectral sequence. It exploits the fact that points in smooth complex
varieties admit good neighborhoods, which are iterated fibrations in smooth
affine curves. The second [SGA43, XVI, §4] uses extensive dévissage and
fibration arguments to reduce to the case of smooth projective curves.

None of these proofs adapt to the setting of Theorem 0.5 as such fibration
arguments cannot be successfully implemented in Stein geometry. However, the
dévissage argument of the second proof can indeed be used in the complex-
analytic setting to yield the following easier relative comparison theorem
over Stein spaces.

######  Theorem 0.6 (Theorem 3.7).

Let $S$ be a Stein space. Let $f:X\to Y$ be a morphism of
$\mathcal{O}(S)$-schemes of finite type and let $\mathbb{L}$ be a torsion
étale abelian sheaf on $X$. Assume that $f$ is proper or that $\mathbb{L}$ is
constructible. Then the base change morphisms
$\mathrm{R}^{k}f_{*}(\mathbb{L})^{\mathrm{an}}\to\mathrm{R}^{k}f^{\mathrm{an}}_{*}(\mathbb{L}^{\mathrm{an}})$
are isomorphisms for $k\geq 0$.

In contrast, our proof of Theorem 0.5 initially takes the point of view of the
first proof of Artin’s theorem. Reductions based on Theorem 0.6 allow us to
assume that $X=\mathrm{Spec}(\mathcal{O}(S))$ and $\mathbb{L}=\mathbb{Z}/m$.
We then consider the Leray spectral sequence comparing the étale and the
classical topology. In order to show that this spectral sequence degenerates,
one has to prove that singular cohomology classes with $\mathbb{Z}/m$
coefficients on analytifications of étale $\mathcal{O}(S)$-schemes are étale-
locally trivial.

We proceed in two steps. First, we show that such cohomology classes become
unramified after pull-back to a finite ramified covering (Proposition 4.3).
This uses Theorem 0.6 in an essential way, as well as a vanishing theorem of
Bhatt [Bha12, Theorem 1.1]. Second, we show that such unramified cohomology
classes may be killed by a further finite ramified covering (Proposition 2.4).
This second step is analytic in nature. It relies on the Oka–Weil
approximation theorem and on Grauert’s bump method as developed by Henkin and
Leiterer [HL98] and Forstnerič [For17].

This procedure unfortunately only shows that the relevant singular cohomology
classes are killed on possibly ramified coverings, not on étale coverings. To
overcome this difficulty, our proof of Theorem 0.5 makes use of Voevodsky’s
qfh topology [Voe96, §3.1], which is finer than the étale topology and allows
for such coverings.

The relative comparison theorem (Theorem 0.6) is an analogue for étale sheaves
of the relative GAGA theorem of Hakim [Hak72, VIII, Théorème 3.2] (see also
[Bin76b, Theorem 4.2] or [AT19, Appendix C]) for coherent sheaves. In the
coherent context, there is no need for an absolute comparison theorem such as
Theorem 0.5, as higher cohomology groups of algebraic coherent sheaves on
affine schemes, or of analytic coherent sheaves on Stein spaces, vanish. This
is a marked difference with the étale setting, in which the absolute
comparison theorem lies deeper.

In this article, we make extensive use of relative algebraic geometry over
complex spaces as initiated by Hakim [Hak72], and developed by Bingener
[Bin76b] when the base is Stein. We refer to §3.1 for our conventions which
follow [Bin76b]. In particular, we use the above-mentioned relative GAGA
theorem, through an application of [Bin76b, (7.2)], in Step 4 of the proof of
Theorem 0.5.

###  0.7. Structure of the article

Section 1 gathers general results concerning Stein spaces. Most of this
material is included to fix our conventions, for the convenience of the
reader, or for lack of appropriate references. Specialists of Stein geometry
might want to skip it. Section 2 contains the main analytic input of our work:
a procedure to kill a singular cohomology class with torsion coefficients on a
Stein compactum, after pull-back to a finite ramified covering. We deduce
Theorem 0.6 from Artin’s comparison theorem and its proof in Section 3, and we
use it to kill the ramification of singular cohomology classes with torsion
coefficients after pull-back to finite ramified coverings in Section 4. The
above results are combined in Section 5 to prove Theorem 0.5, and to derive
cohomological dimension bounds including Theorem 0.4. Applications to sums of
squares are given in Section 6.

###  0.8. Acknowledgements

I thank Jean-Benoît Bost for a helpful conversation and for having made
[Bin76b] available to me, and José F. Fernando and Marco Maculan for useful
comments.

##  1\. Geometry of Stein spaces

Among the many general facts on Stein spaces and their compact subsets that
are collected in this section, let us put forward a descent result for
$\mathcal{O}(S)$-convexity along finite surjective holomorphic maps
(Proposition 1.5), and the correspondence between finite ramified coverings of
connected normal Stein spaces and finite extensions of their meromorphic
function fields (Proposition 1.13).

###  1.1. Complex spaces and coherent sheaves

A complex space is a $\mathbb{C}$-ringed space that is locally isomorphic to a
model complex space defined by a finitely generated ideal sheaf in a domain of
$\mathbb{C}^{N}$ (see [GR84, 1, §1.5]). We assume them to be second countable,
but not necessarily Hausdorff, finite-dimensional or reduced.

As in [GR79], a complex space $S$ is said to be Stein if
$H^{k}(S,\mathcal{F})=0$ for all $k\geq 1$ and all coherent sheaves
$\mathcal{F}$ on $S$. We refer to this property as Cartan’s Theorem B and to
its consequence that coherent sheaves on Stein spaces are globally generated
as Cartan’s Theorem A. Stein spaces are Hausdorff (holomorphic functions
separate distinct $s,s^{\prime}\in S$ as
$H^{1}(S,\mathcal{I}_{\\{s,s^{\prime}\\}})=0$). A complex space $S$ is Stein
if and only if its reduction $S^{\mathrm{red}}$ is [GR79, V, §4.3, Theorem 5].
The following lemma sometimes allows us to reduce problem about Stein spaces
to the case of Stein manifolds.

######  Lemma 1.1.

Let $S$ be a Stein space of dimension $n$. Then there exist a Stein manifold
$S^{\prime}$ of dimension $2n+1$ and a proper injective holomorphic map
$i:S\to S^{\prime}$ such that $i(S)$ is a strong deformation retract of
$S^{\prime}$.

###### Proof.

Let $i:S^{\mathrm{red}}\to\mathbb{C}^{2n+1}$ be a proper injective holomorphic
map (see [Nar60, Theorem 5]). As
$\mathcal{O}(S)\to\mathcal{O}(S^{\mathrm{red}})$ is onto (by Cartan’s Theorem
B), one may extend it to a proper injective holomorphic map
$i:S\to\mathbb{C}^{2n+1}$. Its image ${i(S)\subset\mathbb{C}^{2n+1}}$ is a
Stein analytic subset (by [GR84, 3, §1.3, Proposition] and [GR79, V, §1,
Theorem 1 b)]). The existence of $S^{\prime}$ now follows from [Mih96, Theorem
3.1]. ∎

If $S$ is a complex space and $\mathcal{F}$ is a coherent sheaf on $S$, we
endow $H^{0}(S,\mathcal{F})$ with the canonical Fréchet topology defined in
[GR79, V, §6] and use freely its properties listed in [GR79, V, §6.4]. When
$S$ is reduced and $\mathcal{F}=\mathcal{O}_{S}$, it coincides with the
topology of uniform convergence on compact subsets [GR79, V, §6.6, Theorem 8].

###  1.2. Runge domains

If $K\subset S$ is a compact subset of a complex space, its
$\mathcal{O}(S)$-convex hull is $\widehat{K}_{\mathcal{O}(S)}:=\\{s\in
S\mid|f(s)|\leq\sup_{t\in K}|f(t)|\textrm{ for all }f\in\mathcal{O}(S)\\}.$ By
[GR79, V, §4.2, Theorem 3], a complex space $S$ is Stein if and only if

  1. (i)

the global holomorphic functions $\mathcal{O}(S)$ separates the points of $S$,
and

  2. (ii)

the $\mathcal{O}(S)$-convex hull $\widehat{K}_{\mathcal{O}(S)}\subset S$ of
any compact subset $K\subset S$ is compact.

An open subset $\Omega$ in a Stein space $S$ is said to be Runge if it
satisfies any of the equivalent properties of the following proposition. As we
could not find a precise discussion in the literature when $S$ is possibly
nonreduced, we include a brief proof. Such a generality will be useful in the
proof of Proposition 1.5.

######  Proposition 1.2.

Let $\Omega\subset S$ be an open subset in a Stein space. The following
assertions are equivalent, and hold for $(S,\Omega)$ if and only if they hold
for $(S^{\mathrm{red}},\Omega^{\mathrm{red}})$.

  1. (i)

For all compact subsets $K\subset\Omega$, one has
$\widehat{K}_{\mathcal{O}(\Omega)}=\widehat{K}_{\mathcal{O}(S)}$.

  2. (ii)

The image of the restriction map $\mathcal{O}(S)\to\mathcal{O}(\Omega)$ is
dense, and $\Omega$ is Stein.

  3. (iii)

For any coherent sheaf $\mathcal{F}$ on $S$, the restriction map
$H^{0}(S,\mathcal{F})\to H^{0}(\Omega,\mathcal{F})$ has dense image, and
$\Omega$ is Stein.

###### Proof.

Write $(*)^{\mathrm{red}}$ for assertion $(*)$ for the pair
$(S^{\mathrm{red}},\Omega^{\mathrm{red}})$. Then (i)red, (ii)red and (iii)red
are equivalent by [GR65, VII, A, Corollary 9 and VIII, A, Theorem 11]. The
implication (iii)$\Rightarrow$(ii) is obvious. If (ii) holds, the continuity
and surjectivity of the restriction map
$\mathcal{O}(\Omega)\to\mathcal{O}(\Omega^{\mathrm{red}})$ show that (ii)red
also holds. That (i)red implies (i) follows from the surjectivity of
$\mathcal{O}(S)\to\mathcal{O}(S^{\mathrm{red}})$. Conversely, if (i) holds,
then $\Omega$ is Stein ($\mathcal{O}(\Omega)$ separates the points of $\Omega$
because $\mathcal{O}(S)$ separates the points of $S$) and (i)red follows from
the surjectivity of
$\mathcal{O}(\Omega)\to\mathcal{O}(\Omega^{\mathrm{red}})$.

To prove (iii)${}^{\mathrm{red}}\Rightarrow$(iii), we argue as in [NS77, Lemma
1.10]. The characterization [GR79, V, §6.2, Theorem 4] of the topology on
$H^{0}(\Omega,\mathcal{F})$ shows that it is the initial topology with respect
to the restriction maps $H^{0}(\Omega,\mathcal{F})\to
H^{0}(\Omega^{\prime},\mathcal{F})$ with $\Omega^{\prime}\subset\Omega$
relatively compact. By [Nar62, (1.1)], we may restrict to those
$\Omega^{\prime}$ such that $(\Omega^{\prime})^{\mathrm{red}}$ is Runge in
$\Omega^{\mathrm{red}}$. We may thus replace $\Omega$ with such an
$\Omega^{\prime}$ and assume that $\Omega$ is relatively compact in $S$. Let
$\mathcal{N}\subset\mathcal{O}_{S}$ be the nilradical of $\mathcal{O}_{S}$. As
$\Omega$ is relatively compact in $S$, there exists $k\geq 0$ such that
$\mathcal{N}^{k}=0$ on $\Omega$. Then $\mathcal{F}$ is a successive extension
of
$\mathcal{F}/\mathcal{N}\mathcal{F},\dots,\mathcal{N}^{k-1}\mathcal{F}/\mathcal{N}^{k}\mathcal{F}$
and $\mathcal{N}^{k}\mathcal{F}$. As the set of coherent sheaves satisfying
(iii) is stable by extensions (use [NS77, Lemma 1.9]), we may suppose that
$\mathcal{N}\mathcal{F}=0$ on $\Omega$. As $H^{0}(S,\mathcal{F})\to
H^{0}(S,\mathcal{F}/\mathcal{N}\mathcal{F})$ is onto by Cartan’s Theorem B, we
may replace $\mathcal{F}$ with $\mathcal{F}/\mathcal{N}\mathcal{F}$. Now
$\mathcal{F}$ may be identified with a coherent sheaf on $S^{\mathrm{red}}$
and the required density statement follows from (iii)red. ∎

###  1.3. $\mathcal{O}(S)$-convex compact subsets

A compact subset $K$ of a complex space $S$ is said to be
$\mathcal{O}(S)$-convex if $\widehat{K}_{\mathcal{O}(S)}=K$. In the following
lemma, $\mathcal{C}^{\infty}$ strongly plurisubharmonic (psh) functions are
meant in the sense of [Nar61, §2].

######  Lemma 1.3.

Let $\rho:S\to\mathbb{R}$ be a $\mathcal{C}^{\infty}$ strongly psh function on
a reduced Stein space.

  1. (i)

For $c\in\mathbb{R}$, the open subset $\\{s\in S\mid\rho(s)<c\\}$ of $S$ is
Runge.

  2. (ii)

For $c\in\mathbb{R}$, if $S_{\leq c}:=\\{s\in S\mid\rho(s)\leq c\\}$ is
compact, then it is $\mathcal{O}(S)$-convex.

###### Proof.

Assertion (i) is [Nar61, Theorem 3]. The compactness of $S_{\leq c}$ and (i)
together imply that $S_{\leq c}$ has a basis of Runge neighborhoods in $S$,
hence is $\mathcal{O}(S)$-convex. ∎

We now analyze how $\mathcal{O}(S)$-convexity behaves with respect to finite
morphisms. Our goal is Proposition 1.5 (ii) which will be used in the proof of
Proposition 2.4.

######  Lemma 1.4.

Let $p:T\to S$ be a finite surjective holomorphic map between connected normal
Stein spaces. Let $K\subset S$ be a compact subset. Then

$\widehat{p^{-1}(K)}_{\mathcal{O}(T)}=p^{-1}(\widehat{K}_{\mathcal{O}(S)}).$

###### Proof.

The inclusion $\widehat{p^{-1}(K)}_{\mathcal{O}(T)}\subset
p^{-1}(\widehat{K}_{\mathcal{O}(S)})$ is obvious. To prove the converse
inclusion, we use the norm morphism
$\textrm{N}_{p}:p_{*}\mathcal{O}_{T}\to\mathcal{O}_{S}$ of $p$ (it is defined
on the locus over which $p$ is an unramified finite covering of manifolds by
summing over the fibers and it extends to all of $S$ by Riemann’s extension
theorem [GR84, 7, § 4.2]).

Choose $t\in T\setminus\widehat{p^{-1}(K)}_{\mathcal{O}(T)}$ and set
$s:=p(t)$. Let $\delta$ be the maximal cardinality of the fibers of $p$ over
some compact neighborhood of $K$. By the version [For17, Theorem 2.8.4] of the
Oka–Weil approximation theorem, there exists $g\in\mathcal{O}(T)$ with
$|g|\leq 1$ on $\widehat{p^{-1}(K)}_{\mathcal{O}(T)}$, with $g=0$ on
$p^{-1}(\\{s\\})\setminus\\{t\\}$ and with $g(t)=\delta+1$. Then
$f:=\textrm{N}_{p}(g)\in\mathcal{O}(S)$ satisfies $|f|\leq\delta$ on $K$, and
$|f(s)|>\delta$ (to see it, use that $p$ is open by [GR84, 3, §3.2, Criterion
of Openness]). Hence $s\notin\widehat{K}_{\mathcal{O}(S)}$. ∎

######  Proposition 1.5.

Let $p:T\to S$ be a finite holomorphic map between Stein spaces, and let
$K\subset S$ be a compact subset.

  1. (i)

If $K$ is $\mathcal{O}(S)$-convex, then $p^{-1}(K)$ is
$\mathcal{O}(T)$-convex.

  2. (ii)

If $p^{-1}(K)$ is $\mathcal{O}(T)$-convex and $p$ is surjective, then $K$ is
$\mathcal{O}(S)$-convex.

###### Proof.

Assertion (i) is immediate from the definitions and we now prove (ii). Using
(i), one may replace $T$ with its normalization and hence assume that $T$ is
normal. We may also replace $S$ with its reduction and assume that it is
reduced.

Choose $s\in S\setminus K$. To construct $f\in\mathcal{O}(S)$ such that
$|f|<1$ on $K$ and $|f(s)|=1$, we may first construct it in restriction to the
(finite) union of irreducible components of $S$ intersecting $K\cup\\{s\\}$
and then extend it to $S$ using Cartan’s Theorem B. We may thus assume that
$S$ has finite dimension $n$ and argue by induction on $n$.

Let $\widetilde{S}$ be the normalization of $S$. Using the natural
factorization $T\to\widetilde{S}\to S$ of $p$, we may either assume that $S$
is normal or that $p$ is a normalization morphism. In the first case, we may
suppose that $S$ is connected, and replace $T$ by any of its connected
components dominating $S$. The result then follows from Lemma 1.4.

We now deal with the second case where $p$ is a normalization morphism. Let
$\mathcal{I}\subset\mathcal{O}_{S}$ be the annihilitor of the cokernel of
$\mathcal{O}_{S}\hookrightarrow p_{*}\mathcal{O}_{T}$ (the conductor of $p$).
It is a coherent sheaf of $\mathcal{O}_{S}$-ideals by [GR84, Annex, §4.4]
which is also, in view of its definition, a sheaf of
$p_{*}\mathcal{O}_{T}$-ideals. We let $S^{\prime}\subset S$ and
$T^{\prime}\subset T$ be the (possibly nonreduced) complex subspaces defined
by $\mathcal{I}$. We get a commutative exact diagram of coherent sheaves on
$S$:

(1.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{p_{*}\mathcal{O}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{p_{*}\mathcal{O}_{T^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{S^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

By (i) applied to the inclusion $T^{\prime}\hookrightarrow T$, the subset
$p^{-1}(S^{\prime}\cap K)=T^{\prime}\cap p^{-1}(K)$ of $T^{\prime}$ is
$\mathcal{O}(T^{\prime})$-convex. The induction hypothesis applied to
$p|_{T^{\prime}}:T^{\prime}\to S^{\prime}$ then shows that $S^{\prime}\cap K$
is $\mathcal{O}(S^{\prime})$-convex. Use [Nar62, (1.1)] to choose a Runge
neighborhood $\Omega$ of $S^{\prime}\cap K$ in $S^{\prime}$ such that
$s\notin\Omega$. Apply [Nar62, (1.1)] again to construct a relatively compact
Runge neighborhood $\Theta$ of $p^{-1}(K)$ in $T$ that is disjoint from
$p^{-1}(\\{s\\})$, and such that $T^{\prime}\cap\Theta\subset p^{-1}(\Omega)$.

The restriction map $\mathcal{O}(\Theta)\to\mathcal{O}(T^{\prime}\cap\Theta)$
is continuous, and surjective by Cartan’s Theorem B. It thus follows from the
open mapping theorem for Fréchet spaces that there exists a neighborhood
$U\subset\mathcal{O}(T^{\prime}\cap\Theta)$ of the origin such that for all
$a\in U$, there exists $b\in\mathcal{O}(\Theta)$ such that
$b|_{T^{\prime}\cap\Theta}=a$ and $|b|<1$ on $p^{-1}(K)$. Let
$V\subset\mathcal{O}(\Omega)$ be the inverse image of $U$ by the (continuous)
pull-back map $\mathcal{O}(\Omega)\to\mathcal{O}(T^{\prime}\cap\Theta)$.

Set $\mathcal{F}:=\mathcal{O}_{S^{\prime}}$ if $s\notin S^{\prime}$ and
$\mathcal{F}:=\mathcal{I}_{\\{s\\}}\subset\mathcal{O}_{S^{\prime}}$ if $s\in
S^{\prime}$. By Proposition 1.2 (iii), there exists $c\in
H^{0}(S^{\prime},\mathcal{F})\subset\mathcal{O}(S^{\prime})$ with
$(1+c)|_{\Omega}\in V$. Set ${d:=1+p^{*}c\in\mathcal{O}(T^{\prime})}$ so that
$a:=d|_{T^{\prime}\cap\Theta}\in U$. By our choice of $U$, there exists
$b\in\mathcal{O}(\Theta)$ with
${b|_{T^{\prime}\cap\Theta}=d|_{T^{\prime}\cap\Theta}}$ and $|b|<1$ on
$p^{-1}(K)$. The Oka–Weil approximation theorem [For17, Theorem 2.8.4] now
shows the existence of $e\in\mathcal{O}(T)$ with $e|_{T^{\prime}}=d$, and such
that $|e|<1$ on $p^{-1}(K)$ and $e=1$ on $p^{-1}(\\{s\\})$.

Diagram (1.1) remains exact after taking global sections by Cartan’s Theorem
B. A diagram chase in the resulting diagram shows the existence of
$f\in\mathcal{O}(S)$ with $p^{*}f=e\in\mathcal{O}(T)$. One has $f(s)=1$ and
$|f|<1$ on $K$, so $K$ is $\mathcal{O}(S)$-convex. ∎

######  Remarks 1.6.

(i) In Proposition 1.5 (ii), one cannot remove the assumption that $S$ and $T$
are Stein, even if $f$ is a reduction morphism (see [Sch70, (8.5)]) or a
normalization morphism (see [Mar75, Theorem 3]).

(ii) In the setting of Proposition 1.5, the equality
$\widehat{p^{-1}(K)}_{\mathcal{O}(S)}=p^{-1}(\widehat{K}_{\mathcal{O}(T)})$
does not hold in general, for instance if $T=\\{(x,y)\in\mathbb{C}^{2}\mid
xy=0\\}$, if $p:S\to T$ if the normalization morphism, and if
$K=\\{(x,0)\in\mathbb{C}^{2}\mid|x|=1\\}\subset T$.

###  1.4. Stein compact subsets

If $\mathcal{F}$ is a sheaf on a complex space $S$ and $K\subset S$ is closed,
we let $\mathcal{F}(K)$ denote the set of germs of sections of $\mathcal{F}$
in a neighborhood of $K$.

A compact subset of a complex space $S$ is said to be Stein if it admits a
basis of Stein open neighborhoods. A Stein compactum is the germ of a complex
space along a Stein compact subset. As the intersection of two Stein open
subsets is Stein (see [GR79, p. 127]), an intersection of Stein compact
subsets is again Stein. By [Nar62, (1.1)], an $\mathcal{O}(S)$-convex compact
subset of a Stein space is Stein.

######  Lemma 1.7.

Let $S$ be a Stein space and let $K\subset L\subset S$ be Stein compact
subsets. Then the ring morphisms $\mathcal{O}(S)\to\mathcal{O}(K)$ and
$\mathcal{O}(L)\to\mathcal{O}(K)$ are flat.

###### Proof.

That $\mathcal{O}(S)\to\mathcal{O}(K)$ is flat is proven in [Kuc05, Proof of
Lemma 2.2]. There, the compact subset $K$ is assumed to be
$\mathcal{O}(S)$-convex, but only the fact that it is Stein is used. In
addition, the Stein space $S$ is assumed to be a connected manifold. The proof
extends to our more general setting, replacing references to Hörmander’s book
with applications of Cartan’s Theorems A and B (in the generality of [GR79]).

Let $(U_{i})_{i\in\mathbb{N}}$ be a decreasing basis of Stein open
neighborhoods of $L$ in $S$. By the above, the ring $\mathcal{O}(K)$ is flat
over $\mathcal{O}(U_{i})$ for all $i\in\mathbb{N}$. By the flatness criterion
[Mat80, Theorem 1 (6)], the ring $\mathcal{O}(K)$ is flat over
$\mathrm{colim}_{i\in\mathbb{N}}(\mathcal{O}(U_{i}))=\mathcal{O}(L)$. ∎

We will say that a Stein compact subset $K$ of a Stein space $S$ is excellent
if the ring $\mathcal{O}(K)$ is noetherian. By a theorem of Siu [Siu69,
Theorem 1], this is the case if and only if for every germ $Z$ of closed
analytic subset along $K$, the subset $Z\cap K$ of $K$ has finitely many
connected components. This holds in particular if $K$ is semianalytic (see
[Fri67, Théorème I.9]). Excellent Stein compact subsets are plentiful as the
next lemma shows.

######  Lemma 1.8.

Let $S$ be a Stein space and let $K\subset S$ be a compact subset.

  1. (i)

The subset $K$ has a semianalytic $\mathcal{O}(S)$-convex compact neighborhood
in $S$.

  2. (ii)

If $K$ is Stein, it has a basis of semianalytic Stein compact neighborhoods in
$S$.

###### Proof.

After replacing it with $S^{\mathrm{red}}$, we may assume that $S$ is reduced.
Let $\rho:S\to\mathbb{R}$ be a real-analytic strongly psh exhaustion function
(see [Nar61, Lemma p. 358]). For $c\gg 0$, the semianalytic compact subset
$\\{s\in S\mid\rho(s)\leq c\\}$ is a neighborhood of $K$ which is
$\mathcal{O}(S)$-convex by Lemma 1.3 (ii), proving (i).

If $K$ is Stein, it admits a basis of Stein open neighborhoods each of which
is exhausted by semianalytic Stein compact subsets by (i). This proves (ii). ∎

######  Lemma 1.9.

Let $K$ be a Stein compact subset of a Stein space $S$. There exists a basis
of Stein compact neighborhoods $(L_{j})_{j\in J}$ of $K$ in $S$ such that
$L_{j}$ admits a basis $(U_{i})_{i\in I_{j}}$ of Stein open neighborhoods and
$L_{j}$ is $\mathcal{O}(U_{i})$-convex for $j\in J$ and $i\in I_{j}$.

###### Proof.

Let $U$ be a Stein open neighborhood of $K$ in $S$. By Lemma 1.8 (i), there
exists a compact neighborhood $L$ of $K$ in $U$ that is
$\mathcal{O}(U)$-convex. By [Nar62, (1.1)], the subset $L$ admits a basis of
Runge open neighborhoods $(U_{i})_{i\in I}$ in $U$. It follows that $L$ is
$\mathcal{O}(U_{i})$-convex for all $i\in I$. This concludes. ∎

If $K$ is a compact subset of Stein space $S$, we define
$\mathcal{S}_{K}\subset\mathcal{O}(S)$ to be the subset of holomorphic
functions that do not vanish on $K$, and
$\mathcal{O}(S)_{K}:=\mathcal{O}(S)[\mathcal{S}_{K}^{-1}]$. Such rings are
studied in [Kuc05] and [ABF19], at least if $S$ is reduced.

######  Lemma 1.10.

Let $S$ be a Stein space and let $K\subset S$ be a compact subset.

  1. (i)

If $K$ is $\mathcal{O}(S)$-convex, the ring morphism
$\mathcal{O}(S)_{K}\to\mathcal{O}(K)$ is faithfully flat.

  2. (ii)

The ring $\mathcal{O}(S)_{K}$ is excellent.

  3. (iii)

If $K$ is Stein and excellent, the ring $\mathcal{O}(K)$ is excellent.

###### Proof.

By Lemma 1.7, the ring $\mathcal{O}(K)$ is $\mathcal{O}(S)$-flat, hence
$\mathcal{O}(S)_{K}$-flat. If $K$ is $\mathcal{O}(S)$-convex, the description
given in [ABF19, Corollary 3.3 (v)] of the maximal ideals of
$\mathcal{O}(S)_{K}$ now implies assertion (i) (use [Mat80, Theorem 2]).

Assertion (ii) is proven in [ABF19, Theorem 3.5]. As the noetherianity of
$\mathcal{O}(S)_{K}$ is not explicitly proven there, we give an argument. If
$K^{\prime}$ is an excellent $\mathcal{O}(S)$-convex compact neighborhood of
$K$ in $S$ (which exists by Lemma 1.8 (i)), then $\mathcal{O}(S)_{K}$ is a
localization of $\mathcal{O}(S)_{K^{\prime}}$. We may thus assume that $K$ is
excellent and $\mathcal{O}(S)$-convex. As $\mathcal{O}(K)$ is noetherian, so
is $\mathcal{O}(S)_{K}$ by (i) and [Bou06, I, §3.5, Corollaire]. To dispel any
doubt as to whether the excellence of $\mathcal{O}(S)_{K}$ holds if $S$ is
possibly nonreduced, one can reduce to the reduced case using [KS21, Main
Theorem 1].

Assertion (iii) is proven in [Bin76a, Bemerkung pp.152-153] (there, the Stein
compact subset $K$ is assumed to be semianalytic but only its excellence is
used). ∎

Let $K$ be a Stein compact subset in a Stein space $S$. By [Zam76, Theorem
4.7], for every prime ideal $\mathfrak{p}\subset\mathcal{O}(K)$, there exist
$s\in K$ and a germ $Z$ of closed analytic subset along $K$ which is
essentially irreducible at $s$ in the sense of [Zam76, p. 118], and such that
$\mathfrak{p}=\\{f\in\mathcal{O}(K)\mid f|_{Z}\textrm{ vanishes identically
near }s\\}$.

######  Lemma 1.11.

With the above notation, the following are equivalent.

  1. (i)

The ring $\mathcal{O}(K)_{\mathfrak{p}}$ is regular.

  2. (ii)

The germ of $Z$ at $s$ is not included in the singular locus of $S$.

###### Proof.

Assume first that $\mathcal{O}(K)_{\mathfrak{p}}$ is regular. Use Lemma 1.8
(ii) to find an excellent Stein compact neighborhood $L$ of $K$ such that $Z$
extends to a germ of closed analytic subset along $L$. We still denote by
$\mathfrak{p}$ the corresponding ideal of $\mathcal{O}(L)$. As
$\mathcal{O}(L)_{\mathfrak{p}}\to\mathcal{O}(K)_{\mathfrak{p}}$ is flat by
Lemma 1.7, the ring $\mathcal{O}(L)_{\mathfrak{p}}$ is regular (see [Mat80,
Theorem 51 (i)]). As $\mathcal{O}(L)$ is excellent by Lemma 1.10 (iii), the
singular locus of $\mathrm{Spec}(\mathcal{O}(L))$ is closed, defined by an
ideal $I\subset\mathcal{O}(L)$. One can thus choose $f\in
I\setminus\mathfrak{p}$. As $f\notin\mathfrak{p}$ and $s\in\mathring{L}$, any
neighborhood of $s$ in $S$ contains a point $t\in Z\cap L$ with $f(t)\neq 0$.
If $\mathfrak{m}\subset\mathcal{O}(L)$ is the ideal of functions vanishing at
$t$, the ring $\mathcal{O}(L)_{\mathfrak{m}}$ is regular because $f\in I$. It
is moreover of dimension $d:=\dim_{t}(Z)$ (see [Zam76, Proof of Corollary
4.9]). As $\mathfrak{m}/\mathfrak{m}^{2}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\mathfrak{m}_{S,t}/\mathfrak{m}_{S,t}^{2}$
(see [Kie67, below Lemma 2]), it follows that $\mathcal{O}_{S,t}$ is regular
hence that $S$ is nonsingular at $t$ (see [GR84, 6, §2.1]).

Assume now that (ii) holds. Let $Z^{\prime}$ be an irreducible component of
the germ of $Z$ at $s$ which is not included in $S^{\mathrm{sing}}$. If
$\mathfrak{p}^{\prime}\subset\mathcal{O}_{S,s}$ is the ideal of functions
vanishing on $Z^{\prime}$, then $(\mathcal{O}_{S,s})_{\mathfrak{p}^{\prime}}$
is regular by [Hou61, Théorème 3]. Since
$\mathcal{O}(K)_{\mathfrak{p}}\to(\mathcal{O}_{S,s})_{\mathfrak{p}^{\prime}}$
is a flat local morphism of local noetherian rings (by Lemma 1.7 and [Zam76,
Proposition 4.11]), the ring $\mathcal{O}(K)_{\mathfrak{p}}$ is regular by
[Mat80, Theorem 51 (i)]. ∎

Recall that a ring morphism $A\to B$ is regular if it is flat and the induced
morphism $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ has locally noetherian and
geometrically regular fibers (see [EGA42, Définition 6.8.1]). Lemma 1.12 is
used in the proof of Lemma 3.5.

######  Lemma 1.12.

Let $S$ be a Stein space and let $K\subset L\subset S$ be excellent Stein
compact subsets. Then the ring morphisms $\mathcal{O}(L)\to\mathcal{O}(K)$ and
$\mathcal{O}(S)_{K}\to\mathcal{O}(K)$ are regular.

###### Proof.

The first assertion is proven in [Bin76b, (2.2)] (there, Lemmas 1.7 and 1.11
are used as well-known facts, and $K$ and $L$ are assumed to be semianalytic
but only their excellence is used). We prove the second assertion following
the arguments of _loc_._cit_. As $\mathcal{O}(K)$ is $\mathcal{O}(S)_{K}$-flat
by Lemma 1.7, and in view of [EGA42, Proposition 6.7.4 d)], it suffices to
show that the fibers of
$\mathrm{Spec}(\mathcal{O}(K))\to\mathrm{Spec}(\mathcal{O}(S)_{K})$ are
regular. Fix $\mathfrak{p}\in\mathrm{Spec}(\mathcal{O}(K))$ and let
$\mathfrak{q}$ be its image in $\mathrm{Spec}(\mathcal{O}(S))$. We will show
the regularity of $\mathcal{O}(K)/\mathfrak{q}\,\mathcal{O}(K)$ at
$\mathfrak{p}$. Set $T:=\\{s\in S\mid f(s)=0\textrm{ for all
}f\in\mathfrak{q}\\}$.

The restriction morphism
$\mathcal{O}(K)/\mathfrak{q}\,\mathcal{O}(K)\to\mathcal{O}(T\cap K)$ is
surjective by Cartan’s Theorem B. It is also injective, as for any relatively
compact Stein open neighborhood $U$ of $K$ in $S$, one can find
$(f_{i})_{1\leq i\leq k}$ in $\mathfrak{q}$ such that $\mathcal{O}_{U}^{\oplus
k}\xrightarrow{(f_{i})}\mathcal{I}_{T}|_{U}$ is onto, and apply Cartan’s
Theorem B to show that $\mathcal{I}_{T}(K)=\mathfrak{q}\,\mathcal{O}(K)$.
After replacing $S$ with $T$ and $K$ with $T\cap K$, we may thus assume that
$\mathfrak{q}=0$ (and $S$ is reduced).

Let $s\in K$ and $Z$ be as in Lemma 1.11. As $\mathfrak{q}=0$, no nonzero
element of $\mathcal{O}(S)$ vanishes on the germ of $Z$ at $s$. As $S$ is
reduced, $S^{\mathrm{sing}}$ has a nonzero equation in $\mathcal{O}(S)$, and
hence the germ of $Z$ at $s$ is not included in $S^{\mathrm{sing}}$. By Lemma
1.11, the ring $\mathcal{O}(K)_{\mathfrak{p}}$ is regular, as wanted. ∎

###  1.5. Meromorphic functions and ramified coverings

We let $\mathcal{M}$ denote the sheaf of germs of meromorphic functions on a
complex space (see [GR84, 6, §3.1]). If $S$ is a reduced and irreducible Stein
space, then $\mathcal{M}(S)$ is the fraction field of the domain
$\mathcal{O}(S)$ (because the sheaf of denominators of $h\in\mathcal{M}(S)$
defined in [GR84, 6, §3.2] is coherent and nonzero, hence admits a nonzero
global section). The following proposition is classical in dimension $1$, see
e.g.[Sho94, 1, §4.14, Corollaries 4 and 5].

######  Proposition 1.13.

Let $S$ be a connected normal Stein space. Then the functor

$\left\\{\begin{array}[]{l}\textrm{\hskip 25.00003ptfinite surjective
holomorphic maps }\\\ p:T\to S\textrm{ of connected normal Stein spaces
}\end{array}\right\\}\to\left\\{\begin{array}[]{l}\textrm{finite field
extensions }\\\ \hskip 20.00003pt\mathcal{M}(S)\subset F\end{array}\right\\}$

which associates with $p:T\to S$ the extension
$\mathcal{M}(S)\subset\mathcal{M}(T)$ of meromorphic function fields is an
equivalence of categories.

###### Proof.

We first show that the functor is well-defined. Let $S^{\mathrm{sing}}$ and
$T^{\mathrm{sing}}$ be the singular loci of $S$ and $T$. Then
$Z:=S^{\mathrm{sing}}\cup p(T^{\mathrm{sing}})$ is a closed analytic subset of
$S$ which has codimension $\geq 2$ by [GR84, 6, §5.3]. Set $U:=S\setminus Z$.
The locus $W\subset T\setminus p^{-1}(Z)$ where $p$ is not a local
biholomorphism is an analytic subset of pure codimension $1$ because it is the
zero locus of the Jabobian of $p$. Over $U\setminus p(W)$, the map $p$ is a
finite local biholomorphism. As $U\setminus p(W)$ is connected by [GR84, 9,
§1.2], the map $p|_{p^{-1}(U)}:p^{-1}(U)\to U$ is a $d$-sheeted analytic
covering of complex manifolds in the sense of [GR84, 7, §2.1], for some $d\geq
1$. It then follows from [GR84, 7, §3.1, Corollary 2] that the field extension
$\mathcal{M}(U)\subset\mathcal{M}(p^{-1}(U))$ is finite (of degree $\leq d$).
As $\mathcal{M}(S)=\mathcal{M}(U)$ and $\mathcal{M}(T)=\mathcal{M}(p^{-1}(U))$
by [GR84, 9, §5.2], the field extension $\mathcal{M}(S)\subset\mathcal{M}(T)$
is finite (of degree $\leq d$).

Let $p_{1}:T_{1}\to S$ and $p_{2}:T_{2}\to S$ be two finite surjective
holomorphic maps of connected normal Stein spaces. It follows from [Iss66,
Theorem II] that any morphism $\mathcal{M}(T_{1})\subset\mathcal{M}(T_{2})$ of
$\mathbb{C}$-algebras is induced by a unique holomorphic map $f:T_{2}\to
T_{1}$, and that the former is a morphism of $\mathcal{M}(S)$-algebras if and
only if $p_{2}=p_{1}\circ f$. This shows that the functor is full and
faithful.

We finally prove that the functor is essentially surjective. Let $F$ be a
finite extension of degree $d$ of $\mathcal{M}(S)$. Let
$P(x)=x^{d}+\sum_{i=0}^{d-1}a_{i}x^{i}\in\mathcal{M}(S)[x]$ be an irreducible
polynomial with splitting field $F$. Let $b\in\mathcal{O}(S)$ be a nonzero
element with $ba_{i}\in\mathcal{O}(S)$ for $i\in\\{0,\dots,d-1\\}$, and let
$R\subset\mathbb{P}^{1}(\mathbb{C})\times S$ be the zero locus of
$bX^{d}+\sum_{i=0}^{d-1}ba_{i}X^{i}Y^{d-i}$, where $[X:Y]$ are homogeneous
coordinates of $\mathbb{P}^{1}(\mathbb{C})$. Let $R^{\prime}$ be an
irreducible component of $R$ dominating $S$, let $R^{\prime}\to
R^{\prime\prime}\to S$ be the Stein factorization of the projection
$R^{\prime}\to S$ (see [GR84, 10, §6.1]), and let $T$ be the normalization of
$R^{\prime\prime}$, with projection $p:T\to S$. As the inverse image by $R\to
S$ of a general point of $S$ has cardinality $\leq d$, the same holds for the
inverse image by $\pi$ of a general point of $S$. We deduce from the first
paragraph of this proof that the field extension
$\mathcal{M}(S)\subset\mathcal{M}(T)$ has degree $\leq d$. As the element
$X/Y\in\mathcal{M}(T)$ is annihilated by $P$, we get inclusions
$\mathcal{M}(S)\subset F\subset\mathcal{M}(T)$. A degree argument now shows
that $F$ and $\mathcal{M}(T)$ are isomorphic extensions (of degree $d$) of
$\mathcal{M}(S)$. ∎

######  Remark 1.14.

It follows from the proof of Proposition 1.13 that a degree $d$ finite
extension $\mathcal{M}(S)\subset F$ corresponds to a $d$-sheeted analytic
covering $p:T\to S$.

We deduce at once from Proposition 1.13 the following corollary.

######  Corollary 1.15.

Let $S$ be a normal Stein space of pure dimension $n$ with finitely many
connected components. Associating with $p:T\to S$ the $\mathcal{M}(S)$-algebra
$\mathcal{M}(T)$ induces an equivalence of categories

$\left\\{\begin{array}[]{l}\textrm{finite surjective holomorphic maps }p:T\to
S\\\ \hskip 1.99997pt\textrm{of normal Stein spaces of pure dimension
}n\end{array}\right\\}\to\left\\{\begin{array}[]{l}\textrm{\hskip
8.99994ptfinite \'{e}tale}\\\
\mathcal{M}(S)\textrm{-algebras}\end{array}\right\\}.$

The next lemma will be used in the proofs of Propositions 2.1 (ii) and 4.2.

######  Lemma 1.16.

Let $q:R\to S$ be a finite surjective holomorphic map of connected normal
Stein spaces. There exists a finite surjective holomorphic map
$p^{\prime}:T\to R$ of connected normal Stein spaces and a finite group
$\Gamma$ acting on $T$ and acting trivially on $S$, such that $p:=q\circ
p^{\prime}$ is $\Gamma$-equivariant and $\Gamma$ acts transitively on the
fibers of $p$.

###### Proof.

The field extension $\mathcal{M}(S)\subset\mathcal{M}(R)$ is finite by
Proposition 1.13. Let $\mathcal{M}(S)\subset F$ be its Galois closure, with
Galois group $\Gamma$. Let $p^{\prime}:T\to R$ be the holomorphic map
associated with $\mathcal{M}(R)\subset F$ through the equivalence of
categories of Proposition 1.13. By functoriality, the finite group $\Gamma$
acts on $T$ and $p:=q\circ p^{\prime}$ is $\Gamma$-equivariant. The quotient
complex space $T/\Gamma$ (see [Car57a, Theorem 4]) is connected, normal and
admits a natural finite holomorphic map $T/\Gamma\to S$. One deduces from the
field inclusions
$\mathcal{M}(S)\subset\mathcal{M}(T/\Gamma)\subset\mathcal{M}(T)^{\Gamma}=\mathcal{M}(S)$
that $\mathcal{M}(S)=\mathcal{M}(T/\Gamma)$. In view of Proposition 1.13, the
projection $T/\Gamma\to S$ is a biholomorphism. Consequently, the group
$\Gamma$ acts transitively on the fibers of $p$. ∎

##  2\. Killing cohomology classes on finite coverings

The goal of this section is Proposition 2.4, proven in §2.3 by induction on
the degree of a cohomology class. We cover the base case of the induction in
§2.1 and the induction procedure relies on Grauert’s bump method described in
§2.2.

###  2.1. Finite coverings of $\mathcal{O}(S)$-convex compact subsets

The following proposition will be key in dealing with degree $1$ cohomology
classes.

######  Proposition 2.1.

Let $S$ be a Stein manifold of dimension $n$, let $K\subset S$ be an
$\mathcal{O}(S)$-convex compact subset, let $U\subset S$ be an open
neighborhood of $K$ and let ${f:\widetilde{U}\to U}$ be a surjective finite
local biholomorphism.

  1. (i)

After maybe shrinking $U$, there exist a finite surjective holomorphic map
${q:R\to S}$ and an open embedding $h:\widetilde{U}\hookrightarrow R$ such
that $q\circ h=f$.

  2. (ii)

After maybe shrinking $U$, there exist a finite surjective holomorphic map
${p:T\to S}$ and a holomorphic map $g:p^{-1}(U)\to\widetilde{U}$ with $f\circ
g=p|_{p^{-1}(U)}$.

###### Proof of (i).

By [Nar62, (1.1)], we may assume that $U$ is Stein after shrinking it, hence
that so is $\widetilde{U}$ by [GR79, V, §1, Theorem 1 d)]. By [Nar60, Theorem
3], one can then find an embedding
$i:\widetilde{U}\hookrightarrow\mathbb{C}^{N}$ (with $N=2n+1$). Consider the
embedding $j:\widetilde{U}\hookrightarrow\mathbb{C}^{N}\times U$ defined by
$j(z)=(i(z),f(z))$. Write $\mathrm{pr}_{1}:\mathbb{C}^{N}\times
U\to\mathbb{C}^{N}$ and $\mathrm{pr}_{2}:\mathbb{C}^{N}\times U\to U$ for the
projections. For any continuous $\varepsilon:U\to]0,+\infty[$, define

(2.1) $\Theta_{\varepsilon}:=\\{(a,b)\in\mathbb{C}^{N}\times
U\mid\exists\,c\in\widetilde{U}\textrm{ with }f(c)=b\textrm{ and
}|a-i(c)|<\varepsilon(b)\\}.$

Choose $\varepsilon$ sufficiently small so that the element $c$ in (2.1) is
always unique. The subset $\Theta_{\varepsilon}\subset\mathbb{C}^{N}\times U$
is then a tubular neighborhood of $j(\widetilde{U})$. Define
$\pi:\Theta_{\varepsilon}\to\widetilde{U}$ such that $\pi(a,b)=c$. By [CM86,
Proposition 2.1], there exists a neighborhood $\Omega$ of $j(\widetilde{U})$
in $\Theta_{\varepsilon}$ that is Runge in $\mathbb{C}^{N}\times U$. Finally,
choose $\delta:U\to]0,+\infty[$ continuous sufficiently small so that
$\overline{\Theta_{\delta}}\subset\Omega$.

Define a holomorphic map $\phi:\Omega\to\mathbb{C}^{N}$ by setting
$\phi(z)=\mathrm{pr}_{1}(z)-i\circ\pi(z)$. The map $\phi$ is submersive along
$\overline{\Theta_{\delta}}$, and its zero locus in
$\overline{\Theta_{\delta}}$ is included in $\Theta_{\delta}$ and projects
biholomorphically to $\widetilde{U}$ by $\pi$. All these properties persist
(after maybe shrinking $U$) if one replaces $\phi$ with a holomorphic map
$\phi^{\prime}:\Omega\to\mathbb{C}^{N}$ close enough to $\phi$ on
$\overline{\Theta_{\delta}}\cap\mathrm{pr}_{2}^{-1}(K)$ in the
$C^{0}$-topology (hence in the $C^{1}$-topology by the Cauchy estimates). We
construct such a $\phi^{\prime}$ as follows. First approximate $\phi$ by the
restriction of a holomorphic map $\mathbb{C}^{N}\times U\to\mathbb{C}^{N}$
using that $\Omega$ is Runge in $\mathbb{C}^{N}\times U$. Expand this map as
$N$ power series in $N$ variables with coefficients in $\mathcal{O}(U)$,
truncate these power series to turn them into $N$ polynomials in $N$ variables
with coefficients in $\mathcal{O}(U)$, and then use the Oka–Weil approximation
theorem [GR65, VII, A, Theorem 6] to approximate these coefficients on $K$ by
restrictions of elements of $\mathcal{O}(S)$. Let $\psi:\mathbb{C}^{N}\times
S\to\mathbb{C}^{N}$ be the holomorphic map (polynomial in the first variable)
obtained in this way and set $\phi^{\prime}:=\psi|_{\Omega}$.

Let $\Psi_{1},\dots,\Psi_{N}\in\mathcal{O}(S)[X_{0},\dots,X_{N}]$ be the
homogenizations of the components of $\psi$, and let
$\widehat{R}\subset\mathbb{P}^{N}(\mathbb{C})\times S$ be the locus where they
all vanish, with projection $\hat{q}:\widehat{R}\to S$. The map $\hat{q}$ is
surjective by [Har77, I, Theorem 7.2]. Hence so is the finite holomorphic map
$q:R\to S$ appearing in the Stein factorization [GR84, 10, §6.1] of $\hat{q}$.
By our choice of $\phi^{\prime}$, the intersection of $\widehat{R}$ with
$\Theta_{\delta}$ is biholomorphic (via the map $\pi$) to $\widetilde{U}$. The
resulting open embedding $\hat{h}:\widetilde{U}\to\widehat{R}$ satisfies
$\hat{q}\circ\hat{h}=f$.

The uniqueness of the Stein factorization shows that it is compatible with
base change by open embeddings. As $\hat{h}(\widetilde{U})$ is a connected
component of $\hat{q}^{-1}(U)$ such that
$\hat{q}|_{\hat{h}(\widetilde{U})}:\hat{h}(\widetilde{U})\to U$ is finite (as
$f$ is), we deduce that the composition of $\hat{h}$ with the natural map
$\widehat{R}\to R$ is an open embedding $h:\widetilde{U}\to R$ such that
$q\circ h=f$. ∎

###### Proof of (ii).

Let $q:R\to S$, and $h:\widetilde{U}\hookrightarrow Y$ be as in (i). We may
assume that $R$ is normal after normalizing it, and that $S$ is connected
after replacing it with one of its connected components. After shrinking $U$,
we may suppose that it has finitely many connected components
$U_{1},\dots,U_{r}$. If $p_{i}:T_{i}\to S$ solves our problem for
$f|_{f^{-1}(U_{i})}:f^{-1}(U_{i})\to U_{i}$ instead of $f$, then we may choose
$T$ to be the fiber product $T_{1}\times_{S}\dots\times_{S}T_{r}$. We may thus
assume that $U$ is connected. Replacing $\widetilde{U}$ with one of its
connected components, we may also suppose that $\widetilde{U}$ is connected.
Finally replacing $R$ with its connected component containing
$h(\widetilde{U})$, we reduce to the case where $R$ is connected.

Let $p^{\prime}:T\to R$ and $\Gamma$ be as in Lemma 1.16, and set $p:=q\circ
p^{\prime}$. Let $V$ be a connected component of $p^{-1}(U)$. By the
transitivity of the action of $\Gamma$ on the fibers of $p$, one may choose
$\gamma_{V}\in\Gamma$ such that $\gamma_{V}(V)\subset
p^{\prime-1}(\widetilde{U})$. One then defines
$g(x):=p^{\prime}\circ\gamma_{V}(x)$ for $x\in V$. ∎

###  2.2. Grauert’s bump method

To prove Proposition 2.4, we will use Grauert’s bump method as developed by
Henkin and Leiterer [HL98]. In Proposition 2.3 below, we sum up what we
exactly need, relying on the exposition by Forstnerič [For17].

######  Lemma 2.2.

Let $S$ be a Stein manifold of dimension $n$. Any point $s\in S$ admits an
$\mathcal{O}(S)$-convex compact neighborhood $B\subset S$ such that $B$ admits
a basis of contractible open neighborhoods in $S$.

###### Proof.

By [Nar60, Theorem 3], one may view $S$ as a submanifold of $\mathbb{C}^{N}$
for $N\gg 0$. Then $\rho:S\to\mathbb{R}$ defined by $\rho(x):=|x-s|^{2}$ is a
$\mathcal{C}^{\infty}$ strongly psh exhaustion function, and
$B:=\\{\rho\leq\varepsilon\\}$ works for $0<\varepsilon\ll 1$ (see Lemma 1.3
(ii)). ∎

######  Proposition 2.3.

Let $S$ be a Stein manifold of dimension $n$. Then there exist sequences
$(A_{0},A_{1},\dots)$ and $(B_{0},B_{1},\dots)$ of compact subsets of $S$ such
that:

  1. (i)

the subsets $A_{i}$ and $B_{i}$ are $\mathcal{O}(S)$-convex;

  2. (ii)

the subset $B_{i}$ has a basis of contractible open neighborhoods;

  3. (iii)

one has $A_{0}=\varnothing$ and $A_{i+1}\subset A_{i}\cup B_{i}$;

  4. (iv)

for any compact subset $K\subset S$, one has $K\subset A_{i}$ for $i\gg 0$.

###### Proof.

Let $\rho:S\to\mathbb{R}$ be a $\mathcal{C}^{\infty}$ strongly psh exhaustion
function (see [Nar61, Lemma p. 358]). By [GG73, Proposition 6.13] and [For17,
Lemma 3.10.3], we may assume that its critical points are nondegenerate, have
distinct images by $\rho$, and are nice in the sense of [For17, Definition
3.10.2].

We adopt the following nonstandard terminology. We say that a pair
$(A,A^{\prime})$ of $\mathcal{O}(S)$-convex compact subsets of $S$ is a bump
if there exists an $\mathcal{O}(S)$-convex compact subset $B\subset S$ with a
basis of contractible neighborhoods with $A^{\prime}\subset A\cup B$. The
proof has two parts, which, combined, prove the proposition.

In the first part (the noncritical extension case), we show that if
${c<c^{\prime}}$ are such that $\rho$ has no critical values in
$[c,c^{\prime}]$, then one can go from $A:=\\{\rho\leq c\\}$ to
$A^{\prime}:=\\{\rho\leq c^{\prime}\\}$ by a sequence of bumps containing $A$.
This follows at once from [For17, Proof of Lemma 5.10.3]. Indeed, it is
explained there how to go from $A$ to $A^{\prime}$ by an increasing sequence
of sublevel sets of (varying) $\mathcal{C}^{\infty}$ psh exhaustion functions
(which are $\mathcal{O}(S)$-convex by Lemma 1.3 (ii)), in a way that the
difference between two successive ones is included in an open subset of any
fixed open covering of $A^{\prime}$. In view of Lemma 2.2, one can ensure that
this difference is included in an $\mathcal{O}(S)$-convex compact subset with
a basis of contractible neighborhoods.

In the second part (the critical extension case), we fix a critical value of
$\rho$ (assumed without loss of generality to be equal to $0$), with
associated critical point $p_{0}\in S$. We show that there exist
$c<0<c^{\prime}$ such that one can go from $A:=\\{\rho\leq c\\}$ to
$A^{\prime}:=\\{\rho\leq c^{\prime}\\}$ by a sequence of bumps containing $A$.
To do so, use Lemma 2.2 to choose a compact $\mathcal{O}(S)$-convex
neighborhood of $p_{0}$ in $S$ with a basis of contractible open
neighborhoods. Apply the constructions of [For17, p. 102] with $q=1$ and
$U\subset B$ an open neighborhood of $p_{0}$. In particular, choose $c_{0}>0$
small enough so that [For17, Lemma 3.11.4] applies, and let $0<t_{0}<t_{1}$
and $\tau:\\{\rho<3c_{0}\\}\to\mathbb{R}$ be as in _loc_._cit_. Set
$c:=-t_{0}$ and $c^{\prime}:=c_{0}$. One can go from $A$ to $\\{\tau\leq
t_{1}-t_{0}\\}$ by a bump by [For17, Lemma 3.11.4 (c)]. Arguing as in the
noncritical extension case (using the $\mathcal{C}^{\infty}$ strongly psh
function $\tau$ on $\Omega:=\\{\rho<3c_{0}\\}$, which is legitimate by [For17,
Proposition 3.11.4 (d)]), one sees that one can go from $\\{\tau\leq
t_{1}-t_{0}\\}$ to $\\{\tau\leq 2c_{0}\\}$ by a sequence of bumps (the
involved compact $\mathcal{O}(\Omega)$-convex sets are also
$\mathcal{O}(S)$-convex because $\Omega$ is Runge in $S$ by Lemma 1.3 (i)).
Finally, one can go from $\\{\tau\leq 2c_{0}\\}$ to $A^{\prime}$ by a bump by
[For17, Lemma 3.11.4 (b)]. ∎

###  2.3. Induction on the cohomological degree

We finally reach our goal.

######  Proposition 2.4.

Let $S$ be a Stein space, let $K\subset S$ be an $\mathcal{O}(S)$-convex
compact subset, and let $U\subset S$ be an open neighborhood of $K$. Fix
integers $k,m\geq 1$. For all $\alpha\in H^{k}(U,\mathbb{Z}/m)$, there exists
a finite surjective holomorphic map $p:T\to S$ and an open neighborhood $V$ of
$K$ in $U$ such that $\alpha|_{p^{-1}(V)}=0$ in
$H^{k}(p^{-1}(V),\mathbb{Z}/m)$.

###### Proof.

We argue by induction on $k\geq 1$, which we fix. Using Proposition 1.5 (i),
we may replace $S$ with an irreducible component of its reduction, and assume
that $S$ is irreducible, hence of finite dimension $n$. We further reduce to
the case where $S$ is a manifold as follows. Let $i:S\to S^{\prime}$ be as in
Lemma 1.1, fix a continuous retraction $r:S^{\prime}\to i(S)$ and replace $S$,
$K$, $U$ and $\alpha$ by $S^{\prime}$, $i(K)$, $r^{-1}(U)$ and $r^{*}\alpha$,
noting that $i(K)$ is $\mathcal{O}(i(S))$-convex by Proposition 1.5 (ii),
hence $\mathcal{O}(S^{\prime})$-convex.

Assume first that $k=1$. Let $f:\widetilde{U}\to U$ be the degree $m$
unramified cyclic covering associated with $\alpha$. After maybe shrinking
$U$, choose $p:T\to S$ and $g:p^{-1}(U)\to\widetilde{U}$ with $f\circ
g=p|_{p^{-1}(U)}$ as in Proposition 2.1 (ii). As $f^{*}\alpha=0$ by choice of
$f$, one has $\alpha|_{p^{-1}(U)}=g^{*}f^{*}\alpha=0$, as wanted.

Assume now that $k\geq 2$. After shrinking $U$, we may assume that it is a
Runge domain in $S$ (see [Nar62, (1.1)]). Let $(A_{0},A_{1},\dots)$ and
$(B_{0},B_{1},\dots)$ be as in Proposition 2.3 applied to the Stein manifold
$U$. As $U$ is Runge in $X$, it follows from Proposition 2.3 (i) that the
$A_{i}$ and the $B_{i}$ are $\mathcal{O}(S)$-convex compact subsets of $S$.

We will now show by induction on $i\geq 0$ that there exists an open
neighborhood $V_{i}$ of $A_{i}$ in $U$ and a finite surjective holomorphic map
$p_{i}:T_{i}\to S$ such that $\alpha|_{p_{i}^{-1}(V_{i})}=0$ in
$H^{k}(p_{i}^{-1}(V_{i}),\mathbb{Z}/m)$. When $i=0$, one can take
$V_{0}=\varnothing$.

Assume that we have proven the statement for $i$ and let us prove it for
$i+1$. Let $W_{i}$ be a contractible open neighborhood of $B_{i}$ in $U$ (see
Proposition 2.3 (ii)). Consider the boundary map

$\partial:H^{k-1}(p_{i}^{-1}(V_{i}\cap W_{i}),\mathbb{Z}/m)\to
H^{k}(p_{i}^{-1}(V_{i}\cup W_{i}),\mathbb{Z}/m)$

in the Mayer–Vietoris exact sequence. As $\alpha|_{p_{i}^{-1}(V_{i})}=0$ by
the induction hypothesis on $i$ and
$\alpha|_{p_{i}^{-1}(W_{i})}=(p_{i}|_{p_{i}^{-1}(W_{i})})^{*}(\alpha|_{W_{i}})=0$
because $W_{i}$ is contractible, there exists $\beta\in
H^{k-1}(p_{i}^{-1}(V_{i}\cap W_{i}),\mathbb{Z}/m)$ such that
$\partial(\beta)=\alpha|_{p_{i}^{-1}(V_{i}\cup W_{i})}$. By the induction
hypothesis on $k$ (applied to the $\mathcal{O}(T_{i})$-convex compact subset
$p_{i}^{-1}(A_{i}\cap B_{i})$ of $T_{i}$, see Proposition 1.5 (i)), there
exists a finite surjective holomorphic map $q_{i}:T_{i+1}\to T_{i}$ and an
open neighborhood $\Omega_{i}$ of $p_{i}^{-1}(A_{i}\cap B_{i})$ in
$p_{i}^{-1}(V_{i}\cap W_{i})$ such that $\beta|_{q_{i}^{-1}(\Omega_{i})}=0$.
Let $V_{i}^{\prime}$ and $W_{i}^{\prime}$ be open neighborhoods of $A_{i}$ and
$B_{i}$ in $V_{i}$ and $W_{i}$ respectively, such that
$p_{i}^{-1}(V^{\prime}_{i}\cap W^{\prime}_{i})\subset\Omega_{i}$. Set
$p_{i+1}:=p_{i}\circ q_{i}$ and $V_{i+1}:=V_{i}^{\prime}\cup W_{i}^{\prime}$.
Then $\alpha|_{p_{i+1}^{-1}(V_{i+1})}$ is the image of
$\beta|_{p_{i+1}^{-1}(V^{\prime}_{i}\cap W^{\prime}_{i})}$ by the boundary map

$\partial^{\prime}:H^{k-1}(p_{i+1}^{-1}(V^{\prime}_{i}\cap
W^{\prime}_{i}),\mathbb{Z}/m)\to H^{k}(p_{i+1}^{-1}(V^{\prime}_{i}\cup
W^{\prime}_{i}),\mathbb{Z}/m)$

of the Mayer–Vietoris exact sequence, by compatibility of $\partial$ and
$\partial^{\prime}$. As $\beta|_{p_{i+1}^{-1}(V^{\prime}_{i}\cap
W^{\prime}_{i})}$ vanishes (because $\beta|_{q_{i}^{-1}(\Omega_{i})}=0$), it
follows that $\alpha|_{p_{i+1}^{-1}(V_{i+1})}=0$, as required.

In view of Proposition 2.3 (iv), one can take $V=V_{i}$ and $p=p_{i}$ for
$i\gg 0$. ∎

##  3\. A relative comparison theorem

After studying analytifications of algebraic varieties over Stein algebras in
§3.1, and étale sheaves on them in §3.2, we prove the relative Artin
comparison theorem in Stein geometry in §3.3 (Theorem 3.7).

###  3.1. Analytification of algebraic varieties over Stein algebras

Fix a Stein space $S$. Beware that the ring $\mathcal{O}(S)$ is in general not
noetherian. Let $(P)$ be a property of morphisms of schemes. A morphism
$f:X\to Y$ of $\mathcal{O}(S)$-schemes is said to interiorly satisfy $(P)$ if
there exists an open covering $(U_{i})_{i\in I}$ of $S$ such that the
morphisms $f_{\mathcal{O}(U_{i})}:X_{\mathcal{O}(U_{i})}\to
Y_{\mathcal{O}(U_{i})}$ satisfy $(P)$. We say that an $\mathcal{O}(S)$-scheme
$X$ interiorly satisfies $(P)$ if so does the structural morphism
$X\to\mathrm{Spec}(\mathcal{O}(S))$.

######  Lemma 3.1.

Let $(P)$ be a property of morphisms of schemes that is stable by base change
and fpqc local on the base. Let $S$ be a Stein space and let $f:X\to Y$ be a
morphism of $\mathcal{O}(S)$-schemes that interiorly satisfies $(P)$.

  1. (i)

For $K\subset S$ compact, $f_{\mathcal{O}(K)}:X_{\mathcal{O}(K)}\to
Y_{\mathcal{O}(K)}$ satisfies $(P)$.

  2. (ii)

For $K\subset S$ compact, $f_{\mathcal{O}(S)_{K}}:X_{\mathcal{O}(S)_{K}}\to
Y_{\mathcal{O}(S)_{K}}$ satisfies $(P)$.

  3. (iii)

For $U\subset S$ open and relatively compact,
$f_{\mathcal{O}(U)}:X_{\mathcal{O}(U)}\to Y_{\mathcal{O}(U)}$ satisfies $(P)$.

###### Proof.

To prove (i) and (ii), we may replace $K$ with $\widehat{K}_{\mathcal{O}(S)}$
(by base change), and consequently assume that $K$ is $\mathcal{O}(S)$-convex,
hence Stein. Any $s\in S$ has a basis of Stein compact neighborhoods (see
Lemma 1.8 (ii)). By hypothesis on $f$ and the base change property, each $s\in
K$ has a Stein compact neighborhood $K_{s}$ such that $f_{\mathcal{O}(K_{s})}$
satisfies $(P)$. By compactness of $K$, we may extract a finite family
$(K_{i})_{1\leq i\leq k}$ covering $K$. Replacing $K_{i}$ with $K_{i}\cap K$
(and using the base change property again), we may assume that $K_{i}\subset
K$ for $1\leq i\leq k$. The morphism
$\mathcal{O}(K)\to\prod_{i=1}^{k}\mathcal{O}(K_{i})$ is flat by Lemma 1.7,
hence faithfully flat by [Mat80, Theorem 2 (i)$\Leftrightarrow$(iii)] and the
description of the maximal ideals of $\mathcal{O}(K)$ given in [Zam76,
Corollary 3.3]. Assertion (i) follows since $(P)$ is fpqc local on the base.
So does (ii) by Lemma 1.10 (i). Assertion (iii) follows from (i) applied with
$K=\overline{U}$ (by base change). ∎

An $\mathcal{O}(S)$-scheme interiorly locally of finite type is interiorly
locally of finite presentation (use Lemmas 1.8 (ii) and 3.1 (i)), i.e.,
belongs to the category $\mathbb{K}$ in the terminology of Bingener [Bin76b,
p. 23]. With such an $\mathcal{O}(S)$-scheme $X$, one can associate its
analytification $X^{\mathrm{an}}$ (see [Bin76b, Satz 1.1] when $X$ is of
finite presentation over $\mathcal{O}(S)$; the construction and its properties
extend under our more general hypotheses as indicated in [Bin76b, p. 23]). It
is a (possibly nonseparated) complex space over $S$ endowed with a map
$i_{X}:X^{\mathrm{an}}\to X$ of locally ringed spaces such that the map

$\displaystyle\mathrm{Hom}_{S}(T,X^{\mathrm{an}})$
$\displaystyle\to\mathrm{Hom}_{\mathcal{O}(S)}(T,X)$ $\displaystyle h\hskip
10.00002pt$ $\displaystyle\mapsto\hskip 10.00002pti_{X}\circ h.$

is bijective for all complex spaces $T$ over $S$. In particular, with each
point $x\in X^{\mathrm{an}}$ corresponds a closed point of $X$ with complex
residue field, also denoted by $x$.

This construction is functorial [Bin76b, p. 2] (and we let
$f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}$ denote the holomorphic
map induced by a morphism $f:X\to Y$ of $\mathcal{O}(S)$-schemes interiorly
locally of finite type), commutes with the formation of fiber products
[Bin76b, p. 3], and is compatible with change of the base Stein space [Bin76b,
(1.2)].

Lemmas 3.2 and 3.3 give examples of $\mathcal{O}(S)$-schemes interiorly of
finite type.

######  Lemma 3.2.

Let $Z$ be a closed analytic subspace of a Stein space $S$. Then there exists
an open immersion $j:V\hookrightarrow\mathrm{Spec}(\mathcal{O}(S))$ interiorly
of finite type such that $j^{\mathrm{an}}$ identifies with $S\setminus
Z\hookrightarrow S$. If $Z$ is set-theoretically defined by the vanishing of
finitely many elements of $\mathcal{O}(S)$, one may moreover choose $j$ to be
quasi-compact.

###### Proof.

Set
$V:=\mathrm{Spec}(\mathcal{O}(S))\setminus\mathrm{Spec}(\mathcal{O}(S)/\mathcal{I}_{Z}(S))$,
where $\mathcal{I}_{Z}$ be the ideal sheaf of $Z$ in $S$. Let $U\subset S$ be
a relatively compact Stein open subset. By Lemma 3.4 below,

$V_{U}:=V\times_{\mathrm{Spec}(\mathcal{O}(S))}\mathrm{Spec}(\mathcal{O}(U))=\mathrm{Spec}(\mathcal{O}(U))\setminus\mathrm{Spec}(\mathcal{O}(U)/\mathcal{I}_{Z}(U))$

and $\mathcal{I}_{Z}(U)$ is finitely generated over $\mathcal{O}(U)$. We
deduce that $V_{U}$ is an $\mathcal{O}(U)$-scheme of finite type. The
description of $j^{\mathrm{an}}$ follows from [Bin76b, Proof of Satz 1.1].

If $Z$ is set-theoretically defined by the vanishing of finitely many elements
of $\mathcal{O}(S)$, we apply the above construction after replacing $Z$ with
the complex subspace defined by the vanishing of these equations. This ensures
that $\mathcal{I}_{Z}(S)$ is a finitely generated ideal of $\mathcal{O}(S)$
and hence that $j$ is quasi-compact. ∎

######  Lemma 3.3.

If $p:T\to S$ is a finite holomorphic map of Stein spaces, then the induced
morphism $f:\mathrm{Spec}(\mathcal{O}(T))\to\mathrm{Spec}(\mathcal{O}(S))$ is
interiorly finite and $f^{\mathrm{an}}=p$.

###### Proof.

Let $U\subset S$ be a relatively compact Stein open subset. By Lemma 3.4 below
applied to $\mathcal{F}:=p_{*}\mathcal{O}_{T}$, the $\mathcal{O}(U)$-module
$\mathcal{O}(T)\otimes_{\mathcal{O}(S)}\mathcal{O}(U)$ is finitely generated.
It follows that $f$ is interiorly finite. As
$\mathcal{O}(T)\otimes_{\mathcal{O}(S)}\mathcal{O}(U)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\mathcal{O}(p^{-1}(U))$
by Lemma 3.4, that $f^{\mathrm{an}}=p$ may be verified locally on $S$. We may
thus assume that $S$ is finite-dimensional, in which case it follows from
[Bin76b, (1.3)]. ∎

######  Lemma 3.4.

Let $U$ be a relatively compact Stein open subset of a Stein space $S$. Let
$\mathcal{F}$ be a coherent sheaf on $S$. The $\mathcal{O}(U)$-module
$\mathcal{F}(U)$ is generated by finitely many elements of $\mathcal{F}(S)$,
and $\mathcal{F}(S)\otimes_{\mathcal{O}(S)}\mathcal{O}(U)\to\mathcal{F}(U)$ is
an isomorphism.

###### Proof.

As $U$ is relatively compact, Cartan’s Theorem A implies that there exist
$a_{1},\dots,a_{r}\in\mathcal{F}(S)$ generating $\mathcal{F}$ on $U$. The
first assertion and the surjectivity of
$\mathcal{F}(S)\otimes_{\mathcal{O}(S)}\mathcal{O}(U)\to\mathcal{F}(U)$ follow
from the vanishing of $H^{1}(U,\mathrm{Ker}(\mathcal{O}_{U}^{\oplus
r}\xrightarrow{a_{i}}\mathcal{F}|_{U}))$.

Now, let $\sum_{j=1}^{s}b_{j}\otimes
c_{j}\in\mathcal{F}(S)\otimes_{\mathcal{O}(S)}\mathcal{O}(U)$ be such that
$\sum_{j=1}^{s}b_{j}c_{j}=0$ in $\mathcal{F}(U)$. Consider the exact sequence
$0\to\mathcal{N}\to\mathcal{O}_{S}^{\oplus s}\xrightarrow{b_{j}}\mathcal{F},$
so $(c_{j})\in\mathcal{N}(U)$. By the surjectivity result applied to
$\mathcal{N}$, there exist
$(d_{j,1}),\dots,(d_{j,t})\in\mathcal{N}(S)\subset\mathcal{O}(S)^{\oplus s}$
and $e_{1},\dots,e_{t}\in\mathcal{O}(U)$ with
$\sum_{k=1}^{t}e_{k}d_{j,k}=c_{j}$ for $1\leq j\leq s$. Then ∎

∎ $\sum_{j=1}^{s}b_{j}\otimes c_{j}=\sum_{j=1}^{s}\sum_{k=1}^{t}b_{j}\otimes
e_{k}d_{j,k}=\sum_{k=1}^{t}\Big{(}\sum_{j=1}^{s}b_{j}d_{j,k}\Big{)}\otimes
e_{k}=0.$

###  3.2. Étale sheaves on algebraic varieties over Stein algebras

Let $S$ be a Stein space, and let $X$ be an $\mathcal{O}(S)$-scheme interiorly
locally of finite type. There is a commutative diagram of morphisms of sites

(3.1)
$\textstyle{(X^{\mathrm{an}})_{\mathrm{cl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\varepsilon}$$\textstyle{X^{\mathrm{an}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{X}}$$\textstyle{X_{\mathrm{\acute{e}t}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\hbox
to0.0pt{,\hss}}$

where $X_{\mathrm{\acute{e}t}}$ is the small étale site of the scheme $X$, and
$X^{\mathrm{an}}_{\mathrm{cl}}$ is the site of local isomorphisms of the
topological space $X^{\mathrm{an}}$ (see [SGA43, XI, §4.0]). Indeed, if
$f:Y\to X$ is étale, then $f^{\mathrm{an}}:Y^{\mathrm{an}}\to X^{\mathrm{an}}$
is a local homeomorphism by [Bin76b, Satz 3.1 (1)]. It follows from [SGA41,
III, Théorème 4.1] that $\delta_{*}$ induces an equivalence of topoi.

If $\mathbb{L}$ is an abelian sheaf on $X_{\mathrm{\acute{e}t}}$, we define
$\mathbb{L}^{\mathrm{an}}:=\delta_{*}\varepsilon^{*}\mathbb{L}$. For $x\in
X^{\mathrm{an}}$, one has $\mathbb{L}^{\mathrm{an}}_{x}=\mathbb{L}_{x}$.
Finally, we say that $\mathbb{L}$ is interiorly constructible if there exists
an open covering $(U_{i})_{i\in I}$ of $S$ such that the étale sheaves
$\mathbb{L}|_{X_{\mathcal{O}(U_{i})}}$ are constructible.

######  Lemma 3.5.

Let $K$ be an excellent Stein compact subset of a Stein space $S$. Let $f:X\to
Y$ be a morphism interiorly of finite type of $\mathcal{O}(S)$-schemes
interiorly locally of finite type. Fix a torsion abelian sheaf $\mathbb{L}$ on
$X_{\mathrm{\acute{e}t}}$. Then the base change morphisms
$(\mathrm{R}^{k}f_{*}\mathbb{L})|_{Y_{\mathcal{O}(K)}}\to\mathrm{R}^{k}f_{\mathcal{O}(K),*}(\mathbb{L}|_{X_{\mathcal{O}(K)}})$
are isomorphisms for $k\geq 0$.

###### Proof.

Fix $k\geq 0$. As $\mathcal{O}(S)\to\mathcal{O}(S)_{K}$ is a localization, the
description [SGA42, VIII, Théorème 5.2] of the fibers of higher push-forwards
in étale cohomology implies that the base change morphism

(3.2)
$\mathrm{R}^{k}f_{*}(\mathbb{L})|_{Y_{\mathcal{O}(S)_{K}}}\to\mathrm{R}^{k}f_{\mathcal{O}(S)_{K},*}(\mathbb{L}|_{X_{\mathcal{O}(S)_{K}}})$

is an isomorphism. By Néron–Popescu desingularization (see [Swa98, Theorem
1.1]) and Lemma 1.12, the ring $\mathcal{O}(K)$ is a filtered colimit of
smooth $\mathcal{O}(S)_{K}$-algebras. By the smooth base change theorem
[SGA43, Exp. XVI, Corollaire 1.2] and the commutation of étale cohomology with
filtered limits of schemes with affine transition maps [SGA42, Exp. VII,
Corollaire 5.8], the base change morphism

(3.3)
$(\mathrm{R}^{k}f_{\mathcal{O}(S)_{K},*}(\mathbb{L}|_{X_{\mathcal{O}(S)_{K}}}))|_{Y_{\mathcal{O}(K)}}\to\mathrm{R}^{k}f_{\mathcal{O}(K),*}(\mathbb{L}|_{X_{\mathcal{O}(K)}})$

is also an isomorphism. The lemma follows from (3.2) and (3.3). ∎

######  Lemma 3.6.

Let $S$ be a Stein space. Let $f:X\to Y$ be a morphism interiorly of finite
type of $\mathcal{O}(S)$-schemes interiorly locally of finite type. Let
$\mathbb{L}$ be an interiorly constructible sheaf on
$X_{\mathrm{\acute{e}t}}$. Then $\mathrm{R}^{k}f_{*}\mathbb{L}$ is interiorly
constructible for $k\geq 0$.

###### Proof.

Working locally on $Y$, we may suppose that $X$ and $Y$ are interiorly of
finite type over $\mathcal{O}(S)$. Fix $s\in S$. Let $K$ be an excellent Stein
compact neighborhood of $s$ in $S$ chosen small enough so that
$\mathbb{L}|_{X_{\mathcal{O}(K)}}$ is constructible (see Lemma 1.8 (ii)). As
$f_{\mathcal{O}(K)}:X_{\mathcal{O}(K)}\to Y_{\mathcal{O}(K)}$ is a morphism of
$\mathcal{O}(K)$-schemes of finite type (see Lemma 3.1 (i)), which are
excellent noetherian schemes by Lemma 1.10 (iii), the sheaf
$\mathrm{R}^{k}f_{\mathcal{O}(K),*}(\mathbb{L}|_{X_{\mathcal{O}(K)}})$ is
constructible by [SGA43, XIX, Théorème 5.1] and [Tem08, Theorem 1.1]. It
follows from Lemma 3.5 that
$(\mathrm{R}^{k}f_{*}\mathbb{L})|_{X_{\mathcal{O}(U)}}$ is constructible for
any open neighborhood $U$ of $s$ in $K$. ∎

###  3.3. A relative Artin comparison theorem over Stein spaces

Here is the statement of our relative comparison theorem. In the constructible
case, its proof is entirely parallel to that of [EGA43, XVI, Théorème 4.1].

######  Theorem 3.7.

Let $S$ be a Stein space. Let $f:X\to Y$ be a morphism interiorly of finite
type of $\mathcal{O}(S)$-schemes interiorly locally of finite type. Let
$\mathbb{L}$ be a torsion abelian sheaf on $X_{\mathrm{\acute{e}t}}$. If
either $f$ is interiorly proper or $\mathbb{L}$ is interiorly constructible,
the base change morphisms
$\mathrm{R}^{k}f_{*}(\mathbb{L})^{\mathrm{an}}\to\mathrm{R}^{k}f^{\mathrm{an}}_{*}(\mathbb{L}^{\mathrm{an}})$
are isomorphisms for $k\geq 0$.

###### Proof.

Fix $y\in Y^{\mathrm{an}}$ and let $s$ be the image of $y$ in $S$. We proceed
in several steps.

######  Step 1.

We first deal with the case where $f$ is interiorly proper.

By Lemma 3.1 (i), the base change
$f_{\mathcal{O}_{S,s}}:X_{\mathcal{O}_{S,s}}\to Y_{\mathcal{O}_{S,s}}$ of $f$
is proper. By Lemma 3.5, proper base change [SGA43, XII, Théorème 5.1], and
Artin’s comparison theorem [SGA43, XVI, Théorème 4.1], the base change
morphisms

(3.4)
$(\mathrm{R}^{k}f_{*}\mathbb{L})_{y}\to\mathrm{R}^{k}f_{\mathcal{O}_{S,s},*}(\mathbb{L}|_{X_{\mathcal{O}_{S,s}}})_{y}\to
H^{k}_{\mathrm{\acute{e}t}}(X_{y},\mathbb{L}|_{X_{y}})\to
H^{k}_{\mathrm{\acute{e}t}}(X_{y}^{\mathrm{an}},\mathbb{L}^{\mathrm{an}}|_{X^{\mathrm{an}}_{y}})$

are isomorphisms. As $f^{\mathrm{an}}$ is proper by [Bin76b, Satz 3.1 (5)],
proper base change in topology (see [God58, II, Théorème 4.11.1]) implies that
the base change morphism

(3.5) $\mathrm{R}^{k}f^{\mathrm{an}}_{*}(\mathbb{L}^{\mathrm{an}})_{y}\to
H^{k}_{\mathrm{\acute{e}t}}(X_{y}^{\mathrm{an}},\mathbb{L}^{\mathrm{an}}|_{X^{\mathrm{an}}_{y}})$

is also an isomorphism. Combining (3.4) and (3.5) shows that the base change
morphism
$\mathrm{R}^{k}f_{*}(\mathbb{L})_{y}^{\mathrm{an}}\to\mathrm{R}^{k}f^{\mathrm{an}}_{*}(\mathbb{L}^{\mathrm{an}})_{y}$
is an isomorphism, as wanted.

######  Step 2.

Setup of the proof when $\mathbb{L}$ is interiorly constructible.

By Lemma 3.5, if $U\subset S$ is a Stein open neighborhood of $s$, the base
change morphism
$(\mathrm{R}^{k}f_{*}\mathbb{L})_{y}\to(\mathrm{R}^{k}f_{\mathcal{O}(U),*}(\mathbb{L}|_{X_{\mathcal{O}(U)}}))_{y}$
is an isomorphism. Therefore, to check that
$\mathrm{R}^{k}f_{*}(\mathbb{L})_{y}^{\mathrm{an}}\to\mathrm{R}^{k}f^{\mathrm{an}}_{*}(\mathbb{L}^{\mathrm{an}})_{y}$
is an isomorphism, we may at any time replace $S$ with a Stein open
neighborhood of $s$ in $S$. As one can moreover work locally on $Y$, one may
assume that $f$ is a morphism of finite type between $\mathcal{O}(S)$-schemes
of finite presentation, that $\mathbb{L}$ is constructible, that $Y$ is
separated, and that $X^{\mathrm{an}}$ is finite-dimensional. We argue by
induction on the dimension of $X^{\mathrm{an}}$.

######  Step 3.

We reduce to the case where $f$ is an open immersion of reduced separated
$\mathcal{O}(S)$-schemes of finite presentation such that
$f_{\mathcal{O}_{S,s}}$ has dense image, and $\mathbb{L}=\mathbb{Z}/m$.

Let $K$ be an excellent Stein compact neighborhood of $s$ in $S$ (see Lemma
1.8 (ii)). By [SGA43, IX, Proposition 2.14 (ii) and XIV, Théorème 1.1], the
sheaf $\mathbb{L}|_{X_{\mathcal{O}(K)}}$ has a resolution by finite products
of sheaves of the form $\pi_{*}\mathbb{M}$ with $\pi:Z\to X_{\mathcal{O}(K)}$
finite and $\mathbb{M}$ a constant constructible sheaf on
$Z_{\mathrm{\acute{e}t}}$. Shrinking $S$, we may assume that $\mathbb{L}$
itself admits such a resolution (with finite morphisms $\pi:Z\to X$). By the
spectral sequence of a resolution, we may suppose that $\mathbb{L}$ is of the
form $\pi_{*}\mathbb{M}$. As the higher direct images of morphisms induced by
$\pi$ and $\pi^{\mathrm{an}}$ vanish by [SGA42, VIII, Proposition 5.5] and
[God58, II, Théorème 4.11.1], and as
$\pi_{*}(\mathbb{M})^{\mathrm{an}}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\pi^{\mathrm{an}}_{*}(\mathbb{M}^{\mathrm{an}})$
by Step 1, we may further assume that $\mathbb{L}$ is constant (after
replacing $X$ with $Z$ and $\mathbb{L}$ with $\mathbb{M}$), and then that
$\mathbb{L}\simeq\mathbb{Z}/m$.

Let $\mathcal{U}=(U_{i})_{i\in I}$ be a finite affine covering of $X$.
Computing cohomology using Čech spectral sequences associated with
$\mathcal{U}$, we may replace $f$ with $f|_{U_{J}}$, where $U_{J}:=\cap_{i\in
J}\,U_{i}$ for $J\subset I$. Since $X$ was quasi-separated (as is any
$\mathcal{O}(S)$-scheme of finite presentation), this trick reduces us to the
case where $X$ is separated.

As the étale and classical topologies are insensitive to nilpotents (see
[SGA42, VIII, Théorème 1.1]), we may assume that $X$ and $Y$ are reduced.

Write $f_{\mathcal{O}_{S,s}}$ as the composition
$X_{\mathcal{O}_{S,s}}\xrightarrow{g_{s}}\overline{X}_{s}\xrightarrow{h_{s}}Y_{\mathcal{O}_{S,s}}$
of an open immersion $g_{s}$ with dense image and of a proper morphism $h_{s}$
using Nagata’s compactification theorem [SP, Theorem 0F41]. Use a limit
argument based on [EGA43, Théorèmes 8.8.2 and 8.10.5] to extend, after
shrinking $S$, these morphisms of schemes to a diagram
$X\xrightarrow{g}\overline{X}\xrightarrow{h}Y$ of reduced separated
$\mathcal{O}(S)$-schemes of finite presentation, where $g$ is an open
immersion and $h$ is proper. As Theorem 3.7 holds for $h$ (and any torsion
étale sheaf) by Step 1, the composite functors spectral sequence computing
$\mathrm{R}^{k}(h\circ g)_{*}\mathbb{Z}/m$ reduces us to proving it for $g$.

######  Step 4.

We reduce to the case where moreover $Y_{\mathcal{O}_{S,s}}$ is regular.

Let $\mu_{s}:\widetilde{Y}_{s}\to Y_{\mathcal{O}_{S,s}}$ be a resolution of
singularities (use [Tem08, Theorem 1.1] and Lemma 1.10 (iii)). Set
$\widetilde{X}_{s}:=X\times_{Y_{\mathcal{O}_{S,s}}}\widetilde{Y}_{s}$, let
$j_{s}:U_{s}\hookrightarrow X_{s}$ be a dense open subset over which $\mu_{s}$
is an isomorphism, and let $\tilde{j}_{s}:U_{s}\to\widetilde{X}_{s}$ be the
lift of $j_{s}$. By a limit argument based on [EGA43, Théorèmes 8.8.2 and
8.10.5], the above schemes and morphisms arise by base change, after shrinking
$S$, from a cartesian diagram

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{j}}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu_{X}}$$\scriptstyle{\tilde{f}}$$\textstyle{\widetilde{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y\hbox
to0.0pt{,\hss}}$

whose horizontal arrows are quasi-compact open immersions and whose vertical
arrows are proper of finite presentation. Let $i:Z\to X$ and
$\tilde{i}:\widetilde{Z}\to\widetilde{X}$ be the inclusions of the complements
of $U$ in $X$ and $\widetilde{X}$ respectively. As $j_{s}$ and $\tilde{j}_{s}$
have dense image, both $\dim(Z^{\mathrm{an}})$ and
$\dim((\widetilde{Z})^{\mathrm{an}})$ are $<\dim(X^{\mathrm{an}})$ after
possibly shrinking $S$ by [Bin76b, Aussage 2.8]. The induction hypothesis thus
shows that Theorem 3.7 holds for $(f\circ i,\mathbb{Z}/m)$ and
$(\tilde{f}\circ\tilde{i},\mathbb{Z}/m)$, hence for $(f,i_{*}\mathbb{Z}/m)$
and $(\tilde{f},\tilde{i}_{*}\mathbb{Z}/m)$.

The five lemma applied to long exact sequences induced by the short exact
sequence $0\to j_{!}\,\mathbb{Z}/m\to\mathbb{Z}/m\to i_{*}\mathbb{Z}/m\to 0$
now shows that, to prove Theorem 3.7 for $(f,\mathbb{Z}/m)$, it suffices to
prove it for $(f,j_{!}\,\mathbb{Z}/m)$.

One computes at once using proper base change [SGA43, XII, Théorème 5.1] that
$\mu_{X,*}(\tilde{j}_{!}\,\mathbb{Z}/m)=j_{!}\,\mathbb{Z}/m$ and that
$\mathrm{R}^{q}\mu_{X,*}(\tilde{j}_{!}\,\mathbb{Z}/m)=0$ if $q>0$. Using the
composite functors spectral sequence computing
$\mathrm{R}^{k}(f\circ\mu_{X})_{*}$, we deduce that it suffices to prove
Theorem 3.7 for $(\tilde{f},\tilde{j}_{!}\,\mathbb{Z}/m)$. By another
application of the five lemma based on the short exact sequence
$0\to\tilde{j}_{!}\,\mathbb{Z}/m\to\mathbb{Z}/m\to\tilde{i}_{*}\mathbb{Z}/m\to
0$, it remains to prove Theorem 3.7 for $(\tilde{f},\mathbb{Z}/m)$.

######  Step 5.

We reduce to the case where $Y_{\mathcal{O}_{S,s}}\setminus
f_{\mathcal{O}_{S,s}}(X_{\mathcal{O}_{S,s}})$ is moreover regular.

The noetherian scheme $Y_{\mathcal{O}_{S,s}}\setminus
f_{\mathcal{O}_{S,s}}(X_{\mathcal{O}_{S,s}})$ is excellent by Lemma 1.10
(iii). Stratify it by its regular locus, the regular locus of its singular
locus, etc. Using [EGA43, Théorèmes 8.8.2 and 8.10.5], lift this
stratification to a stratification of ${Y\setminus f(X)}$, after maybe
shrinking $S$. One can thus write $f=f_{r}\circ\dots\circ f_{1}$ where
${f_{i}:X_{i-1}\to X_{i}}$ is a quasi-compact open immersion (with $X_{0}:=X$
and $X_{r}:=Y$), so that $f_{i,\mathcal{O}_{S,s}}$ has dense image and
$(X_{i}\setminus f_{i}(X_{i-1}))_{\mathcal{O}_{S,s}}$ is regular. Use [Bin76b,
Aussage 2.8] to ensure after shrinking $S$ that $\dim((X_{i}\setminus
X_{i-1})^{\mathrm{an}})<\dim(X^{\mathrm{an}})$.

By [SGA43, XIX, Théorème 2.1], one has $\mathbb{Z}/m\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}f_{i,\mathcal{O}_{S,s},*}\mathbb{Z}/m$.
It thus follows from Lemma 3.5 that $\mathbb{Z}/m\to f_{i,*}\mathbb{Z}/m$
becomes an isomorphism after restriction to $(X_{i})_{\mathcal{O}_{S,s}}$.
Using [SGA43, Corollaire 2.7.4], we may suppose after shrinking $S$ that
$\mathbb{Z}/m\to f_{i,*}\mathbb{Z}/m$ is an isomorphism.

Assume that Theorem 3.7 holds for all the $(f_{i},\mathbb{Z}/m)$. We prove by
induction on $i$ that it holds for $(f_{i}\circ\dots\circ
f_{1},\mathbb{Z}/m)$. To do so, it suffices to make use of the composite
functors spectral sequence computing
$\mathrm{R}^{k}(f_{i}\circ(f_{i-1}\circ\dots\circ f_{1}))_{*}\mathbb{Z}/m$,
since $\mathbb{Z}/m\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}f_{i,*}\mathbb{Z}/m$,
and noting that Theorem 3.7 holds for $(f_{i-1}\circ\dots\circ
f_{1},\mathrm{R}^{q}f_{i,*}\mathbb{Z}/m)$ when $q>0$ by the induction
hypothesis, as $\mathrm{R}^{q}f_{i,*}\mathbb{Z}/m$ is interiorly constructible
by Lemma 3.6, and supported on $X_{i}\setminus X_{i-1}$.

######  Step 6.

We apply purity to conclude.

As the theorem is trivial if $y\in f^{\mathrm{an}}(X^{\mathrm{an}})$, we
assume that $y\in Y^{\mathrm{an}}\setminus f^{\mathrm{an}}(X^{\mathrm{an}})$.
After shrinking $Y$, we may assume that $Y_{\mathcal{O}_{S,s}}\setminus
f_{\mathcal{O}_{S,s}}(X_{\mathcal{O}_{S,s}})\to Y_{\mathcal{O}_{S,s}}$ is a
regular immersion of pure codimension $c\geq 1$. Using [Bin76b, (2.7) and
(2.8)], one can ensure after shrinking $S$ that $Y^{\mathrm{an}}\setminus
f^{\mathrm{an}}(X^{\mathrm{an}})\to Y^{\mathrm{an}}$ is an embedding of pure
codimension $c$ of complex manifolds. It follows from Lemma 3.5 and [SGA43,
XIX, Théorèmes 3.2 and 3.4] that
$(\mathrm{R}^{k}f_{*}\mathbb{Z}/m)_{y}=\mathbb{Z}/m$ if $k\in\\{0,2c-1\\}$ and
vanishes otherwise. A direct topological computation shows that
$(\mathrm{R}^{k}f^{\mathrm{an}}_{*}\mathbb{Z}/m)_{y}=\mathbb{Z}/m$ if
$k\in\\{0,2c-1\\}$ and vanishes otherwise. Inspecting the construction of
these isomorphisms shows that they are compatible, and concludes the proof. ∎

######  Remark 3.8.

Theorem 3.7 fails if one drops both the properness and the constructibility
hypotheses, already if $S$ is a point, if
$f:\mathbb{A}^{1}_{\mathbb{C}}\to\mathrm{Spec}(\mathbb{C})$ is the structural
morphism, if $k=0$, and if $\mathbb{L}$ is a direct sum of skyscraper sheaves
at the integers.

##  4\. Killing ramification on finite coverings

The goal of this section is result is Proposition 4.3. It will be used in
conjunction with Proposition 2.4 in the proof of our Theorem 5.1.

###  4.1. Killing ramification locally

We first apply Theorem 3.7 to kill the ramification of cohomology classes in
the local analytic setting.

######  Proposition 4.1.

Let $U$ be an open subset of a connected normal Stein space $S$. Choose $s\in
U$. Let $Z\subset S$ be a closed analytic subset. Fix $\alpha\in
H^{k}(U\setminus(Z\cap U),\mathbb{Z}/m)$ with $k,m\geq 1$. There exist a
finite surjective holomorphic map $q:R\to S$ of connected normal Stein spaces,
a point $r\in R$ with $q(r)=s$, and an open neighborhood $\Theta$ of $r$ in
$q^{-1}(U)$ with $\alpha|_{\Theta\setminus(q^{-1}(Z)\cap\Theta)}=0$ in
$H^{k}(\Theta\setminus(q^{-1}(Z)\cap\Theta),\mathbb{Z}/m)$.

###### Proof.

We may assume that $Z$ is nowhere dense in $S$ (otherwise $Z=S$ and the lemma
is trivial). Let $\mathcal{I}_{Z}\subset\mathcal{O}_{S}$ be the ideal sheaf of
$Z$. Use Cartan’s Theorem A to choose finitely many elements $(f_{i})_{i\in
I}$ of $\mathcal{I}_{Z}(S)$ that generate $\mathcal{I}_{Z,s}$. After replacing
$Z$ with the vanishing locus of the $f_{i}$, we may apply Lemma 3.2 to find a
quasi-compact open immersion
${j:V\hookrightarrow\mathrm{Spec}(\mathcal{O}(S))}$ such that
$j^{\mathrm{an}}$ identifies with $S\setminus Z\hookrightarrow S$. Theorem 3.7
then shows that the base change morphism

(4.1)
$(\mathrm{R}^{k}j_{*}\mathbb{Z}/m)^{\mathrm{an}}\to\mathrm{R}^{k}j^{\mathrm{an}}_{*}\mathbb{Z}/m$

is an isomorphism. Let $\alpha_{s}$ be the image of $\alpha$ in
$(\mathrm{R}^{k}j^{\mathrm{an}}_{*}\mathbb{Z}/m)_{s}$ and consider its inverse
image $\beta_{s}\in(\mathrm{R}^{k}j_{*}\mathbb{Z}/m)_{s}$ by the isomorphism
(4.1).

Let $\beta\in H^{k}_{\mathrm{\acute{e}t}}(X_{V},\mathbb{Z}/m)$ be a
representative of $\beta_{s}$, where $f:X\to\mathrm{Spec}(\mathcal{O}(S))$ is
an integral affine étale neighborhood of $s\in\mathrm{Spec}(\mathcal{O}(S))$
and $X_{V}:=X\times_{\mathrm{Spec}(\mathcal{O}(S))}V$. The scheme
$(X_{V})_{s}:=X_{V}\times_{\mathrm{Spec}(\mathcal{O}(S))}\mathrm{Spec}(\mathcal{O}(S)_{s})$
is étale of finite type over $\mathcal{O}(S)_{s}$ (use Lemma 3.1 (ii)), hence
an excellent noetherian scheme by Lemma 1.10 (ii). By [Bha12, Theorem 1.1],
there exists a finite surjective morphism $\pi_{s}:Y_{s}\to(X_{V})_{s}$ such
that $\beta|_{Y_{s}}=0$ in $H^{k}_{\mathrm{\acute{e}t}}(Y_{s},\mathbb{Z}/m)$
(note that Bhatt’s proof simplifies in our situation, as the group scheme
$\mathbb{Z}/m$ is étale, see [Bha12, Remark 3.3]).

By a limit argument based on [EGA43, Théorèmes 8.8.2 and 8.10.5], there exist
an affine open neighborhood $W$ of $s$ in $\mathrm{Spec}(\mathcal{O}(S))$ and
a finite surjective morphism of finite presentation
$\pi:Y\to(X_{V})\times_{\mathrm{Spec}(\mathcal{O}(S))}W$ such that the base
change of $\pi$ by $\mathrm{Spec}(\mathcal{O}(S)_{s})\to W$ identifies with
$\pi_{s}$. By [SGA42, Exp. VII, Corollaire 5.8], one may assume after
shrinking $W$ that $\beta|_{Y}=0$ in
$H^{k}_{\mathrm{\acute{e}t}}(Y,\mathbb{Z}/m)$. After replacing $X$ with
$X\times_{\mathrm{Spec}(\mathcal{O}(S))}W$, we may finally view $\pi$ as a
finite surjective morphism of finite presentation $\pi:Y\to X_{V}$ such that
$\pi^{*}\beta=0$. As $X_{V}$ is integral, we may assume that so is $Y$ (after
replacing it with an irreducible component that dominates $X_{V}$).

One may factor ${f:X\to\mathrm{Spec}(\mathcal{O}(S))}$ as the composition
$X\xrightarrow{a}\overline{X}\xrightarrow{b}\mathrm{Spec}(\mathcal{O}(S))$ of
a quasi-compact open immersion $a$ and of a finite morphism $b$, by Zariski’s
Main Theorem [EGA43, Théorème 8.12.6]. A second application of Zariski’s Main
Theorem allows us to factor the natural morphism $Y\to\overline{X}$ as the
composition $Y\xrightarrow{c}\overline{Y}\xrightarrow{d}\overline{X}$ of a
quasi-compact open immersion $c$ and a finite morphism $d$. After replacing
$\overline{X}$ with the closure of $X$ in $\overline{X}$ and $\overline{Y}$
with the closure of $Y$ in $\overline{Y}$, one may assume that $\overline{X}$
and $\overline{Y}$ are integral. Here is a diagram summarizing the situation:

$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{c}$$\textstyle{\overline{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{X_{V}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a}$$\scriptstyle{\hskip
7.00002ptf}$$\textstyle{\overline{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j\hskip
14.00003pt}$$\textstyle{\mathrm{Spec}(\mathcal{O}(S))\hbox to0.0pt{.\hss}}$

As the natural morphism $Y\to\overline{Y}\times_{\overline{X}}X_{V}$ is a
proper open immersion with nonempty source and integral target, it is an
isomorphism. Choose $x\in X^{\mathrm{an}}$ with $f^{\mathrm{an}}(x)=s$. As $f$
is étale, the map $f^{\mathrm{an}}$ is a local biholomorphism at $x$ (see
[Bin76b, Satz 3.1 (1)]) so that $X^{\mathrm{an}}$ is normal at $x$, hence
locally irreducible at $x$. As $(X_{V})^{\mathrm{an}}$ is dense in
$X^{\mathrm{an}}$ (because $Z$ is nowhere dense in $S$), and as
$\pi^{\mathrm{an}}$ is finite surjective and $d^{\mathrm{an}}$ is finite by
[Bin76b, Satz 3.1 (2) (5)], the restriction
$(\overline{Y}\times_{\overline{X}}X)^{\mathrm{an}}\to X^{\mathrm{an}}$ of
$d^{\mathrm{an}}$ is finite and surjective. One may thus choose
$y\in(\overline{Y}\times_{\overline{X}}X)^{\mathrm{an}}\subset(\overline{Y})^{\mathrm{an}}$
such that $d^{\mathrm{an}}(y)=x$ and moreover, by local irreducibility of
$X^{\mathrm{an}}$ at $x$, such that the image in
$(\overline{X})^{\mathrm{an}}$ of some irreducible component $C$ of
$(\overline{Y})^{\mathrm{an}}$ through $y$ contains a neighborhood of $x$.

Consider the natural commutative diagram

(4.2)
$\textstyle{(\mathrm{R}^{k}j_{*}\mathbb{Z}/m)_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{R}^{k}j^{\mathrm{an}}_{*}\mathbb{Z}/m)_{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{R}^{k}c_{*}\mathbb{Z}/m)_{y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\mathrm{R}^{k}c^{\mathrm{an}}_{*}\mathbb{Z}/m)_{y}\hbox
to0.0pt{.\hss}}$

As $\pi^{*}\beta=0$, the image of $\beta_{s}$ in
$(\mathrm{R}^{k}c_{*}\mathbb{Z}/m)_{y}$ vanishes. By commutativity of (4.2),
the image of $\alpha_{s}$ in
$(\mathrm{R}^{k}c^{\mathrm{an}}_{*}\mathbb{Z}/m)_{y}$ also vanishes. This
exactly means that there exists an open neighborhood $\Theta^{\prime}$ of $y$
in $(b^{\mathrm{an}}\circ
d^{\mathrm{an}})^{-1}(U)\subset(\overline{Y})^{\mathrm{an}}$ such that
$\alpha|_{Y^{\mathrm{an}}\cap\Theta^{\prime}}=0$. If $\Theta^{\prime}$ is
small enough to be included in $(d^{\mathrm{an}})^{-1}(X^{\mathrm{an}})$, one
has the equality

$Y^{\mathrm{an}}\cap\Theta^{\prime}=\Theta^{\prime}\setminus((b^{\mathrm{an}}\circ
d^{\mathrm{an}})^{-1}(Z)\cap\Theta^{\prime}).$

To conclude, it suffices to take $R$ to be the normalization of
$C\subset\overline{Y}^{\mathrm{an}}$, to let $r$ be any preimage of $y$ in
$R$, and to choose $\Theta$ to be the inverse image of $\Theta^{\prime}$ in
$R$. ∎

######  Proposition 4.2.

Let $S$ be a Stein space, let $U\subset S$ be open, and choose $s\in U$. Let
$Z\subset S$ be a closed analytic subset. Fix $\alpha\in
H^{k}(U\setminus(Z\cap U),\mathbb{Z}/m)$ with $k,m\geq 1$. Then there exist a
finite surjective holomorphic map $p:T\to S$ and an open neighborhood $\Omega$
of $s$ in $U$ such that $\alpha|_{p^{-1}(\Omega\setminus(Z\cap\Omega))}=0$ in
$H^{k}(p^{-1}(\Omega\setminus(Z\cap\Omega)),\mathbb{Z}/m)$.

###### Proof.

Let $\nu:\widetilde{S}\to S$ be the normalization map and let
$(\tilde{s}_{i})_{i\in I}$ be the preimages of $s$ by $\nu$. Assume that the
maps $p_{i}:T_{i}\to\widetilde{S}$ prove the proposition for
$(\widetilde{S},\nu^{-1}(U),\tilde{s}_{i},\nu^{-1}(Z),\nu^{*}\alpha)$ using
disjoint open neighborhoods $\Omega_{i}$ of $\tilde{s}_{i}$ in
$\widetilde{S}$. Then the fiber product $p:T\to S$ of the maps $\nu\circ
p_{i}$ solves our original problem for any neighborhood $\Omega$ of $s$ in $S$
such that $\nu^{-1}(\Omega)\subset\cup_{i\in I}\,\Omega_{i}$. We may thus
assume that $S$ is normal. After replacing $S$ with its connected component
containing $s$, we may also suppose that it is connected.

Choose $q:R\to S$, a point $r\in R$ and $\Theta\subset q^{-1}(U)$ as in
Proposition 4.1. Let $p^{\prime}:T\to R$ and $\Gamma$ be given by Lemma 1.16,
and set $p:=q\circ p^{\prime}$. Let $(t_{j})_{j\in J}$ be the preimages of $s$
by $p$. As $p^{\prime}$ is surjective and $\Gamma$ acts transitively on the
fibers of $p$, one can find, for each $j\in J$, an element
$\gamma_{j}\in\Gamma$ such that $p^{\prime}\circ\gamma_{j}(t_{j})=r$. Let
$\Omega_{j}$ be a neighborhood of $t_{j}$ in $T$ such that
$p^{\prime}\circ\gamma_{j}(\Omega_{j})\subset\Theta$. After shrinking the
$\Omega_{j}$, we may assume that they are disjoint. Choosing $\Omega$ to be a
neighborhood of $s$ in $U$ such that $p^{-1}(\Omega)\subset\cup_{j\in
J}\,\Omega_{j}$ concludes the proof. ∎

###  4.2. Killing ramification globally

We finally globalize Proposition 4.2 thanks to the Čech-to-derived spectral
sequence.

######  Proposition 4.3.

Let $S$ be a Stein space, let $K\subset S$ be a compact subset and let $U$ be
an open neighborhood of $K$ in $S$. Let $Z\subset S$ be a closed analytic
subset, set $V:=S\setminus Z$, and fix $\alpha\in H^{k}(V\cap U,\mathbb{Z}/m)$
for some $k,m\geq 1$. Then, after maybe shrinking $U$, there exist a finite
surjective holomorphic map $p:T\to S$ and a class ${\beta\in
H^{k}(p^{-1}(U),\mathbb{Z}/m)}$ with $\alpha|_{p^{-1}(V\cap
U)}=\beta|_{p^{-1}(V\cap U)}$ in $H^{k}(p^{-1}(V\cap U),\mathbb{Z}/m)$.

###### Proof.

Let $\mathcal{H}^{s}$ be the presheaf on $S$ defined by
$\mathcal{H}^{s}(\Omega):=H^{s}(\Omega,\mathbb{Z}/m)$ for $\Omega\subset S$
open. Note that the presheaf $\mathcal{H}^{0}$ is the constant sheaf
$\mathbb{Z}/m$. Let $\mathcal{U}=(U_{i})_{i\in I}$ be a finite open covering
of $U$ and consider the Čech-to-derived spectral sequence

$E_{2}^{r,s}=\check{H}^{r}(V\cap\mathcal{U},\mathcal{H}^{s})\implies
H^{r+s}(V\cap U,\mathbb{Z}/m)$

associated with the open covering $V\cap\mathcal{U}=(V\cap U_{i})_{i\in I}$ of
$V\cap U$ ([SP, Lemma 03AZ], see [SP, Definition 03AM] for the definition of
Čech cohomology). Let $F^{\bullet}$ be the filtration on its abutment. Let
$l\geq 0$ be maximal with $\alpha\in F^{l}H^{k}(V\cap U,\mathbb{Z}/m)$.

Assume first that $l<k$. If $J=(i_{0},\dots,i_{l})\in I^{l+1}$, write
$U_{J}:=\cap_{t=0}^{l}U_{i_{t}}$. Then $\alpha$ is induced by a class in
$\check{H}^{l}(V\cap\mathcal{U},\mathcal{H}^{k-l})$, represented by a cocycle
$(\gamma_{J})_{J\in I^{l+1}}$, where

$\gamma_{J}\in\mathcal{H}^{k-l}(V\cap U_{J})=H^{k-l}(V\cap
U_{J},\mathbb{Z}/m).$

For each $s\in K$ and $J\in I^{l+1}$, one can apply Proposition 4.2 to find an
open neighborhood $\Omega_{s,J}$ of $s$ in $U_{J}$ and a finite surjective
holomorphic map ${p_{s,J}:T_{s,J}\to S}$ with
$\gamma_{J}|_{p_{s,J}^{-1}(V\cap\Omega_{s,J})}=0$. Intersecting the
$\Omega_{s,J}$ and taking the fiber product of the $p_{s,J}:T_{s,J}\to S$ over
$S$ for varying $J$ yields an open neighborhood $\Omega_{s}$ of $s$ in $U$ and
a finite surjective holomorphic map $p_{s}:T_{s}\to S$. Extract from
$(\Omega_{s})_{s\in K}$ a finite covering of $K$ (which we view, after
shrinking $U$, as a finite covering $\mathcal{U}^{\prime}$ of $U$ which
refines $\mathcal{U}$) and let $p:T\to S$ be the fiber product over $S$ of the
corresponding maps $p_{s}:T_{s}\to S$. Our choices ensure that the image in
$\check{H}^{l}(p^{-1}(V)\cap p^{-1}(\mathcal{U}^{\prime}),\mathcal{H}^{k-l})$
of $[\gamma_{J}]\in\check{H}^{l}(V\cap\mathcal{U},\mathcal{H}^{k-l})$ is
represented by the zero cocycle and hence vanishes. As a consequence, after
replacing $S$, $K$, $V$ and $\mathcal{U}$ with $T$, $p^{-1}(K)$, $p^{-1}(V)$
and $p^{-1}(\mathcal{U}^{\prime})$, we have managed to increase the value of
$l$.

Repeating this procedure finitely many times, we may assume that $l=k$, hence
that $\alpha$ is induced by a class
$\tilde{\alpha}\in\check{H}^{k}(V\cap\mathcal{U},\mathcal{H}^{0})=\check{H}^{k}(V\cap\mathcal{U},\mathbb{Z}/m)$.
We may suppose that $S$ is normal and connected (replace it with a connected
component of its normalization) and that $Z$ is nowhere dense in $S$
(otherwise $Z=S$ and $\alpha=0$). Then, if $j:V\hookrightarrow S$ is the
inclusion morphism, the adjunction morphism $\mathbb{Z}/m\to
j_{*}j^{*}\mathbb{Z}/m$ is an isomorphism (use [GR84, 9, §1.2, Theorem vii)
$\Leftrightarrow$ viii)]). As a consequence, the restriction map
$\check{H}^{k}(\mathcal{U},\mathbb{Z}/m)\to\check{H}^{k}(V\cap\mathcal{U},\mathbb{Z}/m)$
is an isomorphism, and we let
$\tilde{\beta}\in\check{H}^{k}(\mathcal{U},\mathbb{Z}/m)$ be the inverse image
of $\tilde{\alpha}$. The image $\beta\in H^{k}(U,\mathbb{Z}/m)$ of
$\tilde{\beta}$ by the natural morphism
$\check{H}^{k}(\mathcal{U},\mathbb{Z}/m)\to H^{k}(U,\mathbb{Z}/m)$ then has
the required property that $\beta|_{V\cap U}=\alpha$ in $H^{k}(V\cap
U,\mathbb{Z}/m)$. ∎

##  5\. An absolute comparison theorem

After fixing in §5.1 our conventions concerning the Grothendieck topologies
that we will use, we prove our main comparison theorem (Theorem 5.1) in §5.2,
we extend it to the $\mathrm{Gal}(\mathbb{C}/\mathbb{R})$-equivariant setting
in §5.3 and §5.4, and we give applications to cohomological dimension bounds
in §5.5.

###  5.1. The qfh topology

Our proof of Theorem 5.1 requires the use of a Grothendieck topology for
schemes that is finer than the étale topology: Voevodsky’s $\mathrm{qfh}$
topology whose coverings are given by surjective quasi-finite morphisms. The
precise definition that we use is the one given in [Cho22, §6.6]. As explained
in _loc_._cit_., it coincides with Voevodsky’s [Voe96, Definition 3.1.2] in
the noetherian case.

Let $S$ be a Stein space, and let $X$ be a separated $\mathcal{O}(S)$-scheme
of finite type. We will consider the following commutative diagram of sites:

(5.1)
$\textstyle{(X^{\mathrm{an}})_{\mathrm{qc}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta}$$\textstyle{(X^{\mathrm{an}})^{\prime}_{\mathrm{cl}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\textstyle{X_{\mathrm{qfh}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X_{\mathrm{\acute{e}t}})^{\prime}.}$

The site $(X_{\mathrm{\acute{e}t}})^{\prime}$ is the variant of the site
$X_{\mathrm{\acute{e}t}}$ of (3.1) where one restricts the objects to those
étale $X$-schemes that are separated and of finite type over $\mathcal{O}(S)$,
and where one only considers coverings involving finitely many arrows. As
objects of $X_{\mathrm{\acute{e}t}}$ admit finite Zariski coverings by affine
hence separated schemes, the topoi associated with
$(X_{\mathrm{\acute{e}t}})^{\prime}$ and $X_{\mathrm{\acute{e}t}}$ are
equivalent by [SP, Lemma 03A0]. We may thus use them interchangeably for
cohomological computations.

The site $(X^{\mathrm{an}})^{\prime}_{\mathrm{cl}}$ is the variant of the site
$(X^{\mathrm{an}})_{\mathrm{cl}}$ of (3.1) where one restricts the objects to
those topological spaces endowed with a local homeomorphism to
$X^{\mathrm{an}}$ that are Hausdorff (but we impose no further constraints on
coverings). As objects of $(X^{\mathrm{an}})_{\mathrm{cl}}$ admit open
coverings by Hausdorff spaces, the topoi associated with
$(X^{\mathrm{an}})^{\prime}_{\mathrm{cl}}$ and
$(X^{\mathrm{an}})_{\mathrm{cl}}$ are equivalent, again by [SP, Lemma 03A0].

The site $X_{\mathrm{qfh}}$ has as objects those $X$-schemes that are
separated and of finite type over $\mathcal{O}(S)$, and is endowed with the
$\mathrm{qfh}$ topology [Cho22, §6.6], restricting to coverings involving only
finitely many arrows.

The site $(X^{\mathrm{an}})_{\mathrm{qc}}$ has as objects the Hausdorff and
locally compact topological spaces over $X^{\mathrm{an}}$ and is endowed with
the $\mathrm{qc}$ topology [SP, Definition 09X0].

The morphisms of sites in (5.1) are the obvious ones (étale coverings are
$\mathrm{qfh}$ coverings by [SP, Lemma 0ETV] and analytifications of
$\mathrm{qfh}$ coverings are $\mathrm{qc}$ coverings as a consequence of [SP,
Lemma 09X5]).

###  5.2. Artin’s comparison theorem over Stein compact sets

We may now state and prove the main theorem of this article.

######  Theorem 5.1.

Let $S$ be a Stein space. Let ${f:X\to\mathrm{Spec}(\mathcal{O}(S))}$ be an
$\mathcal{O}(S)$-scheme interiorly of finite type. Fix a torsion abelian sheaf
$\mathbb{L}$ on $X_{\mathrm{\acute{e}t}}$. Assume that $f$ is interiorly
proper or that $\mathbb{L}$ is interiorly constructible. If one lets $U$ run
over all Stein open neighborhoods of a Stein compact subset $K$ of $S$, the
base change morphisms

(5.2) $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(U)},\mathbb{L}_{\mathcal{O}(U)})\to\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}((X_{\mathcal{O}(U)})^{\mathrm{an}},\mathbb{L}^{\mathrm{an}})$

are isomorphisms for $k\geq 0$.

###### Proof.

We split the proof in several steps.

######  Step 1.

We first reduce to the case where $X=\mathrm{Spec}(\mathcal{O}(S))$.

By Lemma 1.8 (ii), the morphism (5.9) identifies with the natural morphism

(5.3) $\underset{K\subset
L}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(L)},\mathbb{L}_{\mathcal{O}(L)})\to\underset{K\subset
L}{\mathrm{colim}}\,\,H^{k}((X_{\mathcal{O}(L)})^{\mathrm{an}},\mathbb{L}^{\mathrm{an}}),$

where $L$ runs over all excellent Stein compact neighborhoods of $K$ in $S$,
and $(X_{\mathcal{O}(L)})^{\mathrm{an}}:=(f^{\mathrm{an}})^{-1}(L)$. Both
sides of (5.9) or equivalently (5.3) are computed by (colimits of) Leray
spectral sequences, whose $E_{2}^{p,q}$ terms read respectively

$\underset{K\subset
L}{\mathrm{colim}}\,\,H^{p}_{\mathrm{\acute{e}t}}(\mathrm{Spec}(\mathcal{O}(L)),(\mathrm{R}^{q}f_{*}\mathbb{L})_{\mathcal{O}(L)})\textrm{
\,and \,}\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}(U,(\mathrm{R}^{q}f_{*}\mathbb{L})^{\mathrm{an}}),$

where we used Lemma 3.5 and Theorem 3.7 applied with
${Y=\mathrm{Spec}(\mathcal{O}(S))}$. These terms are isomorphic by the
$X=\mathrm{Spec}(\mathcal{O}(S))$ case of Theorem 5.1 applied to the sheaves
$\mathrm{R}^{q}f_{*}\mathbb{L}$.

######  Step 2.

We further reduce to the case where $\mathbb{L}=\mathbb{Z}/m$ for some $m\geq
1$.

If $K^{\prime}$ is a fixed excellent Stein compact neighborhood of $K$ in $S$
(which exists by Lemma 1.8 (ii)), the sheaf
$\mathbb{L}|_{\mathrm{Spec}(\mathcal{O}(K^{\prime}))}$ is the filtered colimit
of its constructible subsheaves by [SGA43, IX, Proposition 2.9 (iii)].
Consequently, after shrinking $S$, we may assume that $\mathbb{L}$ is the
filtered colimit of its constructible subsheaves.

As both the étale cohomology of $\mathrm{Spec}(\mathcal{O}(L))$ and the sheaf
cohomology of $L$ commute with filtered colimits (see [SGA42, VII, Proposition
3.3] and [God58, II, Théorème 4.12.1]), we may assume that $\mathbb{L}$ is
constructible.

Fix again an excellent Stein compact neighborhood $K^{\prime}$ of $K$ in $S$.
By [SGA43, IX, Proposition 2.14 (ii) and XIV, Théorème 1.1], the sheaf
$\mathbb{L}|_{\mathrm{Spec}(\mathcal{O}(K^{\prime}))}$ admits a resolution by
finite products of sheaves of the form $\pi_{*}\mathbb{M}$, where
$\pi:Z\to\mathrm{Spec}(\mathcal{O}(K^{\prime}))$ is finite and $\mathbb{M}$ is
a constant constructible sheaf on $Z_{\mathrm{\acute{e}t}}$. After shrinking
$S$, we may assume that $\mathbb{L}$ itself admits such a resolution (with
morphisms $\pi:Z\to\mathrm{Spec}(\mathcal{O}(S))$). Using the spectral
sequence of a resolution, we may assume that $\mathbb{L}$ is of the form
$\pi_{*}\mathbb{M}$. As the higher direct images of morphisms induced by $\pi$
and $\pi^{\mathrm{an}}$ vanish by [SGA42, VIII, Proposition 5.5] and [God58,
II, Théorème 4.11.1], and as $\pi_{*}(\mathbb{M})^{\mathrm{an}}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\pi^{\mathrm{an}}_{*}(\mathbb{M}^{\mathrm{an}})$
by Theorem 3.7, we may assume that $\mathbb{L}$ is constant (after replacing
$S$ with $Z^{\mathrm{an}}$ and $\mathbb{L}$ with $\mathbb{M}$), and then that
$\mathbb{L}\simeq\mathbb{Z}/m$, as wanted.

######  Step 3.

We finally assume that $X=\mathrm{Spec}(\mathcal{O}(S))$ and
$\mathbb{L}=\mathbb{Z}/m$ for some $m\geq 1$.

For any Stein open neighborhood $U$ of $K$ in $S$ (resp.any excellent Stein
compact neighborhood $L$ of $K$ in $S$), write
$X_{U}:=\mathrm{Spec}(\mathcal{O}(U))$
(resp.$X_{L}:=\mathrm{Spec}(\mathcal{O}(L))$). Our goal is to show that the
morphisms

(5.4) $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(X_{U},\mathbb{Z}/m)\xrightarrow{\varepsilon^{*}}\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}(X_{U}^{\mathrm{an}},\mathbb{Z}/m)$

are isomorphisms. For any excellent Stein compact neighborhood $L$ of $K$ in
$S$, the change of topology morphisms
$H^{k}_{\mathrm{\acute{e}t}}(X_{L},\mathbb{Z}/m)\to
H^{k}((X_{L})_{\mathrm{qfh}},\mathbb{Z}/m)$ are isomorphisms by [Voe96,
Theorem 3.4.4]. Taking the colimit over all such $L$ and using Lemma 1.8 (ii)
shows that $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(X_{U},\mathbb{Z}/m)\to\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}((X_{U})_{\mathrm{qfh}},\mathbb{Z}/m)$ is an
isomorphism. As the pull-back morphisms
$H^{k}(X_{U}^{\mathrm{an}},\mathbb{Z}/m)\to
H^{k}((X_{U}^{\mathrm{an}})_{\mathrm{qc}},\mathbb{Z}/m)$ are also isomorphisms
by [SP, Lemma 09X4], it follows from diagram (5.1) that (5.4) may be
identified with the morphism

(5.5) $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}((X_{U})_{\mathrm{qfh}},\mathbb{Z}/m)\xrightarrow{\zeta^{*}}\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}((X_{U}^{\mathrm{an}})_{\mathrm{qc}},\mathbb{Z}/m).$

The colimit over $U$ of the Leray spectral sequences of $\zeta$ reads

(5.6) $E_{2}^{p,q}=\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\mathrm{R}^{q}\zeta_{*}\mathbb{Z}/m)\implies\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p+q}(U_{\mathrm{qc}},\mathbb{Z}/m).$

We will prove that the adjunction morphism $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\mathbb{Z}/m)\to
E_{2}^{p,0}$ is an isomorphism in Step 4 and that $E_{2}^{p,q}=0$ for $q>0$ in
Step 5. It follows that the spectral sequence (5.6) degenerates and that (5.5)
hence (5.4) are isomorphisms.

######  Step 4.

One has $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\mathbb{Z}/m)\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\zeta_{*}\mathbb{Z}/m)$.

Let $U_{0}$ be a fixed Stein open neighborhood of $K$ in $S$ and let $Y$ be a
separated $\mathcal{O}(U_{0})$-scheme of finite type. For any Stein open
neighborhood $U$ of $K$ in $U_{0}$, write $Y_{U}:=Y\times_{X_{U_{0}}}X_{U}$.
Consider the adjunction morphism

(5.7) $\underset{K\subset U\subset
U_{0}}{\mathrm{colim}}\,\,H^{0}((Y_{U})_{\mathrm{qfh}},\mathbb{Z}/m)\to\underset{K\subset
U\subset
U_{0}}{\mathrm{colim}}\,\,H^{0}((Y_{U})_{\mathrm{qfh}},\zeta_{*}\mathbb{Z}/m).$

As the étale sheaf $\mathbb{Z}/m$ is already a $\mathrm{qfh}$ sheaf (see [SP,
Lemma 0EW8]), the left-hand side of (5.7) is isomorphic to $\underset{K\subset
U\subset
U_{0}}{\mathrm{colim}}\,\,H^{0}_{\mathrm{\acute{e}t}}(Y_{U},\mathbb{Z}/m)$. In
addition, as the constant sheaf $\mathbb{Z}/m$ for the usual topology is
already a $\mathrm{qc}$ sheaf (see [SP, Lemma 09X3]), the right-hand side of
(5.7) is isomorphic to $\underset{K\subset U\subset
U_{0}}{\mathrm{colim}}\,\,H^{0}((Y_{U})^{\mathrm{an}},\mathbb{Z}/m)$. All in
all, the morphism (5.7) may be identified with

$\underset{K\subset U\subset
U_{0}}{\mathrm{colim}}\,\,H_{\mathrm{\acute{e}t}}^{0}(Y_{U},\mathbb{Z}/m)\to\underset{K\subset
U\subset U_{0}}{\mathrm{colim}}\,\,H^{0}((Y_{U})^{\mathrm{an}},\mathbb{Z}/m),$

and hence is an isomorphism by [Bin76b, (7.2)].

The computation of the cohomology of the site $(X_{U})_{\mathrm{qfh}}$ using
hypercoverings [SP, Proposition 09VZ], the fact that any given $\mathrm{qfh}$
hypercovering of $X_{U}$ involves only finitely many separated
$\mathcal{O}(U)$-schemes of finite type at each simplicial level (as coverings
in $(X_{U})_{\mathrm{qfh}}$ involve only finitely many arrows) and the fact
that (5.7) is an isomorphism for such schemes, together imply that the natural
morphism

$\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\mathbb{Z}/m)\to\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\zeta_{*}\mathbb{Z}/m)$

is an isomorphism, which is what we wanted to prove.

######  Step 5.

One has $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\mathrm{R}^{q}\zeta_{*}\mathbb{Z}/m)=0$
for $q>0$.

Fix $q>0$. We claim that for any Stein open neighborhood $U_{0}$ of $K$ in
$S$, and any separated quasi-finite $\mathcal{O}(U_{0})$-scheme of finite
presentation $Y$, the group

(5.8) $\underset{K\subset U\subset
U_{0}}{\mathrm{colim}}\,\,H^{0}((Y_{U})_{\mathrm{qfh}},\mathrm{R}^{q}\zeta_{*}\mathbb{Z}/m)$

vanishes. Taking this claim for granted, the computation of the cohomology of
the site $(X_{U})_{\mathrm{qfh}}$ using hypercoverings [SP, Proposition 09VZ],
the fact that any given $\mathrm{qfh}$ hypercovering of $X_{U}$ involves only
finitely many separated quasi-finite $\mathcal{O}(U)$-schemes of finite
presentation at each simplicial level, and the fact that (5.8) vanishes for
such schemes, together imply that $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{p}((X_{U})_{\mathrm{qfh}},\mathrm{R}^{q}\zeta_{*}\mathbb{Z}/m)=0$.

We now prove the claim. Using Lemma 1.9, we may assume that $K$ admits a basis
$(U^{\prime}_{i})_{i\in I}$ of Stein open neighborhoods such that $K$ is
$\mathcal{O}(U^{\prime}_{i})$-convex for all ${i\in I}$. We may thus restrict
the colimit (5.8) to such neighborhoods. Fix a Stein open neighborhood $U$ of
$K$ in $U_{0}$ such that $K$ is $\mathcal{O}(U)$-convex and choose a class
$\alpha$ in
$H^{0}((Y_{U})_{\mathrm{qfh}},\mathrm{R}^{q}\zeta_{*}\mathbb{Z}/m)$. There
exists a $\mathrm{qfh}$ covering of $Y_{U}$, which we may choose to be given
by a single morphism $W\to Y_{U}$, such that $\alpha$ is induced by a class
$\beta\in H^{q}(W^{\mathrm{an}},\mathbb{Z}/m)$. By Zariski’s Main Theorem (see
[EGA43, Théorème 8.12.6]), there exists a factorization
$W\xrightarrow{j}\overline{W}\xrightarrow{\pi}X_{U}$ of $W\to X_{U}$ where $j$
is an open immersion and $\pi$ is finite.

By Proposition 4.3 applied to the Stein space $(\overline{W})^{\mathrm{an}}$,
to the Stein compact subset $(\pi^{\mathrm{an}})^{-1}(K)$, and to the
cohomology class $\beta$, there exist, after maybe shrinking $U$, a finite
surjective holomorphic map $p:T\to(\overline{W})^{\mathrm{an}}$ and a class
$\gamma\in H^{q}(T,\mathbb{Z}/m)$ with
$\gamma|_{p^{-1}(W^{\mathrm{an}})}=\beta|_{p^{-1}(W^{\mathrm{an}})}$. As $K$
is $\mathcal{O}(U)$-convex, the compact subset $(\pi^{\mathrm{an}}\circ
p)^{-1}(K)$ is $\mathcal{O}(T)$-convex (see Proposition 1.5 (i)). One can thus
apply Proposition 2.4 to ensure, after modifying
$p:T\to(\overline{W})^{\mathrm{an}}$ and further shrinking $U$, that
$\gamma=0$ and hence that $\beta|_{p^{-1}(W^{\mathrm{an}})}=0$.

By Lemma 3.3, the scheme $\mathrm{Spec}(\mathcal{O}(T))$ is interiorly finite
with analytification isomorphic to $T$ (both when viewed as a
$\overline{W}$-scheme or as an $\mathcal{O}(U)$-scheme), and the structural
morphism $g:\mathrm{Spec}(\mathcal{O}(T))\to\overline{W}$ satisfies
$g^{\mathrm{an}}=p$. Using Lemma 3.1 (iii), one may assume after shrinking $U$
that $g$ is finite of finite presentation. It follows that
$g|_{g^{-1}(W)}:g^{-1}(W)\to W$ is a qfh covering. As
$\beta|_{g^{-1}(W)^{\mathrm{an}}}=0$, we conclude that $\alpha\in
H^{0}((Y_{U})_{\mathrm{qfh}},\mathrm{R}^{q}\zeta_{*}\mathbb{Z}/m)$ is qfh-
locally trivial and hence trivial. ∎

When $f$ is interiorly proper, the statement of Theorem 5.1 takes a simpler
form.

######  Corollary 5.2.

Let $K$ be a Stein compact subset of a Stein space $S$ and let $X$ be an
interiorly proper $\mathcal{O}(S)$-scheme. Fix a torsion abelian sheaf
$\mathbb{L}$ on $X_{\mathrm{\acute{e}t}}$. For ${k\geq 0}$, there are
canonical isomorphisms
$H^{k}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(K)},\mathbb{L}|_{X_{\mathcal{O}(K)}})\to
H^{k}((X_{\mathcal{O}(K)})^{\mathrm{an}},\mathbb{L}^{\mathrm{an}})$.

###### Proof.

The isomorphism (5.9) has the required form by [SGA42, Exp. VII, Corollaire
5.8] and [God58, II, Théorème 4.11.1]. ∎

######  Remark 5.3.

Bingener’s [Bin76b, Theorem 7.4] implies the particular case of Theorem 5.1
where $k=1$, the sheaf $\mathbb{L}$ is constant and
$X=\mathrm{Spec}(\mathcal{O}(S))$.

###  5.3. $G$-equivariant complex spaces

Let $G:=\mathrm{Gal}(\mathbb{C}/\mathbb{R})\simeq\mathbb{Z}/2$ be the Galois
group of $\mathbb{R}$, generated by the complex conjugation $\sigma\in G$. We
use the conventions of [BW21, Appendix A] concerning $G$-equivariant complex-
analytic geometry. In particular, a $G$-equivariant complex space is a complex
space endowed with an action of $G$ (as a locally ringed space) such that
$\sigma\in G$ acts $\mathbb{C}$-antilinearly.

A $G$-equivariant complex space is said to be Stein if so is its underlying
complex space. A $G$-invariant Stein compact subset of a $G$-equivariant
complex space $S$ admits a basis of $G$-invariant Stein open neighborhoods in
$S$ (as the intersection of a Stein open neighborhood of $K$ in $S$ with its
image by $\sigma$ is Stein).

The complex conjugate $S^{\sigma}$ of a complex space $S$ with structural
morphism ${\mu:\mathbb{C}\to\mathcal{O}_{S}}$ is the complex space which is
equal to $S$ as a locally ringed space, with structural morphism
$\mu_{S^{\sigma}}:=\mu\circ\sigma:\mathbb{C}\to\mathcal{O}_{S}$. A
$G$-equivariant complex space can equivalently be described as a complex space
$S$ endowed with an isomorphism $\alpha:S^{\sigma}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}S$ such that
$\alpha\circ\alpha^{\sigma}=\mathrm{Id}_{S}$ (see [BW21, §A.2]).

Note that $\mathcal{O}(S)=\mathcal{O}(S)^{G}\otimes_{\mathbb{R}}\mathbb{C}$. A
morphism $f:X\to Y$ of $\mathcal{O}(S)^{G}$-schemes is said to interiorly have
a property if such is the case for $f_{\mathcal{O}(S)}:X_{\mathcal{O}(S)}\to
Y_{\mathcal{O}(S)}$. If $X$ is an $\mathcal{O}(S)^{G}$-scheme interiorly
locally of finite type, the descent datum on
$X_{\mathcal{O}(S)}=X\times_{\mathrm{Spec}(\mathbb{R})}\mathrm{Spec}(\mathbb{C})$
yields an isomorphism
$\alpha:((X_{\mathcal{O}(S)})^{\mathrm{an}})^{\sigma}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}(X_{\mathcal{O}(S)})^{\mathrm{an}}$
which endows $(X_{\mathcal{O}(S)})^{\mathrm{an}}$ with a structure of
$G$-equivariant complex space: the analytification $X^{\mathrm{an}}$ of $X$.
The site morphism $\varepsilon:(X^{\mathrm{an}})_{\mathrm{cl}}\to
X_{\mathrm{\acute{e}t}}$ of (3.1) is $G$-equivariant for the actions of $G$ by
complex conjugation. In particular, the analytification
$\mathbb{L}^{\mathrm{an}}$ of a sheaf $\mathbb{L}$ on
$X_{\mathrm{\acute{e}t}}$ is naturally a $G$-equivariant sheaf on
$X^{\mathrm{an}}$. We will say that $\mathbb{L}$ is interiorly constructible
if so is $\mathbb{L}|_{X_{\mathcal{O}(S)}}$.

We state for later use in the proof of Theorem 6.1 a $G$-equivariant analogue
of Corollary 1.15, which appears in [BW21, Proposition A.5] when $S$ has
dimension $1$.

######  Proposition 5.4.

Let $S$ be a normal $G$-equivariant Stein space of pure dimension $n$ with
finitely many connected components. Associating with $p:T\to S$ the
$\mathcal{M}(S)^{G}$-algebra $\mathcal{M}(T)^{G}$ induces an equivalence of
categories

$\left\\{\begin{array}[]{l}\textrm{finite surjective $G$-equivariant
holomorphic}\\\ \textrm{ maps }p:T\to S\textrm{ of normal $G$-equivariant}\\\
\textrm{ \hskip 15.00002ptStein spaces of pure dimension
}n\end{array}\right\\}\to\left\\{\begin{array}[]{l}\textrm{\hskip
13.00005ptfinite \'{e}tale}\\\
\mathcal{M}(S)^{G}\textrm{-algebras}\end{array}\right\\}.$

###### Proof.

The proposition follows from Corollary 1.15 by using the above description of
$G$-equivariant complex analytic spaces as complex analytic spaces $S$ endowed
with an isomorphism $\alpha:S^{\sigma}\ext@arrow
0359\arrowfill@\relbar\relbar{\hbox{$\rightarrow$}}{}{\,\sim\,}S$ such that
$\alpha\circ\alpha^{\sigma}=\mathrm{Id}_{S}$. ∎

###  5.4. Artin’s comparison theorem over $G$-equivariant Stein compact sets

Here is an extension of Theorem 5.1 to the $G$-equivariant setting, which goes
back to Cox [Cox79, Theorem 1.1] when $S$ is a point (see also [Sch94,
(15.3.2)]).

######  Theorem 5.5.

Let $S$ be a $G$-equivariant Stein space. Let
${f:X\to\mathrm{Spec}(\mathcal{O}(S)^{G})}$ be an $\mathcal{O}(S)^{G}$-scheme
interiorly of finite type. Fix a torsion abelian sheaf $\mathbb{L}$ on
$X_{\mathrm{\acute{e}t}}$. Assume that $f$ is interiorly proper or that
$\mathbb{L}$ is interiorly constructible. If one lets $U$ run over all
$G$-invariant Stein open neighborhoods of a $G$-invariant Stein compact subset
$K$ of $S$, the base change morphisms

(5.9) $\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(U)^{G}},\mathbb{L}_{\mathcal{O}(U)^{G}})\to\underset{K\subset
U}{\mathrm{colim}}\,\,H_{G}^{k}((X_{\mathcal{O}(U)^{G}})^{\mathrm{an}},\mathbb{L}^{\mathrm{an}})$

are isomorphisms for $k\geq 0$.

###### Proof.

The $G$-equivariant site morphism
$\varepsilon:((X_{\mathcal{O}(U)})^{\mathrm{an}})_{\mathrm{cl}}\to(X_{\mathcal{O}(U)})_{\mathrm{\acute{e}t}}$
induces a morphism between the Hochschild–Serre spectral sequences of [Sch94,
Remark 10.9]

$\displaystyle
E_{2}^{p,q}=H^{p}(G,H^{q}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(U)},\mathbb{L}_{\mathcal{O}(U)}))$
$\displaystyle\implies$ $\displaystyle
H^{p+q}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(U)^{G}},\mathbb{L}_{\mathcal{O}(U)^{G}})\textrm{\hskip
3.00003pt and }$ $\displaystyle
E_{2}^{p,q}=H^{p}(G,H^{q}((X_{\mathcal{O}(U)})^{\mathrm{an}},\mathbb{L}^{\mathrm{an}}))$
$\displaystyle\implies$ $\displaystyle
H^{p+q}_{G}((X_{\mathcal{O}(U)^{G}})^{\mathrm{an}},\mathbb{L}^{\mathrm{an}}),$

which is an isomorphism on page $2$ by Theorem 5.1, and hence an isomorphism.
∎

###  5.5. Cohomological dimension

A scheme $X$ is said to have cohomological dimension $\leq n$ if the étale
cohomology of any torsion abelian sheaf on $X_{\mathrm{\acute{e}t}}$ vanishes
in degree $>n$. Our next theorem controls the cohomological dimension of
affine varieties over Stein compacta. It is a generalization in Stein geometry
of the bounds on the cohomogical dimension of complex and real affine
varieties obtained in [SGA43, XIV, Corollaire 3.2] and [Sch94, Corollary
7.21].

######  Theorem 5.6.

Let $K$ be a $G$-invariant Stein compact subset in a $G$-equivariant Stein
space $S$. Let $X$ be an affine $\mathcal{O}(S)^{G}$-scheme interiorly of
finite type with $(X^{\mathrm{an}})^{G}=\varnothing$. Then
$X_{\mathcal{O}(K)^{G}}$ has étale cohomological dimension
$\leq\dim(X^{\mathrm{an}})$.

###### Proof.

We may suppose that $\dim(X^{\mathrm{an}})<\infty$. We wish to show the
vanishing of the degree $k$ cohomology of a torsion abelian sheaf on
$(X_{\mathcal{O}(K)^{G}})_{\mathrm{\acute{e}t}}$ if $k>\dim(X^{\mathrm{an}})$.
By [SP, Lemma 03SA (2)], any such sheaf is a filtered colimit of constructible
sheaves. In view of [SGA42, VII, Proposition 3.3], we may thus only consider
constructible sheaves. By [SGA43, IX, Corollaire 2.7.4], any constructible
abelian sheaf on $(X_{\mathcal{O}(K)^{G}})_{\mathrm{\acute{e}t}}$ is of the
form $\mathbb{L}|_{X_{\mathcal{O}(K)^{G}}}$ for some constructible abelian
sheaf $\mathbb{L}$ on $X_{\mathrm{\acute{e}t}}$, after possibly shrinking $S$.

Let $K^{\prime}$ be an excellent Stein compact neighborhood of $K$ in $S$ (see
Lemma 1.8 (i)). By [SGA42, IX, Proposition 2.14 (ii) and XIV, Théorème 1.1],
the sheaf $\mathbb{L}|_{X_{\mathcal{O}(K^{\prime})^{G}}}$ has a resolution by
finite products of sheaves of the form $\pi_{*}A$, where $A$ is a finite
abelian group and ${\pi:Y\to X_{\mathcal{O}(K^{\prime})^{G}}}$ is finite.
After shrinking $S$, we may assume that $\mathbb{L}$ itself has such a
resolution (with finite morphisms $\pi:Y\to X$). Using the spectral sequence
of a resolution, we may assume that $\mathbb{L}$ is of the form $\pi_{*}A$.
Letting $U$ run over all $G$-invariant Stein open neighborhoods of $K$ in $S$,
one computes

$\displaystyle
H^{k}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(K)^{G}},\mathbb{L}|_{X_{\mathcal{O}(K)^{G}}})$
$\displaystyle=\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(X_{\mathcal{O}(U)^{G}},\mathbb{L}|_{X_{\mathcal{O}(U)^{G}}})$
(5.10) $\displaystyle=\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{\mathrm{\acute{e}t}}(Y_{\mathcal{O}(U)^{G}},A)$
$\displaystyle=\underset{K\subset
U}{\mathrm{colim}}\,\,H^{k}_{G}((Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}},A)$

where we used successively [SGA42, Exp. VII, Corollaire 5.8], the vanishing of
the higher direct images of $\pi$ [SGA42, VIII, Proposition 5.5], and Theorem
5.5.

After shrinking $S$, we may assume that the $\mathcal{O}(S)^{G}$-scheme $Y$ is
affine of finite type, hence a closed subscheme of
$\mathbb{A}^{N}_{\mathcal{O}(S)^{G}}$ for $N\gg 0$. It follows that
$(Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}}$ may be realized as a $G$-invariant
closed complex subspace of $\mathbb{C}^{N}\times U$ (see [Bin76b, Proof of
Satz 1.1]). It is thus a $G$-equivariant Stein space. The long exact sequence
of $G$-equivariant cohomology associated with the short exact sequence of
$G$-modules $0\to A\to A[G]\to A\to 0$ yields, for any $l\geq 0$, exact
sequences

(5.11) $H^{l}((Y_{\mathcal{O}(U)})^{\mathrm{an}},A)\to
H^{l}_{G}((Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}},A)\to
H^{l+1}_{G}((Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}},A).$

By a theorem of Hamm [Ham83, Satz 1], the Stein space
$(Y_{\mathcal{O}(U)})^{\mathrm{an}}$ has the homotopy type of a CW complex of
dimension
$\leq\dim((Y_{\mathcal{O}(U)})^{\mathrm{an}})\leq\dim(X^{\mathrm{an}})$, and
hence $H^{l}((Y_{\mathcal{O}(U)})^{\mathrm{an}},A)=0$ for
$l>\dim(X^{\mathrm{an}})$. In view of (5.5), of (5.11), and of the latter
vanishing, it suffices to show that
$H^{l}_{G}((Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}},A)=0$ for $l\gg 0$. But
$H^{l}_{G}((Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}},A)=H^{l}((Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}}/G,A)$
because $((Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}})^{G}=\varnothing$. As the
real-analytic space $(Y_{\mathcal{O}(U)^{G}})^{\mathrm{an}}/G$ has real
dimension $\leq 2\dim(X^{\mathrm{an}})$, it has the homotopy type of a CW
complex of finite dimension (combine [GMT86, VI, Theorem 2.7] and [Loj64,
Theorem 2]), and the desired vanishing follows. ∎

The following application of Theorem 5.6 extends an algebraic result [CTP90,
Proposition 1.2.1] attributed to Ax by Colliot-Thélène and Parimala.

######  Theorem 5.7.

Let $K$ be a $G$-invariant Stein compact subset of a $G$-equivariant Stein
space $S$. Let $X$ be an $\mathcal{O}(S)^{G}$-scheme interiorly of finite type
with ${(X^{\mathrm{an}})^{G}=\varnothing}$. If $X_{\mathcal{O}(K)^{G}}$ is
integral, its function field has cohomological dimension
$\leq\dim(X^{\mathrm{an}})$.

###### Proof.

We may assume that $X$ is affine (after replacing it with an affine open
subset $X^{\prime}\subset X$ such that $X^{\prime}_{\mathcal{O}(K)^{G}}$ is
nonempty). Let $(U_{i})_{i\in\mathbb{N}}$ be a decreasing basis of
$G$-invariant Stein open neighborhoods of $K$ in $S$. If $i\in\mathbb{N}$ and
$h\in\mathcal{O}(X_{\mathcal{O}(U_{i})^{G}})$ is nonzero in restriction to
$X_{\mathcal{O}(K)^{G}}$, we define $Y_{i,h}$ to be the distinguished affine
open subset of $X_{\mathcal{O}(U_{i})^{G}}$ where $h\neq 0$. Then the function
field $F$ of $X_{\mathcal{O}(K)^{G}}$ can be written as the filtered colimit
of rings
$F=\underset{i,h}{\mathrm{colim}}\,\,\mathcal{O}((Y_{i,h})_{\mathcal{O}(K)^{G}})$.
By Theorem 5.6 applied to the affine $\mathcal{O}(U_{i})^{G}$-scheme
$Y_{i,h}$, the scheme $(Y_{i,h})_{\mathcal{O}(K)^{G}}$ has cohomological
dimension $\leq\dim(X^{\mathrm{an}})$. We deduce from [SP, Lemma 0F0R] that
$F$ has cohomological dimension $\leq\dim(X^{\mathrm{an}})$. ∎

If $(X^{\mathrm{an}})^{G}\neq\varnothing$, one may sometimes apply Theorem 5.7
to a Zariski open subset $X^{\prime}$ of $X$ such that
$((X^{\prime})^{\mathrm{an}})^{G}=\varnothing$. The next corollary is obtained
in this way.

######  Corollary 5.8.

Let $K$ be a $G$-invariant Stein compact subset of a $G$-equivariant Stein
space $S$. If $S^{G}$ is contained in a nowhere dense closed analytic subset
${Z\subset S}$ and $\mathcal{O}(K)^{G}$ is a domain, then
$\mathrm{Frac}(\mathcal{O}(K)^{G})$ has cohomological dimension $\leq\dim(S)$.

###### Proof.

Define
$V:=\mathrm{Spec}(\mathcal{O}(S)^{G})\setminus\mathrm{Spec}(\mathcal{O}(S)^{G}/\mathcal{I}_{Z}(S)^{G})$.
By Lemma 3.2 and its proof, the $\mathcal{O}(S)^{G}$-scheme $V$ is interiorly
of finite type and the analytification of the open immersion
$V\hookrightarrow\mathrm{Spec}(\mathcal{O}(S)^{G})$ identifies with
$S\setminus Z\hookrightarrow S$. It now suffices to apply Theorem 5.7 with
$X=V$ to conclude. ∎

If $S$ is a complex space, then the disjoint union $T:=S\sqcup S^{\sigma}$ may
be endowed with a structure of $G$-equivariant complex space, by letting
$\sigma\in G$ exchange the two factors (and act by the identity on the
underlying locally ringed spaces). One then has $T^{G}=\varnothing$ and
$\mathcal{O}(T)^{G}\simeq\mathcal{O}(S)$. Using this trick, one can formally
deduce non-$G$-equivariant statements from $G$-equivariant ones. For instance,
Theorems 5.6 and 5.7 and Corollary 5.8 immediately imply the following.

######  Theorem 5.9.

Let $K$ be a Stein compact subset of a Stein space $S$. Let $X$ be an affine
$\mathcal{O}(S)$-scheme interiorly of finite type. Then $X_{\mathcal{O}(K)}$
has étale cohomological dimension $\leq\dim(X^{\mathrm{an}})$.

######  Theorem 5.10.

Let $K$ be a Stein compact subset of a Stein space $S$. Let $X$ be an
$\mathcal{O}(S)$-scheme interiorly of finite type. If $X_{\mathcal{O}(K)}$ is
integral, then its function field has cohomological dimension
$\leq\dim(X^{\mathrm{an}})$.

######  Corollary 5.11.

Let $K$ be a Stein compact subset of a Stein space $S$. If $\mathcal{O}(K)$ is
a domain, then $\mathrm{Frac}(\mathcal{O}(K))$ has cohomological dimension
$\leq\dim(S)$.

######  Remarks 5.12.

(i) One could have given direct proofs of Theorems 5.9 and 5.10 and Corollary
5.11, similar to the proofs of Theorems 5.6 and 5.7 and Corollary 5.8
(replacing Theorem 5.5 by Theorem 5.1).

(ii) In the setting of Corollary 5.11, if $S$ has pure dimension $n$ and $K$
is nonempty, the cohomological dimension of $\mathrm{Frac}(\mathcal{O}(K))$ is
equal to $n$. To see it, choose ${s\in K}$. The local ring morphism
$\mathcal{O}(K)_{s}\to\mathcal{O}_{S,s}$ is faithfully flat by Lemma 1.7 and
[Mat80, Theorem 2]. It thus follows from [Bou06, I, §3.5, Corollaire] and
[Mat80, Theorem 19] that $\mathcal{O}(K)_{s}$ is noetherian of dimension $n$.
By [SGA43, Corollaire 2.5], the field
$\mathrm{Frac}(\mathcal{O}(K))=\mathrm{Frac}(\mathcal{O}(K)_{s})$ has
cohomological dimension $\geq n$.

##  6\. Applications to Hilbert’s $17$th problem

We may now present our applications to sums of squares of analytic functions.

###  6.1. Sums of squares

The following theorem is an analogue of Artin’s solution to Hilbert’s $17$th
problem [Art27] on $G$-equivariant Stein compacta.

######  Theorem 6.1.

Let $K$ be a $G$-invariant Stein compact subset of a reduced $G$-equivariant
Stein space $S$. Let $f\in\mathcal{O}(K)^{G}$ be nonnegative on a neighborhood
of $K^{G}$ in $S^{G}$. Then $f$ is a sum of squares in $\mathcal{M}(K)^{G}$.

###### Proof.

Using [GR84, 8, §1.3, Proposition], we may replace $S$ by its normalization
and hence assume that it is normal. Let $K^{\prime}$ be a Stein compact
neighborhood of $K$ in $M$ such that $f\in\mathcal{O}(K^{\prime})^{G}$ (see