# The six functors for Zariski-constructible sheaves in rigid geometry

Bhargav Bhatt  Bhargav Bhatt
Department of Mathematics, University of Michigan
Ann Arbor, MI 48109, USA<EMAIL_ADDRESS>and  David Hansen  David Hansen
Max Planck Institute for Mathematics
Vivatsgasse 7, Bonn 53111, Germany<EMAIL_ADDRESS>
###### Abstract.

We prove a generic smoothness result in rigid analytic geometry over a
characteristic zero nonarchimedean field. The proof relies on a novel notion
of generic points in rigid analytic geometry which are well-adapted to
“spreading out” arguments, in analogy with the use of generic points in scheme
theory. As an application, we develop a six functor formalism for Zariski-
constructible étale sheaves on characteristic zero rigid spaces. Among other
things, this implies that characteristic zero rigid spaces support a well-
behaved theory of perverse sheaves.

###### Key words and phrases:

Rigid analytic spaces, étale cohomology, generic smoothness, six functors

###### 1991 Mathematics Subject Classification:

14G22, 14F20

###### Contents

  1. 1 Introduction
  2. 2 Generic flatness and generic smoothness
    1. 2.1 Shilov and weakly Shilov points
    2. 2.2 Tools involving the cotangent complex
    3. 2.3 Regularity of Shilov fibers
    4. 2.4 Generic flatness
    5. 2.5 Generic smoothness
  3. 3 The six functors for Zariski-constructible sheaves
    1. 3.1 Definition of Zariski-constructible sheaves
    2. 3.2 Zariski-constructible sheaves via algebraic geometry
    3. 3.3 Pushforward, $\otimes$, and $\mathop{R\mathscr{H}\mathrm{om}}$
    4. 3.4 Verdier duality
    5. 3.5 Miscellany
    6. 3.6 Adic coefficients
  4. 4 Perverse sheaves
    1. 4.1 Finite coefficients
    2. 4.2 $\mathbf{Z}_{\ell}$-coefficients
    3. 4.3 $\mathbf{Q}_{\ell}$-coefficients
    4. 4.4 Intersection cohomology
    5. 4.5 Some conjectures

##  1\. Introduction

In this paper, we prove a new generic smoothness result for morphisms of rigid
analytic spaces (regarded as adic spaces, always), and apply it to set up a
$6$ functor formalism for étale cohomology of rigid analytic spaces with
coefficients in Zariski-constructible sheaves. Our first geometric result is
the following (see also Remark 1.4 for an alternative approach through
[Duc18]).

######  Theorem 1.1 (Generic smoothness, Theorems 2.21, 2.27 and 2.29).

Fix a nonarchimedean field $K$, and let $f:X\to Y$ be a quasicompact map of
rigid analytic spaces over $\mathop{\mathrm{Spa}}K$ with $Y$ reduced.

  1. (1)

If $Y$ is geometrically reduced (which is automatic from reducedness if $K$
has characteristic $0$, [Con99, Lemma 3.3.1]), there is a dense open subset
$U\subset Y$ such that $f^{-1}(U)\to U$ is flat.

  2. (2)

If $\mathrm{char}\,K=0$ and $X$ is smooth, there is a dense open subset
$U\subset Y$ such that $f^{-1}(U)\to U$ is smooth. If moreover $f$ is proper,
then the maximal such $U$ is Zariski-open.

In classical algebraic geometry, results like this are easily proved by
spreading out from generic points in $Y$. In non-archimedean geometry, at
least from the point of view of topology, there are far too many generic
points: all rank one points of $Y$, and in particular all classical rigid
points, are generic in the sense of locally spectral spaces. Moreover, at most
of these points, spreading out cannot work naively, due to the subtle mixture
of completions and integral closures which arise when computing the stalks and
residue fields of $\mathcal{O}_{Y}$ and $\mathcal{O}_{Y}^{+}$. Our main new
observation in the proof of Theorem 1.1 is that there is nevertheless a
reasonable rigid analytic analog of generic points from algebraic geometry,
given as follows.

######  Definition 1.2 (Weakly Shilov points, §2.1).

Fix a nonarchimedean field $K$ with a pseudouniformizer $t\in K^{\circ}$ and
residue field $k$. A rank one point $x$ in a rigid space $X/K$ is _weakly
Shilov_ if any one of the following equivalent conditions is satisfied:

  1. (1)

There is an open affinoid subset $\mathop{\mathrm{Spa}}(A,A^{\circ})\subset X$
such that $x$ lies in the Shilov boundary of
$\mathop{\mathrm{Spa}}(A,A^{\circ})$.

  2. (2)

There is an open affinoid subset $\mathop{\mathrm{Spa}}(A,A^{\circ})\subset X$
containing $x$ such that the map $A^{\circ}\to K_{x}^{+}$ identifies111As $x$
is a rank $1$ point, the ring $K_{x}^{+}$ is a rank $1$ valuation ring, and
identifies with the subring $\mathcal{O}_{K_{x}}\subset K_{x}$ of power
bounded elements of the valued field $K_{x}$. $K_{x}^{+}$ with a $t$-completed
localization of $A^{\circ}$.

  3. (3)

The transcendence degree of the secondary residue field
$K_{x}^{+}/\mathfrak{m}$ over $k$ equals the local dimension of $X$ at $x$.

  4. (4)

(Applicable only if $X$ is quasiseparated and quasi-paracompact.) There exists
a formal model $\mathfrak{X}$ of $X$ such that the specialization map
$\mathop{\mathrm{sp}}:|X|\to|\mathfrak{X}_{k}|$ carries $x$ to the generic
point of an irreducible component of $\mathfrak{X}_{k}$.

######  Example 1.3.

If $X=\mathop{\mathrm{Spa}}K\\!\left\langle T\right\rangle$ is the closed unit
disc, the weakly Shilov points are exactly the points of Type 2 in the usual
nomenclature, i.e. the points defined by the Gauss norms on closed subdisks.

Weakly Shilov points are closely related to divisorial valuations as
considered in birational geometry. For our purposes, the utility of these
points arises by combining the characterizations (1) and (2) above: the former
implies such points are dense in $X$, while the latter (roughly) makes these
points amenable to the same commutative algebra arguments as generic points in
algebraic geometry. The proof of Theorem 1.1 proceeds by making this idea
precise: indeed, our arguments show that the subsets $U$ in Theorem 1.1 can be
chosen to contain all the weakly Shilov points of $Y$.

######  Remark 1.4 (Obtaining Theorem 1.1 from Ducros’ work).

Theorem 1.1 can also be deduced from Ducros’ [Duc18] if one switches to
Berkovich spaces; we indicate the argument for Theorem 1.1 (2) in the proper
case (which is the most essential one for this paper) in Remark 2.30. Note
that the idea of using Abhyankar points in [Duc18] seems functionally
equivalent to our idea of weakly Shilov points. On the other hand, our
differential approach to a key step (see Theorem 2.18) differs from the
function theoretic approach of the corresponding [Duc18, Theorem 6.3.7]. We
were unaware of [Duc18] when working on Theorem 1.1, and thank Brian Conrad
for bringing [Duc18] to our attention.

Let us now turn to the application of this result to étale cohomology of rigid
spaces. Recall that for any rigid space $X/K$, work of Huber [Hub96] and
Berkovich [Ber93] shows that the derived category $D(X,\mathbf{Z}/n)$ of étale
$\mathbf{Z}/n$-sheaves admits a reasonable 6-functor formalism (at least for
$n$ invertible on $K$). However, unlike in the case of schemes, it is much
more subtle to isolate a reasonable subcategory of “constructible” complexes
which are stable under the 6 operations.222For instance, constructible sheaves
in the sense of Huber’s work [Hub96], while having many wonderful categorical
properties, do not capture the same geometric intuition as the corresponding
notion in algebraic or complex geometry. Indeed, even the skyscraper sheaf at
a classical point, perhaps the simplest example of a proper pushforward, is
not Huber-constructible. Relatedly, analytifications of algebraically
constructible sheaves on algebraic varieties are almost never Huber-
constructible. In [Han20], the second author proposed that the following
notion should yield the desired theory.

######  Definition 1.5 (Zariski-constructible sheaves, Definition 3.1).

Fix a rigid space $X/K$ and $n>0$. An étale sheaf $\mathscr{F}$ of
$\mathbf{Z}/n$-modules is called _Zariski-constructible_ if $X$ admits a
locally finite stratification $X=\coprod_{i\in I}X_{i}$ into Zariski locally
closed subsets $X_{i}$ such that $\mathscr{F}|X_{i}$ is locally constant with
finite stalks for all $i$. Write $D^{(b)}_{zc}(X,\mathbf{Z}/n)\subset
D(X_{\mathop{\mathrm{\acute{e}t}}},\mathbf{Z}/n)$ for the full subcategory of
the derived category of $\mathbf{Z}/n$-module sheaves on
$X_{\mathop{\mathrm{\acute{e}t}}}$ spanned by complexes that have Zariski-
constructible cohomology sheaves and are locally bounded on $X$.

The paper [Han20] only showed the stability of $D_{zc}(X,\mathbf{Z}/n)$ by the
6 operations in some very limited situations. The techniques in the present
paper yield this stability in satisfactory generality over characteristic zero
fields. The resulting formalism, which can be regarded as the rigid analytic
analog of the classical theory of analytically constructible sheaves on
complex analytic spaces (see, e.g., [Ver76, §2]), is summarized as follows:

######  Theorem 1.6 (The $6$ functor formalism for Zariski-constructible
sheaves).

Let $K$ be a characteristic zero nonarchimedean field of residue
characteristic $p\geq 0$. For an integer $n\geq 1$, the assignment $X\mapsto
D^{(b)}_{zc}(X,\mathbf{Z}/n)$ enjoys the following properties:

  1. (1)

Pullback: For any map $f:X\to Y$, the pullback $f^{*}$ preserves
$D^{(b)}_{zc}$ (Proposition 3.4).

  2. (2)

Proper pushforward: For a proper map $f:X\to Y$, the pushforward $Rf_{*}$
preserves $D^{(b)}_{zc}$ (Theorem 3.10).

  3. (3)

More pushforwards: For a Zariski-compactifiable map $f:X\to Y$, the
pushforwards $Rf_{*}$ and $Rf_{!}$ carry lisse objects in $D^{(b)}_{zc}$ into
$D^{(b)}_{zc}$ (Corollary 3.11).

  4. (4)

$!$-pullback: Given a map $f:X\to Y$, if either $f$ is finite or $(n,p)=1$,
the pullback $Rf^{!}$ preserves $D^{(b)}_{zc}$ (Corollary 3.12).

  5. (5)

Verdier duality: There is a natural dualizing complex $\omega_{X}\in
D^{(b)}_{zc}(X,\mathbf{Z}/n)$ such that the functor
$\mathbf{D}_{X}(-):=\mathop{R\mathscr{H}\mathrm{om}}(-,\omega_{X})$ induces an
anti-equivalence on $D^{(b)}_{zc}$ satisfying biduality (Theorem 3.21).

  6. (6)

$\otimes$ and $\mathop{R\mathscr{H}\mathrm{om}}$: Given $\mathscr{F}\in
D^{(b)}_{zc}(X,\mathbf{Z}/n)$ with locally finite Tor dimension (e.g., if $n$
is a prime), the functors $\mathscr{F}\otimes^{L}_{\mathbf{Z}/n}(-)$ and
$\mathop{R\mathscr{H}\mathrm{om}}(\mathscr{F},-)$ preserves $D^{(b)}_{zc}$
(Corollary 3.14).

Moreover, proper base change holds (Theorem 3.15), and all of these operations
are compatible with extensions of the nonarchimedean base field and with
analytification of algebraic varieties (Proposition 3.16, Theorem 3.21,
Proposition 3.24). Finally, all of these results admit extensions to
$\mathbf{Z}_{\ell}$-coefficients (Theorem 3.36).

Let us make a couple of remarks. First, we do not assume any conditions on $p$
relative to the coefficient ring (except in the case of $Rf^{!}$, but see
Remark 3.23), so this result generalizes some previously known finiteness
theorems in $p$-adic Hodge theory (and uses them as input). Secondly, due to
the poor behaviour of Zariski closures in rigid geometry, it is unreasonable
to expect arbitrary Zariski-constructible sheaves to be stable under
pushforward (Warning 3.2 (1)), so one cannot do much better than (3) above
(although see Proposition 3.26); similar issues also occur in complex
geometry.

Next, we briefly comment on the proofs. Preservation under $f^{\ast}$ and
$\otimes$ is straightforward and is stated for completeness. The first key new
result is the preservation of Zariski-constructibility under $Rf_{\ast}$ for
proper $f$. This was raised as a conjecture in [Han20, Conjecture 1.14]. Here
we reduce it to the known statement that $Rf_{\ast}$ preserves locally
constant constructible complexes when $f$ is both smooth and proper. This
reduction relies on Temkin’s embedded resolution of singularities for quasi-
excellent $\mathbf{Q}$-schemes, results of the second author [Han20, Theorem
1.6] on extending branched covers across Zariski-open immersions (building on
previous work of Bartenwerfer [Bar76] and Lütkebohmert [Lüt93]), and (most
crucially) Theorem 1.1.(2).

For the remaining stabilities in Theorem 1.6, we largely reduce them to
analogous results for schemes (e.g., by replacing an affinoid
$\mathrm{Spa}(A)$ with the scheme $\mathrm{Spec}(A)$). The classical results
from SGA4 treat only finite type objects, and are not sufficient for our
purposes. However, Gabber’s work on étale cohomology of excellent schemes
[ILO14] is presented in exactly the right amount of generality, provided we
are allowed to localize our questions to affinoids. The latter is possible
thanks to our second key new result (which, in particular, settles [Han20,
Conjecture 1.12]).

######  Theorem 1.7 (Locality of Zariski-constructibility, Theorem 3.5).

For abelian sheaves on characteristic zero rigid spaces, the property of being
Zariski-constructible is an étale-local property.

Using this toolkit, one can imitate many standard constructions with
constructible sheaves found in complex or algebraic geometry. As an example,
we show that characteristic zero rigid spaces support a theory of perverse
sheaves which has the same pleasant formal properties as its algebraic
counterpart [BBD82] (except that we need to restrict to qcqs spaces when
working with $\mathbf{Q}_{\ell}$-coefficients).

######  Theorem 1.8 (Perverse sheaves in rigid geometry, Theorem 4.2 and
Theorem 4.11).

Let $K$ be a characteristic zero nonarchimedean field with residue
characteristic $p\geq 0$ and let $X/K$ be a rigid space. Fix a prime $\ell$
and a coefficient ring
$\Lambda\in\\{\mathbf{Z}/\ell^{n}\mathbf{Z},\mathbf{Q}_{\ell}\\}$. If
$\Lambda=\mathbf{Q}_{\ell}$, then assume that $X$ is qcqs and define
$D^{(b)}_{zc}(X,\mathbf{Q}_{\ell}):=D^{(b)}_{zc}(X,\mathbf{Z}_{\ell})\otimes_{\mathbf{Z}_{\ell}}\mathbf{Q}_{\ell}$.

There is a naturally defined perverse $t$-structure on
$D^{(b)}_{zc}(X,\Lambda)$, with abelian heart denoted
$\mathop{\mathrm{Perv}}(X,\Lambda)$. This construction has the following
stability properties:

  1. (1)

Duality: $\mathrm{Perv}(X,\Lambda)$ is Verdier self-dual inside
$D^{(b)}_{zc}(X,\Lambda)$.

  2. (2)

Finite pushforward: $\mathrm{Perv}(-,\Lambda)$ is stable under $f_{*}$ for
$f:Y\to X$ a finite map.

  3. (3)

Intermediate extensions: There is a notion of intermediate extension of lisse
sheaves defined on Zariski-locally closed subsets of $X$.

  4. (4)

Finiteness: If $X$ is quasi-compact, then $\mathop{\mathrm{Perv}}(X,\Lambda)$
is noetherian and artinian.

  5. (5)

Nearby cycles: The nearby cycles functor associated with a formal model of $X$
is perverse $t$-exact when $p\neq\ell$.

Moreover, this construction is compatible with the usual constructions in
algebraic geometry under analytification (in the proper case for
$\Lambda=\mathbf{Q}_{\ell}$), and is compatible with extensions of the ground
field.

As an application, we can define an intersection cohomology complex on any
qcqs characteristic $0$ rigid space, and the resulting intersection cohomology
groups have reasonable properties:

######  Corollary 1.9 (Intersection cohomology of rigid spaces, Theorem 4.13).

Let $K$ be a characteristic zero nonarchimedean field of residue
characteristic $p\geq 0$; let $C/K$ be a completed algebraic closure and let
$\ell$ be any prime. Let $X/K$ be a qcqs rigid space $X/K$.

  1. (1)

Existence of intersection cohomology: There are naturally defined $\ell$-adic
intersection cohomology groups $IH^{n}(X_{C},\mathbf{Q}_{\ell})$. These are
finitely generated $\mathbf{Q}_{\ell}$-modules if $\ell\neq p$ or if $X$ is
proper.

  2. (2)

GAGA: If $X=\mathcal{X}^{an}$ for a proper $K$-scheme $\mathcal{X}$, then
$IH^{*}(X_{C},\mathbf{Q}_{\ell})\simeq
IH^{*}(\mathcal{X}_{C},\mathbf{Q}_{\ell})$.

  3. (3)

Poincaré duality: If $X$ is proper and equidimensional of dimension $d$ and
$\ell\neq p$, there is a natural Poincaré duality isomorphism

$IH^{n}(X_{C},\mathbf{Q}_{\ell})^{\ast}\cong
IH^{-n}(X_{C},\mathbf{Q}_{\ell})(d).$

We end this paper by formulating some conjectures in §4.5, roughly predicting
that deep known results on the intersection cohomology of algebraic varieties
over $K$ carry forth to the rigid context.

### Conventions

We follow the convention that the term “nonarchimedean field” is reserved for
valued fields carrying a rank $1$ valuation.

If $K$ is a nonarchimedean field, we use the terms $K$-affinoid algebra and
topologically of finite type (henceforth abbreviated tft) $K$-algebra
synonymously; recall that these are exactly the Banach $K$-algebras that can
receive a continuous surjection from a Tate algebra $K\langle
x_{1},...,x_{n}\rangle$. Likewise, we say “rigid space over $K$” and “adic
space locally of tft over $\mathop{\mathrm{Spa}}K$” interchangeably. If $A$ is
a tft $K$-algebra, we write
$\tilde{A}:=A^{\circ}/A^{\circ\circ}=(A^{\circ}/t)^{red}$; this is a
$k$-algebra of finite type [BGR84, Corollary 3, §6.3.4], where $k$ is the
residue field of $K$ and $t$ is a pseudouniformizer (i.e., any nonzero element
of $K^{\circ\circ}-\\{0\\}$).

We warn the reader that $A^{\circ}$ need not be tfp over $\mathcal{O}_{K}$
even when $A$ is reduced; this pathology does not occur if either $K$ is
discrete, or if $K$ is stable field with $|K^{*}|$ divisible (if $K$ is
algebraically closed); see [BGR84, §3.6] for the definition of stability, and
[BGR84, §6.4] for the finiteness properties.

We write $\mathop{\mathrm{Spa}}A=\mathop{\mathrm{Spa}}(A,A^{\circ})$ for any
Huber ring $A$.

Say $X$ is an analytic adic space and $x\in X$. We write $K_{x}$ for the
completed residue field of $X$ at $x$. This is a nonarchimedean field. We
shall write $|\cdot|_{x}$ for the associated valuation on functions, though
note that is only well-defined up to equivalence. However, if $A$ is a tft
$K$-algebra and $x\in\mathop{\mathrm{Spa}}A$ is a rank one point, there is a
unique $\mathbf{R}_{\geq 0}$-valued representative of the associated
equivalence class which extends the fixed norm on the base field $K$, i.e. a
unique representative such that $|t|_{x}=|t|_{K}$. We always choose this
representative.

Given $X$ and $x$ as above, the secondary residue field of $X$ at a point $x$
is the quotient $\tilde{K}_{x}=K_{x}^{+}/\mathfrak{m}_{x}$. Here $K_{x}^{+}$
is the valuation subring of the residue field $K_{x}$ defined as the completed
image of the map $\mathcal{O}_{X,x}^{+}\to K_{x}$; when $x$ is a rank $1$
point, the ring $K_{x}^{+}$ coincides with the subring
$\mathcal{O}_{K_{x}}\subset K_{x}$ of power bounded elements. Recall that if
$y\prec x$ is any specialization, then $K_{x}\cong K_{y}$ and
$K_{y}^{+}\subset K_{x}^{+}$ under this identification.

For any tft $K$-algebra $A$, we shall write
$\mathop{\mathrm{sp}}:\mathop{\mathrm{Spa}}A\to\mathop{\mathrm{Spec}}\tilde{A}$
for the specialization map, given by taking the center of the valuation. This
is a continuous, closed, and spectral map of spectral spaces.

If $f:X\to Y$ is a map of rigid spaces over $K$, we say $f$ is _Zariski-
compactifiable_ if it admits a factorization $f=\overline{f}\circ j$, where
$j:X\to X^{\prime}$ is a Zariski-open immmersion and
$\overline{f}:X^{\prime}\to Y$ is a proper morphism. We say a rigid space $X$
over $K$ is Zariski-compactifiable if the structure map
$f:X\to\mathop{\mathrm{Spa}}K$ is so.

### Acknowledgements

In April 2020, Bogdan Zavyalov asked DH whether [Han20, Conjecture 1.14] was
within reach, and the present paper grew directly out of that conversation. We
thank Bogdan heartily for this crucial initial stimulus, and for some helpful
comments on previous drafts of this paper. We are also grateful to Brian
Conrad, Johan de Jong, Haoyang Guo, Mattias Jonsson, Shizhang Li, and Jacob
Lurie for useful conversations and exchanges, and to the anonymous referees
for a number of comments that helped improve the exposition. Bhatt was
partially supported by the NSF (#1801689, #1952399, #1840234), a Packard
fellowship, and the Simons Foundation (#622511).

##  2\. Generic flatness and generic smoothness

In this section, we introduce the notion of weakly Shilov points, and prove
our main geometric results on generic smoothness.

###  2.1. Shilov and weakly Shilov points

Fix a complete nonarchimedean field $K$ and a pseudouniformizer $t\in
K^{\circ}$. Recall the following basic example of a “non-classical” point of a
standard $K$-affinoid.

######  Example 2.1.

If $X=\mathop{\mathrm{Spa}}K\\!\left\langle T\right\rangle$ is the one-
dimensional affinoid ball over $K$, then the Gauss point $X$ is given by the
$t$-adic norm on $K\langle T\rangle$. To describe this norm ring
theoretically, recall that the standard formal for $X$ is given by the formal
closed disc $\mathfrak{X}:=\mathrm{Spf}(\mathcal{O}_{K}\langle T\rangle)$. The
special fibre $\mathfrak{X}_{s}=\mathrm{Spec}(\mathcal{O}_{K}/t[T])$ has a
unique a generic point $\overline{\eta}$; the $t$-completed localization of
$\mathcal{O}_{K}\langle T\rangle$ at $\overline{\eta}$ is a rank one
$t$-complete and $t$-torsionfree valuation ring $V$ equipped with a map
$\mathcal{O}_{K}\langle T\rangle\to V$. The resulting valuation on $K\langle
T\rangle$ is the Gauss point $\eta$. Moreover, the canonical map
$X\to\mathfrak{X}$ given by taking the center of the valuation carries $\eta$
to $\overline{\eta}$.

We now isolate a general class of points with properties similar to the Gauss
point from Example 2.1.

######  Proposition 2.2.

Fix a tft $K$-algebra $A$ and a rank one point
$x\in\mathrm{Spa}(A,A^{\circ})$. The following are equivalent:

  1. (1)

$\mathop{\mathrm{sp}}(x)$ is the generic point of an irreducible component of
$\mathop{\mathrm{Spec}}\tilde{A}$.

  2. (2)

$\\{x\\}=\mathop{\mathrm{sp}}^{-1}(\mathop{\mathrm{sp}}(x))$ as subsets of
$\mathop{\mathrm{Spa}}A$.

  3. (3)

$K_{x}^{+}$ is a $t$-completed ind-Zariski localization of $A^{\circ}$. More
precisely, $K_{x}^{+}$ is the $t$-completed local ring of
$\mathop{\mathrm{Spec}}A^{\circ}$ at the point
$\mathop{\mathrm{sp}}(x)\in\mathop{\mathrm{Spec}}\tilde{A}\subset\mathop{\mathrm{Spec}}A^{\circ}$.

  4. (4)

The seminorm $|\cdot|_{x}$ belongs to the Shilov boundary of $A$ in the sense
of $K$-Banach algebras.

###### Proof.

We begin with two (well-known) observations on the affinoid adic space
$\mathrm{Spa}(A,A^{\circ})$; the first concerns finding suitable rings of
definition, while the second concerns a description via formal models.

First, we claim that there exists an open and bounded tfp
$\mathcal{O}_{K}$-subalgebra $A_{0}\subset A^{0}$ such that
$\mathrm{Spec}(\tilde{A})\to\mathrm{Spec}(A_{0}/tA_{0})$ is a universal
homeomorphism and such that $A^{\circ}$ is the integral closure of $A_{0}$ in
$A$. In fact, the second part follows from [BGR84, Remark following 6.3.4/1].
For the first, using Noether normalization, we can choose a surjective map
$T\to A$, where $T=K\langle x_{1},...,x_{n}\rangle$ is a Tate algebra. By
[BGR84, 6.3.4/2], the map $\tilde{T}\to\tilde{A}$ is module finite. Setting
$A^{\prime}_{0}=\mathrm{im}(T^{\circ}\to A^{\circ})$, we learn that
$A_{0}^{\prime}/(A^{\circ\circ}\cap A_{0}^{\prime})\to\tilde{A}$ is module
finite and injective, with $A_{0}^{\prime}$ being open, bounded, and tfp over
$\mathcal{O}_{K}$. Enlarging $A_{0}^{\prime}$ inside $A^{\circ}$ by adding
finitely topological generators of $\tilde{A}$ over $A_{0}^{\prime}$, we find
an open bounded tfp $\mathcal{O}_{K}$-subalgebra $A_{0}\subset A^{\circ}$ such
that $A_{0}/(A^{\circ\circ}\cap A_{0})\to\tilde{A}$ is bijective. But we also
know that $\sqrt{tA_{0}}=A^{\circ\circ}\cap A_{0}$: any element of the right
side is topologically nilpotent and in $A_{0}$, and must thus have a large
enough power inside $tA_{0}$. Thus, we have found the subalgebra $A_{0}$
indicated at the start of this paragraph.

Next, we also recall an alternative description of the locally ringed space
$(\mathrm{Spa}(A,A^{\circ}),\mathcal{O}^{+})$. Consider the category of all
proper maps $f_{i}:X_{i}\to\mathrm{Spec}(A_{0})$ of schemes which are
isomorphisms after inverting $t$. For each $f_{i}$, let $X_{i,t=0}\subset
X_{i}$ be the special fibre (regarded merely as a closed subset). Set
$Z=\lim_{i}X_{i,t=0}$, so $Z$ is a spectral space. Let $\pi_{i}:Z\to X_{i}$ be
the structure map, and define the structure sheaf $\mathcal{O}_{Z}$ of $Z$ via
$\mathcal{O}_{Z}:=\mathop{\mathrm{colim}}_{i}\pi_{i}^{-1}\mathcal{O}_{X_{i}}$.
Then it is a basic fact that $\mathrm{Spa}(A,A^{\circ})=Z$ as topological
spaces, and $\mathcal{O}^{+}$ identifies with the $t$-adic completion of
$\mathcal{O}_{Z}$. For future use, we remark that, by passing to a cofinal
subsystem, we may (and do) assume that each $X_{i}$ is $\mathcal{O}_{K}$-flat,
i.e., $\mathcal{O}_{X_{i}}$ is $t$-torsionfree. This condition implies that
the generic fibre $X_{i}[1/t]$ is dense in each $X_{i}$, and thus all the
transition maps $X_{i}\to X_{j}$ in the system are surjective: their image is
a closed set containing a dense open. In particular, $\mathcal{O}_{Z}$ is
$t$-torsionfree. By spectrality, the maps $\pi_{i}:Z\to X_{i,t=0}$ are also
surjective for all $i$. Finally, we also remark that $\mathcal{O}_{Z}$ is
integrally closed in $\mathcal{O}_{Z}[1/t]$ by generalities on blowups.

Using the preceding two observations, we prove the equivalences.

$(1)\Rightarrow(2)$: Fix $x\in\mathrm{Spa}(A)$ with
$\mathop{\mathrm{sp}}(x)\in\mathop{\mathrm{Spec}}\tilde{A}$ being a generic
point. Using a suitable Noether normalization, we then learn that
$\tilde{A}_{\mathop{\mathrm{sp}}(x)}$ is a rank $1$ valuation ring with
pseudouniformizer $t$: this ring is the integral closure of a rank $1$
$t$-complete valuation ring in a finite extension of its fraction field. Any
point $y\in\mathop{\mathrm{sp}}^{-1}(\mathop{\mathrm{sp}}(x))$ is represented
by an equivalence class of maps $A_{0}\to V$ where $V$ is a $t$-complete
$t$-torsionfree valuation ring with the property that the closed point of
$\mathrm{Spec}(V)$ is carried to $\mathop{\mathrm{sp}}(x)$. But any such map
factors uniquely as $A_{0}\to\tilde{A}_{\mathop{\mathrm{sp}}(x)}\to V$. Thus,
taking $V$ to be the $t$-completion of $\tilde{A}_{\mathop{\mathrm{sp}}(x)}$
gives the unique such map up to equivalence, showing that $y=x$.

$(2)\Rightarrow(3)$: Write $x_{i}\in X_{i}$ for the image of $x$, so
$\mathcal{O}^{+}_{\mathrm{Spa}(A,A^{+}),x}$ and
$\mathop{\mathrm{colim}}\mathcal{O}_{X_{i},x_{i}}$ identify after
$t$-completion. Now if $x=\mathop{\mathrm{sp}}^{-1}(\mathop{\mathrm{sp}}(x))$,
then $x_{i}=f_{i}^{-1}(\mathop{\mathrm{sp}}(x))$ is the unique preimage of
$\mathop{\mathrm{sp}}(x)$. The map
$A_{0,\mathop{\mathrm{sp}}(x)}\to\mathcal{O}_{X_{i},x_{i}}$ is then integral
(by properness of $X_{i}\to\mathrm{Spec}(A_{0})$, base changed to
$\mathrm{Spec}(A_{0,\mathop{\mathrm{sp}}(x)}$) and an isomorphism after
inverting $t$. Taking a colimit, we learn that
$A_{0,\mathop{\mathrm{sp}}(x)}\to\mathop{\mathrm{colim}}_{i}\mathcal{O}_{X_{i},x_{i}}$
is integral and an isomorphism after inverting $t$. But the target is also
integrally closed in its $t$-localization, so it must coincide with
$A^{\circ}_{\mathop{\mathrm{sp}}(x)}$; here we implicitly use that
$\mathrm{Spec}(\tilde{A})\simeq\mathrm{Spec}(A_{0}/t)$ to identify
$\mathop{\mathrm{sp}}(x)$ with a point of $\mathrm{Spec}(A^{\circ})$, as well
as the fact that $A^{\circ}$ is the integral closure of $A_{0}$. Thus, we have
shown that the $t$-completion of
$A^{\circ}_{\mathop{\mathrm{sp}}(x)}\to\mathcal{O}^{+}_{\mathrm{Spa}(A,A^{+}),x}$
is an isomorphism. As the $t$-completion of the target is $K_{x}^{+}$, the
claim follows.

$(3)\Rightarrow(2)$: This is clear from the description of points of
$\mathrm{Spa}(A,A^{\circ})$ as equivalence classes of maps $A^{\circ}\to V$ to
$t$-complete and $t$-torsionfree valuation rings.

$(2)\Rightarrow(1)$: Assume that $x$ is the unique preimage of
$\mathop{\mathrm{sp}}(x)$ and yet that
$\mathop{\mathrm{sp}}(x)\in\mathrm{Spec}(\tilde{A})$ is not a generic point;
we shall obtain a contradiction. Let $\overline{y}\in\mathrm{Spec}(\tilde{A})$
be a generic point of an irreducible component
$Y\subset\mathrm{Spec}(\tilde{A})$ containing $\mathop{\mathrm{sp}}(x)$.
Applying the implications $(1)\Rightarrow(3)$ to a lift
$y\in\mathrm{Spa}(A,A^{+})$ of $\overline{y}$, we learn that $\overline{y}$
has a unique lift $y\in\mathrm{Spa}(A,A^{+})$, and that $K_{y}^{+}$ is the
$t$-completed local ring of $A^{\circ}_{y}$. In particular, the residue field
of $K_{y}^{+}$ identifies with function field $K(Y)$ of the irreducible
component $Y$. Choose a valuation subring $\overline{V}\subset K(Y)$ that is a
$\tilde{A}$-algebra and has center
$\mathop{\mathrm{sp}}(x)\in\mathrm{Spec}(\tilde{A})$; this valuation has rank
$\geq 1$ as $\overline{y}\neq\mathop{\mathrm{sp}}(x)$. Let $V\subset
K_{y}^{+}$ be the preimage of $\overline{V}$. Then $V$ is a $t$-complete
valuation ring of rank $\geq 2$, is an $A^{\circ}$-algebra, and has center
$\mathop{\mathrm{sp}}(x)$ on $\mathrm{Spec}(\tilde{A})$. The resulting map
$A^{\circ}\to V$ then gives a point $x^{\prime}\in\mathrm{Spa}(A,A^{\circ})$
with rank $\geq 2$ and image $\mathop{\mathrm{sp}}(x)$ in
$\mathrm{Spec}(\tilde{A})$. But $x$ is the unique preimage of
$\mathop{\mathrm{sp}}(x)$, so $x=x^{\prime}$. We now obtain a contradiction as
$x$ had rank $1$ by assumption, while $x^{\prime}$ has rank $\geq 2$ by
construction.

$(1)\Longleftrightarrow(4)$: This is proven in [Ber90, 2.4.4]. ∎

######  Remark 2.3.

One may ask if the equivalences proven in Proposition 2.2 continue to hold
true for the affinoid adic space $\mathrm{Spa}(A,A^{+})$ attached to any
complete Tate ring $(A,A^{+})$. Inspection of the proof shows that the
equivalence of $(2)$ and $(3)$ and the implication $(2)\Rightarrow(1)$ hold
true in general. On the other hand, there is a perfectoid affinoid algebra
where $(1)\Rightarrow(2)$ and $(1)\Rightarrow(3)$ in Proposition 2.2 fail, as
we explain next. We are not aware of a broader class of algebras (than the
$K$-affinoid ones) where the Proposition 2.2 holds true.

Take a complete and algebraically closed extension $C/\mathbf{Q}_{p}$ whose
algebraically closed residue field $k$ has transcendence degree $\geq 1$ over
$\mathbf{F}_{p}$. Write $V=\mathcal{O}_{C}$, and let $W\subset V$ be the
preimage of $\overline{\mathbf{F}_{p}}\subset k$. Then $W$ is a $p$-complete
and $p$-torsionfree local ring that is integrally closed in $W[1/p]=V[1/p]$.
Moreover, we have $\sqrt{pW}=\sqrt{pV}$ with $W/\sqrt{pW}\to V/\sqrt{pV}$
identifying with the map $\overline{\mathbf{F}_{p}}\subset k$. A lift $x\in
V\subset W[1/p]$ of any element of $k-\overline{\mathbf{F}_{p}}$ has the
property that $x,x^{-1}\notin W$, so $W$ is not a valuation ring. Endowing $W$
with the $p$-adic topology, we obtain a complete uniform Tate ring
$(W[1/p],W)$ with $W$ perfectoid. The special fibre
$\widetilde{W[1/p]}=W/\sqrt{pW}$ identifies with $\overline{\mathbf{F}_{p}}$,
so $\mathop{\mathrm{Spec}}(\widetilde{W[1/p]})$ has a unique point which is
thus a generic point. On the other hand, the local ring of $\mathrm{Spec}(W)$
at this point is simply $W$, which is not a valuation ring. This shows that
the implication $(1)\Rightarrow(3)$ in Proposition 2.2 fails in this example.
To show that $(1)\Rightarrow(2)$ also fails in this example, one calculates
that $\mathrm{Spa}(W[1/p],W)$ identifies with the Riemann-Zariski space of
$k$, which has more than $1$ element; we omit the argument.

######  Definition 2.4.

Let $A$ be a tft $K$-algebra. A _Shilov point_ $x\in\mathop{\mathrm{Spa}}A$ is
any point satisfying the equivalent conditions of Proposition 2.2.

Recall that in an analytic adic space, every rank one point is generic in the
sense of spectral spaces. For algebraic purposes, the following class of
points is more relevant.

######  Definition 2.5.

Let $X$ be any rigid space over a non-archimedean field $K$. A point $x\in X$
is _weakly Shilov_ if there is an open affinoid subset $x\in W\subset X$ such
that $x$ is a Shilov point of $W$. In particular, such a point has rank one.

######  Example 2.6.

For $X=\mathrm{Spa}(K\langle T\rangle)$ the closed unit disc, the weakly
Shilov points are exactly the points of type $2$ (see [Sch12, Example 2.20]
for the classification of points on $X$). This follows from the
characterization in Proposition 2.9, since the secondary residue field of $x$
is algebraic over $K$ when $x$ has type $1$, $3$ or $4$ (and $x$’s of type 5
have rank $2$, so they are not Shilov).

######  Remark 2.7.

The valuations isolated in Definition 2.5 are sometimes also called divisorial
valuations in birational geometry (see Proposition 2.9 (3)).

######  Lemma 2.8.

Let $X$ be a rigid space over a nonarchimedean field $K$. Then weakly Shilov
points are dense in $X$.

###### Proof.

By definition, any open affinoid $U\subset X$ contains a weakly Shilov point.
∎

We also have the following alternative characterization of weakly Shilov
points.

######  Proposition 2.9.

Let $X$ be a rigid space over a nonarchimedean field $K$ and let $x\in X$ be a
rank one point. The following are equivalent:

  1. (1)

$x$ is weakly Shilov.

  2. (2)

The transcendence degree of the secondary residue field $\tilde{K}_{x}$ over
$K^{\circ}/\mathfrak{m}$ equals the local dimension $\dim_{x}X$ of $X$ at $x$.

  3. (3)

(Applicable only when $X$ is quasiseparated and quasi-paracompact) There
exists a formal model $\mathfrak{X}$ such that $x$ maps to the generic point
of an irreducible component of $\mathfrak{X}_{s}$.

###### Proof.

$(1)\Longleftrightarrow(2)$ follows by combining Lemme 4.4 and Corollaire 4.15
in [Poi13].

$(3)\Rightarrow(1)$: if there exists a formal model $\mathfrak{X}$ as in (3),
then taking $W\subset X$ be the preimage of any formal affine open
$\mathfrak{X}$ containing the image of $x$ shows that $x$ is weakly Shilov.

$(1)\Rightarrow(3)$: Assume $x\in X$ is weakly Shilov. We can then find an
affinoid open $\mathrm{Spa}(R)\subset X$ containing $x$ such that
$x\in\mathrm{Spa}(R)$ is Shilov. By Raynaud, there is a formal model
$\mathfrak{X}$ of $X$ such that $\mathrm{Spa}(R)\subset X$ is the preimage of
a formal affine open $\mathrm{Spf}(R_{0})\subset\mathfrak{X}$, cf. [Bos14,
Theorem 8.4.3]. By passing to a refinement, we can assume that the map
$R_{0}/t\to\tilde{R}$ gives a homeomorphism of spectra; see first paragraph of
the proof of Proposition 2.2. As $x$ is weakly Shilov, its image in
$\mathrm{Spec}(\tilde{R})$ is a generic point. But then its image in
$\mathrm{Spec}(R_{0}/t)$ is also a generic point since
$\mathrm{Spec}(\tilde{R})\to\mathrm{Spec}(R_{0}/t)$ is a homeomorphism by
construction. As $\mathrm{Spf}(R_{0})\subset\mathfrak{X}$ is a formal open
immersion, it follows that $x$ gives a generic point of $\mathfrak{X}_{s}$ as
well ∎

######  Corollary 2.10.

Say $X$ is a rigid space over a nonarchimedean field $K$, and $Z\subset X$ is
a nowhere dense Zariski closed set. Then $Z$ does not contain any weakly
Shilov point of $X$. In particular, if $X$ is reduced, then any weakly Shilov
point of $X$ lies in the regular locus of $X$.

###### Proof.

As $Z\subset X$ is a nowhere dense Zariski closed set, we have
$\dim_{x}(Z)<\dim_{x}(X)$ for all $x\in X$. The first statement now follows
from the characterization of weak Shilov points in Proposition 2.9 (2). The
second statement follows from the first statement applied to the locus
$Z\subset X$ of points that are not regular, which is a nowhere dense Zariski
closed set by excellence considerations. ∎

###  2.2. Tools involving the cotangent complex

To prove our generic smoothness result, it will be convenient to use the
analytic cotangent complex as this provides a homologically well-behaved
object detecting smoothness in non-noetherian situations (such as
topologically finitely presented algebras over non-discrete valuation rings).
In this subsection, we recall some results on this object.

######  Notation 2.11.

Fix a complete nonarchimedean field $K$ with valuation ring $V\subset K$ and a
pseudouniformizer $t\in V$. For any map $A\to B$ of $V$-algebras, write
$L^{an}_{B/A}$ for the derived $t$-completion of $L_{B/A}$. A tft (or
topologically finite type) $V$-algebra is a $V$-algebra $A$ of the form
$V[x_{1},...,x_{n}]^{\wedge}/I$ (where the completion is $t$-adic). If
moreover $I$ is finitely generated, we say that $A$ is tfp (or topologically
finitely presented). The class of tfp $V$-algebras has good properties such as
coherence and classical $t$-adic completenesss, see [GR03, Proposition 7.1.1].
Moreover, any tft $V$-algebra which is $t$-torsion-free is in fact tfp, see
[FGK11, Corollary 7.3.6].

######  Theorem 2.12 (Gabber-Ramero).

Say $A\to B$ is a map of tfp $V$-algebras. Then the following hold true:

  1. (1)

$L^{an}_{B/A}$ is a pseudocoherent $B$-complex.

  2. (2)

Assume that $A\to B$ induces a smooth map of relative dimension $n$ on taking
adic generic fibres. Then $L^{an}_{B/A}[1/t]$ is a finite projective
$B[1/t]$-module of rank $n$.

  3. (3)

If $A\to B$ is surjective, then $L_{B/A}\simeq L^{an}_{B/A}$.

###### Proof.

(1) is [GR03, Theorem 7.1.31].

(2) is the key assertion checked in the proof of [GR03, Theorem 7.2.39].

(3) is [GR03, Theorem 7.1.29].

∎

######  Lemma 2.13.

Let $R$ be a finitely presented flat $V$-algebra.

  1. (1)

$L_{R/V}$ is a pseudocoherent $R$-complex.

  2. (2)

If $R$ is also $V$-finite, then $L_{R/V}$ is derived $t$-complete.

###### Proof.

For (1), by noetherian approximation, we can write $V\to R$ as the base change
of a finitely presented flat map $V_{0}\to R_{0}$ of finitely generated
$\mathbf{Z}$-algebras along some map $V_{0}\to V$. Then $L_{R_{0}/V_{0}}$ is
pseudocoherent, and $L_{R_{0}/V_{0}}\otimes_{R_{0}}^{L}R\simeq L_{R/V}$ by Tor
independent base change, so $L_{R/V}$ is also pseudocoherent.

For (2), observe that if $R$ is $V$-finite, then $R$ is a finite free
$V$-module (as any finitely presented flat $V$-module is finite free). In
particular, a pseudocoherent $R$-complex is also pseudocoherent as a
$V$-complex. The claim now follows as any pseudocoherent $V$-complex is
derived $t$-complete (since $V$ itself is so). ∎

######  Corollary 2.14.

Let $E/K$ be a finite extension, and let $W_{0}\subset E^{\circ}$ be an open
tfp $V$-subalgebra.

  1. (1)

$L_{W_{0}/V}$ is pseudocoherent so $L_{W_{0}/V}\simeq L^{an}_{W_{0}/V}$,
whence $L_{E/K}\simeq L^{an}_{W_{0}/V}[1/t]$.

  2. (2)

$H_{i}(L^{an}_{W_{0}/V}[1/t])=0$ for $i\neq 0,1$.

###### Proof.

For (1): since $W_{0}\subset E^{\circ}$, choosing monic equations defining
generators of $W_{0}$ shows that any such $W_{0}$ is a finitely presented flat
$V$-algebra. Lemma 2.13 then implies that $L_{W_{0}/V}$ is pseudocoherent and
thus already derived $t$-complete, so $L_{W_{0}/V}\simeq L^{an}_{W_{0}/V}$.
The last part of (1) follows by inverting $t$ and noting that formation of
cotangent complexes commutes with localization.

(2) follows from (1) and a general fact: the cotangent complex of any
extension of fields only has homology in degrees $1$ and $0$. Indeed, this
follows from transitivity triangles and the fact that any field is ind-smooth
over its prime subfield (which is perfect) by generic smoothness in algebraic
geometry. ∎

We need the following result later.

######  Theorem 2.15 (Quillen).

Fix a noetherian ring $A$ and a maximal ideal $\mathfrak{m}$ with residue
field $k=A/\mathfrak{m}$. If $L_{k/A}$ is concentrated in degree $-1$, then
$A$ is regular at $\mathfrak{m}$.

###### Proof.

[Qui, Corollary 10.5] shows that $\mathfrak{m}$ is generated by a regular
sequence if $H_{2}(L_{k/A})=0$. As the maximal ideal of a noetherian local
ring is generated by a regular sequence exactly when the ring is regular, the
claim follows. ∎

The following lemma will be useful later as well.

######  Lemma 2.16.

Say $A$ is a reduced Jacobson noetherian ring. Let $M\to N$ be a map of
finitely generated $A$-modules such that $M/\mathfrak{m}M\to N/\mathfrak{m}N$
is injective for every maximal ideal $\mathfrak{m}$. If $M$ is a locally free
$A$-module, then $M\to N$ is injective.

###### Proof.

The question is local, so we can assume $M=A^{\oplus n}$ is a free module. Let
$x=(x_{1},...,x_{n})\in M$ lie in the kernel of $M\to N$. The hypothesis
implies that $x_{i}\in\mathfrak{m}$ for every maximal ideal
$\mathfrak{m}\subset A$. But the intersection of all maximal ideals of $A$ is
its nilradical (as $A$ is Jacobson) which is $0$ (as $A$ is reduced). So
$x=0$, as wanted. ∎

###  2.3. Regularity of Shilov fibers

In this subsection, we establish the key technical ingredient behind our
generic smoothness result, using the analytic cotangent complex.

######  Notation 2.17.

Fix a complete nonarchimedean field $C$ with valuation ring
$\mathcal{O}_{C}\subset C$ and a pseudouniformizer
$t\in\mathcal{O}_{C}$.333Contrary to modern notational conventions, we are not
assuming $C$ is algebraically closed. We hope this causes no confusion.

Our goal is the following.

######  Theorem 2.18.

Let $T\to R$ be a map of tfp $\mathcal{O}_{C}$-algebras. Assume
$\mathrm{Spa}(R[1/t])$ is smooth over $C$. Let $V$ denote the $t$-completed
Zariski local ring of $T$ at a generic point
$\eta\in\mathrm{Spec}(T/t)\subset\mathrm{Spec}(T)$, and set
$R_{V}=R\widehat{\otimes}_{T}V$. Then $R_{V}[1/t]$ is regular.

###### Proof.

Our strategy is to first simplify $T$, and then argue that $R_{V}[1/t]$ is
regular using the cotangent complex.

First, observe that the statement of the theorem is Zariski local around
$\eta\in\mathrm{Spf}(T)$, so we may shrink $\mathrm{Spf}(T)$ around $\eta$ to
assume $\mathrm{Spec}(T/t)$ is irreducible with generic point $\eta$. If
$T^{\prime}=\mathcal{O}_{C}\langle x_{1},...,x_{n}\rangle\to T$ is a finite
injective map, then $\mathrm{Spec}(T/t)\to\mathrm{Spec}(T^{\prime}/t)$ is
finite and surjective. As both these schemes are irreducible, the image of the
generic point $\eta\in\mathrm{Spec}(T/t)$ is the generic point
$\eta^{\prime}\in\mathrm{Spec}(T^{\prime}/t)$, and $\eta$ is the unique
preimage of $\eta^{\prime}$. In particular, we must have
$T\widehat{\otimes}_{T^{\prime}}V^{\prime}\simeq V$ by base change. But then
$R\widehat{\otimes}_{T^{\prime}}V^{\prime}\simeq R_{V}$. Thus, we may replace
$T$ with $T^{\prime}$ to assume that $T$ is a standard Tate algebra over
$\mathcal{O}_{C}$. In particular, $T$ is formally smooth over
$\mathcal{O}_{C}$. In this case, we have $T=T[1/t]^{\circ}$, so $V$ is
actually a $t$-complete and $t$-torsionfree rank one valuation ring (either by
explicit calculation, or as explained in Proposition 2.2).

Next, we describe $L^{an}_{V/\mathcal{O}_{C}}$. By definition, the ring $V$ is
a $t$-completed ind-Zariski localization of $T$. As the formation of the
$t$-completed cotangent complex is compatible with $t$-completed localizations
and $t$-completed filtered colimits, we learn that
$L^{an}_{V/\mathcal{O}_{C}}\simeq
L^{an}_{T/\mathcal{O}_{C}}\widehat{\otimes}_{T}^{L}V$. But $T/\mathcal{O}_{C}$
is formally smooth, so $L^{an}_{T/\mathcal{O}_{C}}$ is a finite projective
$T$-module placed in degree $0$, whence $L^{an}_{V/\mathcal{O}_{C}}$ is a
finite free $V$-module placed in degree $0$.

We now begin proving the theorem. Since $V$ is a $t$-complete and
$t$-torsionfree rank $1$ valuation ring, the ring $K:=V[1/t]$ is a
nonarchimedean field extension of $C$ with $K^{\circ}=V$. As $R_{V}$ is a tfp
$V$-algebra, we can (and will) regard $R_{V}[1/t]$ as an affinoid $K$-algebra.
To show regularity of $R_{V}[1/t]$, it is enough to show that the local rings
of $R_{V}[1/t]$ at all closed points are regular: the regular locus in
$R_{V}[1/t]$ is open (as affinoid $K$-algebras are noetherian and excellent)
and the maximal ideals are dense (as affinoid $K$-algebras are Jacobson). A
closed point is given by a quotient $R_{V}[1/t]\to E$ where $E/K$ is a finite
extension; fix such a point. By Theorem 2.15, it suffices to show that
$L_{E/R_{V}[1/t]}$ has homology only in degrees $0,1$. (In fact, there is no
$H_{0}$ as $R_{V}[1/t]\to E$ is surjective, but it will be convenient to
formulate things this way.) Let $W_{0}\subset E^{\circ}$ be the image of
$R_{V}$ under this map. As $E^{\circ}$ is the integral closure of $V$ in $E$,
the ring $W_{0}$ is a finitely presented finite flat $V$-algebra. Since the
map $R_{V}\to W_{0}$ is a surjection of tfp $V$-algebras, we have
$L_{W_{0}/R_{V}}\simeq L^{an}_{W_{0}/R_{V}}$ by Theorem 2.12 (3), and hence
$L_{E/R_{V}[1/t]}\simeq L^{an}_{W_{0}/R_{V}}[1/t]$ by inverting $t$, so it is
enough to show that $L^{an}_{W_{0}/R_{V}}[1/t]$ has homology only in degrees
$0,1$. Consider the transitivity triangle for $\mathcal{O}_{C}\to R_{V}\to
W_{0}$:

$L^{an}_{R_{V}/\mathcal{O}_{C}}\widehat{\otimes}_{R_{V}}^{L}W_{0}\to
L^{an}_{W_{0}/\mathcal{O}_{C}}\to L^{an}_{W_{0}/R_{V}}.$ (1)

As $R\to R_{V}$ is a $t$-completed Zariski localization, we have

$L^{an}_{R/\mathcal{O}_{C}}{\otimes}_{R}^{L}R_{V}\simeq
L^{an}_{R/\mathcal{O}_{C}}\widehat{\otimes}_{R}^{L}R_{V}\simeq
L^{an}_{R_{V}/\mathcal{O}_{C}},$

where the first isomorphism is from the pseudocoherence of
$L^{an}_{R/\mathcal{O}_{C}}$ coming from Theorem 2.12 (1). The same reasoning
also allows us to drop the completion on the leftmost term in (1). Inverting
$t$ in (1) then gives a triangle

$L^{an}_{R_{V}/\mathcal{O}_{C}}[1/t]{\otimes}_{R_{V}[1/t]}^{L}E\to
L^{an}_{W_{0}/\mathcal{O}_{C}}[1/t]\to L^{an}_{W_{0}/R_{V}}[1/t].$ (2)

Our task was to show that the term on the right has homology only in degrees
$0,1$. The term on the left is a finite projective $E$-module in degree $0$:
indeed, $L^{an}_{R/\mathcal{O}_{C}}[1/t]$ is a finite projective
$R[1/t]$-module by Theorem 2.12 (2) and the smoothness assumption on
$\mathrm{Spa}(R[1/t])$, and we have
$L^{an}_{R/\mathcal{O}_{C}}\otimes_{R[1/t]}^{L}R_{V}[1/t]\simeq
L^{an}_{R_{V}/\mathcal{O}_{C}}[1/t]$ by the reasoning explained above. By the
long exact sequence, it thus suffices to check that
$L^{an}_{W_{0}/\mathcal{O}_{C}}$ has homology only in degrees $0,1$. For this,
we consider transitivity triangle for $\mathcal{O}_{C}\to V\to W_{0}$:

$L^{an}_{V/\mathcal{O}_{C}}\widehat{\otimes}_{V}^{L}E\to
L^{an}_{W_{0}/\mathcal{O}_{C}}[1/t]\to L^{an}_{W_{0}/V}[1/t].$

Now $L^{an}_{V/\mathcal{O}_{C}}[1/t]$ is a finite free $K$-module as explained
previously, so the term on the left is a finite free $E$-module. It remains to
observe that $L^{an}_{W_{0}/V}[1/t]$ has homology in degrees $0,1$ by
Corollary 2.14. ∎

######  Remark 2.19.

In Theorem 2.18, one cannot strengthen the conclusion from regularity to
smoothness. For example, the Frobenius map on the Tate algebra over any
characteristic $p$ nonarchimedean field satisfies the hypothesis of Theorem
2.18, but does not have a single smooth fibre.

######  Remark 2.20.

The proof of Theorem 2.18 relies on the analytic cotangent complex. When $C$
is discretely valued, it is possible to prove Theorem 2.18 by avoiding the
cotangent complex and using instead Popescu’s desingularization theorem.
Indeed, one first observes that $R$ is an excellent noetherian ring: by
Elkik’s theorem, we can write $R$ as the $t$-completion of a finite type
$\mathcal{O}_{C}$-algebra, so $R$ is excellent since $\mathcal{O}_{C}$ is so.
But then the maps $R\to R\otimes_{T}T_{\eta}\to R_{V}$ are regular: the first
map is a localization, while the second one is the completion of an excellent
ring. By Popescu’s theorem, the map $R\to R_{V}$ is ind-smooth. It follows
that $R_{V}[1/t]$ must be regular since $R[1/t]$ is so. In fact, this
reasoning shows that any property of $R[1/t]$ that is local for the smooth
topology passes to $R_{V}[1/t]$. We do not know how to prove the analogous
statement in the general case.

###  2.4. Generic flatness

Fix a nonarchimedean base field $K$. Our goal in this section is the following
theorem.

######  Theorem 2.21.

Let $f:X\to Y$ be a quasicompact map of rigid spaces over $K$. Assume that $Y$
is geometrically reduced. Let $\mathrm{Fl}_{X/Y}\subset Y$ be the maximal open
subset such that $X\times_{Y}U\to U$ is flat. Then $\mathrm{Fl}_{X/Y}$
contains all weakly Shilov points of $Y$. In particular, $\mathrm{Fl}_{X/Y}$
is a dense open subset of $Y$.

Observe that even the non-emptiness of $\mathrm{Fl}_{X/Y}$ is not clear a
priori. Moreover, if $K$ has characteristic $0$, then it suffices to assume
that $Y$ is reduced: as in algebraic geometry, this implies geometric
reducedness in characteristic $0$ (see [Con99, Lemma 3.3.1]).

######  Lemma 2.22.

Let $A$ be a ring with a locally nilpotent ideal $J\subset A$, and let $f:R\to
S$ be any map of finitely presented flat $A$-algebras. Then $f$ is flat if and
only if $\overline{f}:R/JR\to S/JS$ is flat.

###### Proof.

“Only if” is clear. Conversely, suppose $\overline{f}$ is flat. If $J$ is
nilpotent (e.g. if $A$ is Artinian), then flatness of $f$ follows from
Proposition 0.8.3.7 in [FK18].

In the general case, write $A$ as a filtered colimit
$A\simeq\mathop{\mathrm{colim}}_{i\in I}A_{i}$ where $A_{i}\subset A$ is a
filtered system of $\mathbf{Z}$-algebras of finite type. Set $J_{i}=J\cap
A_{i}$, so $J_{i}\subset A_{i}$ is a nilpotent ideal. By standard
approximation arguments, there is an index $i_{0}\in I$ such that $f$ is the
base change along $A_{i_{0}}\to A$ of a map $f_{i_{0}}:R_{i_{0}}\to S_{i_{0}}$
of finitely presented $A_{i_{0}}$-algebras. For all $i\geq i_{0}$, write
$f_{i}:R_{i}\to S_{i}$ for the evident base change of $f_{i_{0}}$. By two
applications of [Gro66, Théorème 11.2.6], $R_{i}$ and $S_{i}$ are flat
$A_{i}$-algebras for all sufficiently large $i$.

Next, note that $\overline{f}$ is the colimit of the diagrams
$\overline{f_{i}}:R_{i}/J_{i}R_{i}\to S_{i}/J_{i}S_{i}$. Since $\overline{f}$
is flat by assumption, $\overline{f_{i}}$ is flat for all sufficiently large
$i$ by another application of [Gro66, Théorème 11.2.6]. But $J_{i}$ is
nilpotent, so flatness of $\overline{f_{i}}$ implies flatness of $f_{i}$ by
the special case of the lemma treated in the first paragraph of the proof.
Therefore $f_{i}$ is flat for all sufficiently large $i$, so $f$ is flat. ∎

Recall that an adic ring $A$ admitting a finitely generated ideal of
definition $I$ is called _topologically universally (t.u.) rigid-noetherian_
if the scheme $\mathop{\mathrm{Spec}}A\left\langle
T_{1},\dots,T_{n}\right\rangle\smallsetminus V(IA\left\langle
T_{1},\dots,T_{n}\right\rangle)$ is Noetherian for all $n\geq 0$. If $A$ is a
tft $K$-algebra, then any ring of definition $A_{0}\subset A$ is t.u. rigid-
Noetherian.

######  Proposition 2.23.

Let $A$ be a t.u. rigid-Noetherian ring, and let $J\subset A$ be any open
ideal consisting of topologically nilpotent elements. Let $f:R\to S$ be a
morphism of flat and topologically finitely presented $A$-algebras. Then $f$
is flat if and only if $\overline{f}:R/JR\to S/JS$ is flat.

###### Proof.

“Only if” is clear. For the converse, choose a finitely generated ideal of
definition $I\subset A$ contained in $J$; let $\overline{J}_{n}\subset
A/I^{n}$ be the image of $J$, so $\overline{J}_{n}$ is locally nilpotent.
Since $R/JR\to S/JR$ is flat, the previous lemma implies that $R/I^{n}R\to
S/I^{n}S$ is flat for all $n\geq 1$. By Corollary 0.8.3.9 in [FK18], we then
deduce that $R\to S$ is flat. ∎

######  Proposition 2.24.

Let $K$ be a nonarchimedean field, and let $f:A\to B$ be a map of tft
$K$-algebras with $A$ geometrically reduced. Then there is a (nonempty)
rational subset $U\subset\mathop{\mathrm{Spa}}A$ containing the Shilov
boundary such that $\mathop{\mathrm{Spa}}B\times_{\mathop{\mathrm{Spa}}A}U\to
U$ is flat.

###### Proof.

By (the proof of) Theorem 1.3 in [BLR95], we can find a finite étale Galois
extension $K^{\prime}/K$ such that the unit ball
$A_{K^{\prime}}^{\circ}\subset
A_{K^{\prime}}\overset{def}{=}A\otimes_{K}K^{\prime}$ is topologically
finitely presented over $K^{\prime\circ}$ and the special fiber
$A_{K^{\prime}}^{\circ}/K^{\prime\circ\circ}A_{K^{\prime}}^{\circ}$ is
(geometrically) reduced. Choose an open tfp $K^{\prime\circ}$-algebra
$B_{0}\subset B\otimes_{K}K^{\prime}$ such that
$(f\otimes_{K}K^{\prime})(A_{K^{\prime}}^{\circ})\subset B_{0}$. Let
$k^{\prime}$ be the residue field of $K^{\prime\circ}$. Then
$A_{K^{\prime}}^{\circ}\otimes_{K^{\prime\circ}}k^{\prime}\to
B_{0}\otimes_{K^{\prime\circ}}k^{\prime}$ is a map of finite-type
$k^{\prime}$-algebras with reduced source, so there exists a non-zero-divisor
$f\in A_{K^{\prime}}^{\circ}\otimes_{K^{\prime}\circ}k^{\prime}$ such that
$(A_{K^{\prime}}^{\circ}\otimes_{K^{\prime\circ}}k^{\prime})[1/f]\to(B_{0}\otimes_{K^{\prime\circ}}k^{\prime})[1/f]$
is flat. Choose any lift $\tilde{f}\in A_{K^{\prime}}^{\circ}$, and let $C$ be
the $\pi$-adic completion of $A_{K^{\prime}}^{\circ}[1/\tilde{f}]$; similarly,
let $D$ be the $\pi$-adic completion of $B_{0}[1/\tilde{f}]$. Applying the
previous proposition with $A=K^{\prime\circ}$, $J=K^{\prime\circ\circ}$,
$R=C$, and $S=D$, we deduce that the map $C\to D$ is flat. Then
$\mathop{\mathrm{Spa}}C[1/\pi]\to\mathop{\mathrm{Spa}}A_{K^{\prime}}$ is the
inclusion of the Laurent domain $U(\frac{1}{\tilde{f}})$, and
$\mathop{\mathrm{Spa}}B_{K^{\prime}}\times_{\mathop{\mathrm{Spa}}A_{K^{\prime}}}U(\frac{1}{\tilde{f}})\cong\mathop{\mathrm{Spa}}D[1/\pi]$
by design, so
$\mathop{\mathrm{Spa}}B_{K^{\prime}}\times_{\mathop{\mathrm{Spa}}A_{K^{\prime}}}U(\frac{1}{\tilde{f}})\to
U(\frac{1}{\tilde{f}})$ is flat. Moreover, $U(\frac{1}{\tilde{f}})$ contains
all the Shilov points of $\mathop{\mathrm{Spa}}A_{K^{\prime}}$ by
construction.

It remains to undo the base change from $K$ to $K^{\prime}$. For this, let
$h\in A^{\circ}$ be the image of $\tilde{f}$ under the norm map
$A_{K^{\prime}}\to A$, and let $U=U(\frac{1}{h})\subset\mathop{\mathrm{Spa}}A$
be the associated Laurent domain. We claim that $U$ satisfies the conclusions
of the theorem. Indeed, writing
$\pi:\mathop{\mathrm{Spa}}A_{K^{\prime}}\to\mathop{\mathrm{Spa}}A$ for the
evident finite étale map, it is clear from the definitions that
$\pi^{-1}(U)=\cap_{g\in\mathrm{Gal}(K^{\prime}/K)}U(\frac{1}{\tilde{f}})g$ as
subsets of $\mathop{\mathrm{Spa}}A_{K^{\prime}}$. Since each $g$-translate of
$U(\frac{1}{\tilde{f}})$ still contains all Shilov points of
$\mathop{\mathrm{Spa}}A_{K^{\prime}}$, we see that $\pi^{-1}(U)$ contains all
Shilov points of $\mathop{\mathrm{Spa}}A_{K^{\prime}}$, and so $U$ contains
all Shilov points of $\mathop{\mathrm{Spa}}A$. Moreover, the results in the
previous paragraph show that
$f_{U}:\mathop{\mathrm{Spa}}B\times_{\mathop{\mathrm{Spa}}A}U\to U$ becomes
flat after base change along the surjective finite étale map $\pi^{-1}(U)\to
U$, so $f_{U}$ is flat by [War17, Proposition 3.1.12]. This concludes the
proof. ∎

###### Proof of Theorem 2.21.

It suffices to check that $\mathrm{Fl}_{X/Y}$ contains an open neighborhood of
every weakly Shilov point $y\in Y$. The formation of $\mathrm{Fl}_{X/Y}$
commutes with base change along open immersions $Y^{\prime}\to Y$ in the
evident sense, so we’re reduced to showing that if $Y$ is affinoid, then
$\mathrm{Fl}_{X/Y}$ contains an open neighborhood of every Shilov point of
$Y$. For this, cover $X$ by finitely many open affinoid subsets $X_{i}$. Then
each $\mathrm{Fl}_{X_{i}/Y}\subset Y$ contains all Shilov points of $Y$ by
Proposition 2.24. Since $\mathrm{Fl}_{X/Y}=\cap_{i}\mathrm{Fl}_{X_{i}/Y}$, we
deduce that $\mathrm{Fl}_{X/Y}$ contains all Shilov points of $Y$. ∎

###  2.5. Generic smoothness

In this subsection, we combine Theorem 2.18 and Theorem 2.21 to prove our main
generic smoothness result. We begin by translating Theorem 2.18 into geometric
language.

######  Theorem 2.25.

Fix a nonarchimedean field $K$, and let $f:X=\mathop{\mathrm{Spa}}B\to
Y=\mathop{\mathrm{Spa}}A$ be a map of $K$-affinoid rigid spaces. Suppose that
$X$ is smooth over $\mathop{\mathrm{Spa}}K$. Then for any Shilov point $y\in
Y$, the adic fiber $X_{y}=\mathop{\mathrm{Spa}}(B\widehat{\otimes}_{A}K_{y})$
is regular. In particular, if $K$ has characteristic zero, then
$X_{y}\to\mathop{\mathrm{Spa}}K_{y}$ is smooth.

###### Proof.

The first part follows immediately from Theorem 2.18 thanks to the
characterization of Shilov points in Proposition 2.2 once one observes that
the local rings of $\mathrm{Spa}(R)$ for a regular tft $K$-algebra $R$ are
regular. The last statement follows as a rigid analytic space over a
characteristic zero nonarchimedean field is smooth exactly when all of its
local rings are regular. ∎

We also need the following result, relating smoothness and fibral smoothness.

######  Lemma 2.26.

Let $f:X\to Y$ be any map of rigid spaces, and let $x\in X$ be any point with
image $y=f(x)$. Then the following are equivalent

  1. (1)

$f$ is smooth at $x$.

  2. (2)

$f$ is flat in a neighborhood of $x$ and
$X_{y}=X\times_{Y}\mathop{\mathrm{Spa}}(K_{y},K_{y}^{+})\to\mathop{\mathrm{Spa}}(K_{y},K_{y}^{+})$
is smooth at $x$.

###### Proof.

This follows from (the proof of) Lemma 2.9.2 in [War17]. ∎

######  Theorem 2.27.

Let $f:X\to Y$ be any quasicompact map of rigid spaces in characteristic zero.
Assume that $X$ is smooth and $Y$ is reduced. Then there is a dense open
subset $U\subset Y$ such that $f^{-1}(U)\to U$ is smooth.

######  Remark 2.28.

Consideration of standard examples (for instance, quasi-elliptic fibrations in
characteristics $2$ and $3$) shows that no result like this can hold in
positive characteristic, not even with the weaker conclusion that
$f^{-1}(U)\to U$ is smooth up to a universal homeomorphism.

###### Proof.

Let $y\in Y$ be any weakly Shilov point. By Theorem 2.21, we can choose some
open subset $U(y)\subset Y$ containing $y$ such that $X\times_{Y}U(y)\to U(y)$
is flat. Moreover, by Theorem 2.25, the entire fiber $X_{y}$ is smooth over
$y$. Applying Lemma 2.26 and the openness of the smooth locus in the source,
we deduce that every point $x\in f^{-1}(y)$ admits a quasicompact open
neighborhood $W_{x}$ in $X$ such that $W_{x}\to Y$ is smooth. Forming a
suitable union of the $W_{x}$’s and using the quasicompacity of $f$, we deduce
that there is a quasicompact open neighborhood $W\subset X$ of the fiber
$f^{-1}(y)$ such that $W\to Y$ is smooth. Shrinking $U(y)$ further, we may
assume by an easy quasicompacity argument that $f^{-1}(U(y))\subset W$. In
particular, $X\times_{Y}U(y)\to U(y)$ is smooth.

Since weakly Shilov points are dense in $Y$, setting
$U=\cup_{y\,\mathrm{weakly\,Shilov}}U(y)$ concludes the proof. ∎

In the proper case, we can do even better.

######  Theorem 2.29.

Let $f:X\to Y$ be a proper map of rigid spaces in characteristic zero, with
$X$ smooth and $Y$ reduced. Then the maximal open subset $S_{f}\subset Y$ over
which $f$ becomes smooth is a dense Zariski-open subset.

###### Proof.

Let $W\subset X$ be the Zariski-open subset where $X\to Y$ is smooth, and let
$Z=X-W$ be the complement regarded as a rigid space with its induced reduced
structure. The composite morphism $g:Z\to Y$ is proper, so by Kiehl’s results
$g_{\ast}\mathcal{O}_{Z}$ is a coherent sheaf on $Y$. Since
$\mathrm{Supp}(g_{\ast}\mathcal{O}_{Z})=f(Z)$, we deduce that $f(Z)\subset Y$
is Zariski-closed, so then $S_{f}=Y-f(Z)$ is Zariski-open, and density follows
from the previous theorem. ∎

######  Remark 2.30 (Deducing Theorem 2.29 from [Duc18]).

Let us indicate how to prove the Berkovich variant of Theorem 2.29, using
results from [Duc18].444We thank a referee for encouraging us to flesh out
this deduction (and in fact for providing a complete argument). Precisely,
given a nonarchimedean base field $K$ of characteristic $0$, we claim that if
$f:X\to Y$ is a proper map of $K$-analytic spaces in the sense of Berkovich
with $X$ quasi-smooth and $Y$ reduced, then $f$ is smooth over a dense
Zariski-open subset of $Y$. To see this, let $T\subset Y$ be image under $f$
of the non-smooth locus of $f$; the latter is Zariski-closed by [Duc18,
Theorem 10.7.2], so the former is Zariski-closed by properness (as in the
proof of Theorem 2.29). As $f$ is smooth over the Zariski-open subset
$Y-T\subset T$, it suffices to show that $Y-T$ is dense in $Y$. In fact, we
claim that $Y-T$ contains all the Abhyankar points $y\in Y$. By [Duc18,
Theorem 10.3.7], the map $f$ is flat at $y$, so by [Duc18, Theorem 5.3.4], it
suffices to note that $f^{-1}(y)$ is quasi-smooth (or equivalently regular, as
$K_{y}$ has characteristic $0$) by [Duc18, Theorem 6.3.7].

##  3\. The six functors for Zariski-constructible sheaves

In this section, we use the geometric results of §2 to develop the six functor
formalism for Zariski-constructible sheaves in rigid analytic geometry over a
characteristic $0$ field.

###  3.1. Definition of Zariski-constructible sheaves

In this subsection we briefly review the definition and basic properties of
Zariski-constructible sheaves on rigid analytic spaces. Most of this material
is taken from [Han20]; the exception is Theorem 3.5.

######  Definition 3.1.

Let $X$ be a rigid analytic space over a nonarchimedean field $K$, and let
$\Lambda$ be a finite commutative ring.

  1. (1)

An étale sheaf $\mathscr{F}\in\mathrm{Sh}(X,\Lambda)$ is _lisse_ there exists
an étale cover $\\{U_{i}\to X\\}$ such that $\mathscr{F}|_{U_{i}}$ is the
constant sheaf associated to a finitely generated $\Lambda$-module.

  2. (2)

A complex $A\in D(X,\Lambda)$ is lisse if the cohomology sheaves
$\mathcal{H}^{n}(A)$ are lisse for all $n$. We write
$D_{lis}(X,\Lambda)\subset D(X,\Lambda)$ for the full subcategory spanned by
lisse complexes.

  3. (3)

An étale sheaf $\mathscr{F}\in\mathrm{Sh}(X,\Lambda)$ is _Zariski-
constructible_ if $X$ admits a locally finite stratification
$X=\coprod_{i\in}X_{i}$ into Zariski locally closed subsets $X_{i}$ such that
$\mathscr{F}|X_{i}$ is a lisse sheaf of $\Lambda$-modules for all $i\in I$. We
write $\mathrm{Sh}_{zc}(X,\Lambda)$ for the full subcategory of Zariski-
constructible sheaves.

  4. (4)

A complex $A\in D(X,\Lambda)$ is Zariski-constructible if the cohomology
sheaves $\mathcal{H}^{n}(A)$ are Zariski-constructible for all $n$. We write
$D_{zc}(X,\Lambda)$ for the full subcategory spanned by Zariski-constructible
complexes.

One has the bounded below variant $D^{+}_{zc}(X,\Lambda)$; similarly for
$D^{-}$ and $D^{b}$. Finally, let $D^{(b)}_{zc}(X,\Lambda)\subset
D_{zc}(X,\Lambda)$ denotes the full triangulated subcategory of complexes
which are locally bounded; similarly for $D^{(-)}$ and $D^{(+)}$. The natural
$\infty$-categorical refinements of $D(X,\Lambda),D^{(b)}_{zc}(X,\Lambda)$,
etc. shall be denoted
$\mathcal{D}(X,\Lambda),\mathcal{D}^{(b)}_{zc}(X,\Lambda)$, etc., as usual.

######  Warning 3.2.

Let us record some subtleties concerning this notion.

  1. (1)

Given a Zariski-open immersion $j:U\to X$ and a Zariski-constructible sheaf
$\mathscr{F}$ on $U$, the extension $j_{!}\mathscr{F}$ can fail to be Zariski-
constructible $X$, unlike the situation in algebraic geometry (see next
example). The main problem is that the operation of taking Zariski-closures in
$X$ of Zariski-closed subsets of $U$ is poorly behaved in general (e.g., it
does something non-trivial over $U$); this issue does not arise if
$\mathscr{F}$ is itself locally constant.

######  Example 3.3.

Let $\mathscr{F}$ be the direct sum of skyscraper sheaves supported at an
infinite discrete set of classical points in $(\mathbf{A}^{1})^{an}$, and let
$j:(\mathbf{A}^{1})^{an}\to(\mathbf{P}^{1})^{an}$ be the standard open
immersion. Then $j_{!}\mathscr{F}$ is not Zariski-constructible on
$(\mathbf{P}^{1})^{an}$: any Zariski-closed set of $(\mathbf{P}^{1})^{an}$
must be either finite or all of $(\mathbf{P}^{1})^{an}$ by rigid GAGA.

This phenomenon should not be regarded as a pathology: similar examples occur
in complex analytic geometry as well, and are a natural consequence of the
non-quasi-compactness of affine space in any kind of analytic geometry.

  2. (2)

Huber’s book [Hub96] defines a notion of “constructible” sheaves that is very
well-behaved from a topos theoretic perspective. However, these sheaves are
typically not Zariski-constructible; for instance, if $j$ is the qcqs open
immersion defined by including a closed disc of radius (say) $1/2$ inside a
closed disc of radius $1$, then $j_{!}\Lambda$ is constructible in Huber’s
sense but is not Zariski-constructible. In fact, the overlap between these two
notions is exactly the lisse sheaves.

Next, we record some simple stability properties of this notion.

######  Proposition 3.4.

  1. (1)

$\mathrm{Sh}_{zc}(X,\Lambda)$ is a weak Serre subcategory of
$\mathrm{Sh}(X,\Lambda)$, and $D_{zc}(X,\Lambda)$ is a thick triangulated
subcategory of $D(X,\Lambda)$.

  2. (2)

_(Devissage)_ A sheaf $\mathscr{F}\in\mathop{\mathrm{Sh}}(X,\Lambda)$ is
Zariski-constructible iff there is a dense Zariski-open subset $U\subset X$
such that $\mathscr{F}|U$ is lisse and $\mathscr{F}|(X\smallsetminus U)$ is
Zariski-constructible.

  3. (3)

Zariski-constructibility is stable under $f^{\ast}$ for $f$ any morphism of
rigid spaces, and under $f_{\ast}$ for finite morphisms.

  4. (4)

A sheaf $\mathscr{F}\in\mathop{\mathrm{Sh}}(X,\Lambda)$ is Zariski-
constructible iff $\mathscr{F}|X_{i}$ is Zariski-constructible for all
irreducible components $X_{i}\subset X$.

###### Proof.

(1)-(3) are proved in [Han20]. For (4), one direction is clear. For the other
direction, let $f_{0}:\tilde{X}\to X$ be the normalization of $X$, and let
$f_{1}:\tilde{X}\times_{X}\tilde{X}\to X$ be the evident map. The hypothesis
guarantees that $f_{0}^{\ast}\mathscr{F}$ (and then also
$f_{1}^{\ast}\mathscr{F}$) is Zariski-constructible, since
$\tilde{X}=\coprod_{i}\tilde{X}_{i}$ is the disjoint union of normalizations
of the irreducible components of $X$. Since $f_{0}$ and $f_{1}$ are both
finite, pushforward along these maps preserves Zariski-constructibility by
(3). The exact sequence $0\to\mathscr{F}\to
f_{0\ast}f_{0}^{\ast}\mathscr{F}\to f_{1\ast}f_{1}^{\ast}\mathscr{F}$ now
exhibits $\mathscr{F}$ as the kernel of a map between Zariski-constructible
sheaves, so we conclude by (1). ∎

In view of Warning 3.2 (1), the following result on the analytic (or even
étale) locality of the notion of Zariski-constructibility is somewhat
surprising:

######  Theorem 3.5.

Let $X$ be a rigid space over a characteristic zero nonarchimedean field $K$,
equipped with an étale $\Lambda$-sheaf $\mathscr{F}$. If there exists an étale
cover $\\{U_{i}\\}$ of $X$ such that $\mathscr{F}|_{U_{i}}$ is Zariski-
constructible, then $\mathscr{F}$ is Zariski-constructible.

In particular, the assignment carrying a rigid space $X$ to the
$\infty$-category $\mathcal{D}^{(b)}_{zc}(X,\Lambda)$ is a stack for the étale
topology.

We shall prove this result in §3.2. Finally, the following result is often
very useful.

######  Proposition 3.6.

If $X$ is a quasicompact rigid space over a nonarchimedean field $K$ of
characteristic zero, then $D^{b}_{zc}(X,\Lambda)$ is the thick triangulated
subcategory of $D(X,\Lambda)$ generated by objects of the form $f_{\ast}M$ for
$f:Y\to X$ a finite morphism and $M$ a constant constructible $\Lambda$-sheaf
on $Y$.

###### Proof.

By induction on $\dim X$ and devissage, it’s enough to show that if $j:U\to X$
is a dense Zariski-open and $\mathscr{F}$ is lisse and killed by a prime
$\ell$, then $j_{!}{\mathscr{F}}$ lies in the claimed subcategory. For this,
choose (as in [Sta18, Tag 0A3R]) a finite étale cover $g:U^{\prime}\to U$ of
prime-to-$\ell$ degree such that $g^{\ast}\mathscr{F}$ is an iterated
extension of copies of $\mathbf{F}_{\ell}$. Then $\mathscr{F}$ is a summand of
$g_{\ast}g^{\ast}\mathscr{F}$, so $\mathscr{F}$ is a summand of an iterated
extension of copies of $g_{\ast}\mathbf{F}_{\ell}$. Now extend $g$ to a finite
cover $f:X^{\prime}\to X$ as in [Han20], so $j_{!}\mathscr{F}$ is a summand of
an iterated extension of copies of
$\mathscr{G}=j_{!}g_{\ast}\mathbf{F}_{\ell}=f_{\ast}j^{\prime}_{!}\mathbf{F}_{\ell}$,
where $j^{\prime}:U^{\prime}\to X^{\prime}$ is the evident map. Letting
$i:Z\to X^{\prime}$ be the complement of $j^{\prime}$, the exact sequence
$0\to f_{\ast}j^{\prime}_{!}\mathbf{F}_{\ell}\to
f_{\ast}\mathbf{F}_{\ell}\to(f\circ i)_{\ast}\mathbf{F}_{\ell}\to 0$ shows
that $\mathscr{G}$ lies in the desired subcategory. ∎

###  3.2. Zariski-constructible sheaves via algebraic geometry

In this subsection, we describe Zariski-constructible sheaves on affinoids
purely in terms of algebraic geometry, and deduce that the property of being
Zariski-constructible is étale local.

Let $K$ be a characteristic zero nonarchimedean field. Recall from [Han20]
that for any affinoid $K$-algebra $A$ and any scheme $\mathcal{X}$ locally of
finite type over $\mathop{\mathrm{Spec}}A$, there is a naturally associated
rigid space $X=\mathcal{X}^{an}$ over $\mathop{\mathrm{Spa}}A$, and a natural
map
$X_{\mathop{\mathrm{\acute{e}t}}}\to\mathcal{X}_{\mathop{\mathrm{\acute{e}t}}}$
of sites, inducing a $t$-exact pullback functor
$\mu_{X}:D(\mathcal{X},\mathbf{Z}/n)\to D(X,\mathbf{Z}/n)$ carrying $D_{c}$
into $D_{zc}$. Here we change notation slightly from [Han20], and write
$(-)^{an}$ interchangeably for $\mu_{X}^{\ast}(-)$.

######  Proposition 3.7 (Algebraization of Zariski-constructible sheaves over
affinoids).

Fix an affinoid $K$-algebra $A$, and write
$\mathcal{S}=\mathop{\mathrm{Spec}}\,A$ and $S=\mathop{\mathrm{Spa}}A$.

  1. (1)

If $f:\mathcal{X}\to\mathcal{Y}$ is any finite type map of finite type
$\mathcal{S}$-schemes, then for any $\mathscr{F}\in
D^{b}_{c}(\mathcal{X},\mathbf{Z}/n)$ the natural base change map
$(Rf_{\ast}\mathscr{F})^{an}\to Rf^{an}_{\ast}\mathscr{F}^{an}$ is an
isomorphism. In particular, $Rf^{an}_{\ast}\mathscr{F}^{an}$ lies in
$D^{b}_{zc}(Y,\mathbf{Z}/n)$.

  2. (2)

If $\mathcal{X}$ is any finite type $\mathcal{S}$-scheme, the functor
$(-)^{an}:D^{b}_{c}(\mathcal{X},\mathbf{Z}/n)\to D^{b}_{zc}(X,\mathbf{Z}/n)$
is fully faithful. If $\mathcal{X}$ is proper over $\mathcal{S}$, it is an
equivalence of categories.

  3. (3)

If $\mathcal{X}$ is any finite type $\mathcal{S}$-scheme, the fully faithful
functor $(-)^{an}:D^{b}_{c}(\mathcal{X},\mathbf{Z}/n)\to
D^{b}_{zc}(X,\mathbf{Z}/n)$ from (2) identifies the full subcategory of lisse
objects on both sides.

Note that part (2) applies notably when $\mathcal{X}=\mathcal{S}$.

###### Proof.

  1. (1)

This is exactly [Han20, Theorem 1.8].

  2. (2)

The full faithfulness is a special case of [Han20, Theorem 1.10.ii]. Essential
surjectivity in the case where $\mathcal{X}/\mathcal{S}$ is proper can be
checked on hearts. We can also assume that $\mathcal{X}$ and $X$ are reduced.
Arguing by induction on $\dim\mathcal{X}$ as in the proof of [Han20, Theorem
1.7], one reduces to checking that any sheaf on $X$ of the form
$j_{!}\mathscr{F}$ is in the essential image of $(-)^{an}$; here $j:U\subset
X$ is the inclusion of any normal Zariski-open subset and $\mathscr{F}$ is any
lisse sheaf. To do this, first note that the complement $Z=X-U$ algebraizes to
a closed subscheme $\mathcal{Z}\subset\mathcal{X}$ by relative rigid GAGA, so
then $\mathcal{U}=\mathcal{X}-\mathcal{Z}$ is an algebraization of $U$. We’re
now reduced to proving that the analytification functor
$\mathrm{F\acute{E}t}(\mathcal{U})\to\mathrm{F\acute{E}t}(U)$ is an
equivalence of categories. We explain the construction of an essential
inverse. Suppose $V\to U$ is any finite étale map. By [Han20, Theorem 1.6],
this extends uniquely to a branched covering $V^{\prime}\to X^{n}$, where
$X^{n}$ is the normalization of $X$. By relative rigid GAGA again, this
algebraizes to a branched covering $\mathcal{V}^{\prime}\to\mathcal{X}^{n}$,
and then
$\mathcal{V}:=\mathcal{V}^{\prime}\times_{\mathcal{X}^{n}}\mathcal{U}\to\mathcal{U}$
is the desired algebraization of $V$.

  3. (3)

Since we already have full faithfulness, it suffices to prove essential
surjectivity on the hearts, i.e., we want to realize a lisse sheaf on $X$ as
the analytification of a unique lisse sheaf on $\mathcal{X}$. By uniqueness
and Zariski/analytic descent for lisse sheaves in algebraic/analytic geometry,
we may assume that $\mathcal{X}$ is separated (or even affine). By finite
descent for lisse sheaves in both algebraic and analytic geometry, we may also
assume $\mathcal{X}$ is normal. In this case, we can realize $\mathcal{X}$ as
an an open subscheme of a normal proper $\mathcal{S}$-scheme
$\overline{\mathcal{X}}$; the argument used in the proof of (2) now yields the
desired algebraization.

∎

We shall use the above description to prove Theorem 3.5 by a topological
argument. To run this argument, we need a couple of lemmas in pure algebraic
geometry on the existence and properties of the maximal open set where a
constructible sheaf is lisse.

######  Lemma 3.8.

Let $X$ be a scheme and let $\mathscr{F}$ be a constructible sheaf on $X$.
Then there exists a maximal open subset $U_{\mathscr{F}}\subset X$ such that
$\mathscr{F}|_{U_{\mathscr{F}}}$ is locally constant. Moreover,
$U_{\mathscr{F}}$ is given by either of the following equivalent descriptions:

  1. (1)

The set of all $x\in X$ such that $\mathscr{F}|_{X_{x}}$ is locally constant.
(Here $X_{x}$ is the local scheme of $X$ at $x$.)

  2. (2)

The set of all $x\in X$ admitting an open neighborhood $x\in U\subset X$ with
$\mathscr{F}|_{U}$ being locally constant.

In particular, $U_{\mathscr{F}}$ contains all the generic points of $X$.

###### Proof.

As local constancy is a local property, the collection of all opens $V\subset
X$ such that $\mathscr{F}|_{V}$ is locally constant is stable under taking
unions. Taking the union of all such opens then gives the maximal open
$U_{\mathscr{F}}$ such that $\mathscr{F}|_{U_{\mathscr{F}}}$ is open. It is
also clear from this description that $U_{\mathscr{F}}$ agrees with the set in
(2). The set in (2) is trivially contained in the set in (1). Conversely, as
the functor sending a scheme $Y$ to its category of locally constant sheaves
(resp. constructible sheaves) is locally finitely presented (i.e., carries
cofiltered limits of affine schemes to direct limits), the set in (1) is also
contained in the set in (2), so the two sets coincide. The final statement is
clear from the description of $U_{\mathscr{F}}$ given by the set in (1). ∎

######  Lemma 3.9.

The formation of the open set $U_{\mathscr{F}}\subset X$ associated with a
pair $(X,\mathscr{F})$ as in Lemma 3.8 is compatible with pullback along
universally generalizing maps of schemes.

###### Proof.

Let $f:Y\to X$ be a universally generalizing map of schemes. We have an
obvious containment $f^{-1}(U_{\mathscr{F}})\subset U_{f^{*}\mathscr{F}}$, and
we must show it is an equality. Assume towards contradiction that there exists
some $y\in U_{f^{*}\mathscr{F}}-f^{-1}(U_{\mathscr{F}})$. Thus,
$(f^{*}\mathscr{F})|_{Y_{y}}$ is locally constant but
$\mathscr{F}|_{X_{f(y)}}$ is not locally constant. There is then a
specialization $x_{1}\rightsquigarrow x_{2}$ of geometric points of $X_{f(y)}$
such that the corresponding cospecialization map
$\mathscr{F}_{x_{2}}\to\mathscr{F}_{x_{1}}$ is not an isomorphism. Choose an
absolutely integrally closed valuation ring $V$ and a map
$\eta_{X}:\mathrm{Spec}(V)\to X_{f(y)}$ that witnesses the specialization
$x_{1}\rightsquigarrow x_{2}$, so $\eta_{X}^{*}(\mathscr{F}|_{X_{f(y)}})$ is
not locally constant. Now that the map $Y_{y}\to X_{f(y)}$ is universally
generalizing (it factors as $Y_{y}\to X_{f(y)}\times_{X}Y\to X_{f(y)}$, with
both maps being universally generalizing) and surjective (all points of
$X_{f(y)}$ specialize to $y$, so surjectivity follows from the universally
generalizing property). By stability of universally generalizing surjective
maps under base change, we can replace $V$ with an extension if necessary to
lift $\eta_{X}$ to a map $\eta_{Y}:\mathrm{Spec}(V)\to Y_{y}$. But then we
have
$\eta_{X}^{*}(\mathscr{F}|_{X_{f(y)}})=\eta_{Y}^{*}(f^{*}\mathscr{F}|_{Y_{y}})$;
this is a contradiction as the left side is not locally constant by choice of
$\eta_{X}$, while the right side is locally constant by choice of $y$. ∎

###### Proof of Theorem 3.5.

Let us first give the argument when $\\{U_{i}\\}$ is a cover of $X$ for the
analytic topology. We proceed by induction on $\dim(X)$. The $\dim(X)=0$ case
is clear: $X$ is a disjoint union of points in this case. In general, as
Zariski-constructibility is stable under pullback, we may assume each
$U_{i}=\mathrm{Spa}(A_{i})$ is affinoid. Write
$\mathcal{U}_{i}=\mathrm{Spec}(A_{i})$ for the obvious algebraization of
$U_{i}$. The map $U_{i}\to\mathcal{U}_{i}$ identifies constructible
$\mathbf{Z}/n$-sheaves on the target with Zariski-constructible
$\mathbf{Z}/n$-sheaves on the source by Proposition 3.7, so there is a unique
constructible $\mathbf{Z}/n$-sheaf $\mathscr{F}_{i}$ over $\mathcal{U}_{i}$
descending $\mathscr{F}|_{U_{i}}$. Let
$\mathcal{V}_{i}:=\mathcal{U}_{i,\mathscr{F}_{i}}\subset\mathcal{U}_{i}$ be
the maximal Zariski open over which $\mathcal{F}_{i}$ is locally constant as
in Lemma 3.8, let $V_{i}\subset U_{i}$ be its Zariski-open preimage, and let
$Z_{i}\subset U_{i}$ be the Zariski-closed complement of $V_{i}$ (regarded as
a reduced rigid space); note that $Z_{i}\subset V_{i}$ is nowhere dense as
$\mathcal{V}_{i}\subset\mathcal{U}_{i}$ contains all the generic points. As
the natural algebraizations of the maps given by rational localizations of
affinoids are universally generalizing, Lemma 3.9 implies that for all $i,j$,
the Zariski-open subsets $V_{i}\cap(U_{i}\cap U_{j})$ and $V_{j}\cap(U_{i}\cap
U_{j})$ of $U_{i}\cap U_{j}$ agree, and consequently their complements also
agree. By descent for coherent ideal sheaves applied to the ideal sheaves
$\mathcal{I}_{Z_{i}}\subset\mathcal{O}_{U_{i}}$ of the $Z_{i}$’s, there is a
unique Zariski-closed subset $Z\subset X$ such that $Z\cap U_{i}=Z_{i}$. As
$Z_{i}$ is nowhere dense in $U_{i}$ for all $i$, we must have
$\dim(Z)<\dim(X)$. Moreover, the sheaf $\mathscr{F}$ is lisse over $X-Z$ by
construction. Induction on dimension shows that $\mathscr{F}|_{Z}$ is also
Zariski-constructible, so we win.

To adapt this argument to the étale topology, note the proof above has two
essential ingredients:

  1. (1)

For each pair of indices $i,j$, if $V\subset U_{i}\cap
U_{j}=U_{i}\times_{X}U_{j}$ is an open affinoid, then the natural
algebraization of $V\to U_{i}$ (resp. $V\to U_{j}$) is a universally
generalizing map of affine schemes.

  2. (2)

Descent for coherent sheaves holds true with respect to the cover
$\\{U_{i}\\}$.

These properties are also true for étale covers of $X$, so the descent claim
also holds true in the étale topology. Indeed, étale descent for coherent
sheaves on rigid spaces is [dJvdP96, Corollary 3.2.3], while the first
property reduces to the well-known fact that for any étale map of affinoid
rigid spaces, the associated ring map is flat. ∎

###  3.3. Pushforward, $\otimes$, and $\mathop{R\mathscr{H}\mathrm{om}}$

In this subsection, we prove some of our main stability properties for
Zariski-constructible sheaves. Until further notice, we fix a characteristic
zero nonarchimedean base field $K$ of residue characteristic $p$ (with $p=0$
allowed). The first main result in this section is the following theorem,
which was conjectured by the second author [Han20, Conjecture 1.14].

######  Theorem 3.10 (Proper direct images).

Let $f:X\to Y$ be a proper map of rigid spaces over $K$. Then $Rf_{*}$
preserves $D^{(b)}_{zc}(-,\mathbf{Z}/n)$.

Note that $p|n$ is allowed here. We caution the reader that if $f:X\to Y$ is a
proper map of rigid spaces with $Y$ irreducible, then in contrast with the
situation for algebraic varieties, it is not always true that $X$ has finitely
many irreducible components, or that the fibers of $f$ have bounded
dimension.555 For a simple example where both conditions fail, let
$Y=(\mathop{\mathrm{Spec}}K[T])^{an}$ be the rigid affine line, set
$X_{n}=(\mathbf{P}^{n})^{an}$ and let $f_{n}:X_{n}\to Y$ be the map which
factors over the inclusion of the closed point $T=p^{-n}$. Then $f=\coprod
f_{n}:\coprod X_{n}\to Y$ is proper. Thus, it is necessary to use $D^{(b)}$
and not $D^{b}$ in the above formulation, and similarly in many other places
in this section.

###### Proof.

By Theorem 3.5, we may assume $Y=\mathrm{Spa}(A)$ for an affinoid $K$-algebra
$A$. In particular, $X$ and $Y$ are both qcqs. We may clearly also assume that
$Y$ is reduced. Our task is to show that $Rf_{*}$ preserves
$D^{b}_{zc}(-,\mathbf{Z}/n)$ in this situation. As $X$ is quasi-compact, we
may apply Proposition 3.6, so it suffices to show that $Rf_{*}\mathbf{Z}/n\in
D^{b}_{zc}$. In fact, as the fibres of $f$ have bounded dimension (e.g., by
checking on formal models), we know by cohomological dimension estimates and
proper base change that $Rf_{*}$ has finite cohomological dimension, so it is
enough to show that $Rf_{*}\mathbf{Z}/n\in D_{zc}$, i.e., that each
$R^{i}f_{*}\mathbf{Z}/n$ is Zariski-constructible on $Y$. By proper base
change and induction on $\dim(Y)$, it suffices to find a dense open $U\subset
Y$ such that $(R^{i}f_{*}\mathbf{Z}/n)|_{U}$ is locally constant. As $K$ has
characteristic $0$, Temkin’s [Tem18, Theorem 1.1.13 (i)] gives a proper
hypercover $\epsilon:X^{\bullet}\to X$ with each $X^{i}$ being $K$-smooth.
Cohomological descent enables us to compute $Rf_{*}\mathbf{Z}/n$ as the
totalization of $Rg_{*}\mathbf{Z}/n$, where $g=f\circ\epsilon$. As
$\mathcal{H}^{i}$ of the totalization of a cosimplicial object
$K(\bullet):\Delta\to\mathcal{D}^{\geq 0}$ only depends on truncated
cosimplicial object $K(\bullet)|_{\Delta_{\leq i+1}}$, we can replace
$X^{\bullet}$ with the finite diagram $X^{\leq i+1}$ and then by each $X^{j}$
to assume that $X$ is smooth. As $Y$ is reduced, our generic smoothness result
(Theorem 2.29) yields a Zariski-dense Zariski-open $U\subset X$ such that
$f:X\to Y$ is smooth over $U$. It then suffices to show that
$R^{i}g_{*}\mathbf{Z}/n$ is locally constant when $g$ is both proper and
smooth. To check this, we may assume $n=\ell$ is a prime. Now if $\ell\neq p$,
the claim reduces to [Hub96, Corollary 6.2.3], while the claim for $\ell=p$
reduces to [SW20, Theorem 10.5.1]. ∎

As a consequence of Theorem 3.10, we also get some additional stability
results. First, locally constant sheaves are carried to Zariski-constructible
complexes via pushforward along a fairly general class of maps.

######  Corollary 3.11 (Direct images of lisse complexes).

Let $f:X\to Y$ be a Zariski-compactifiable map of rigid spaces. Then $Rf_{!}$
and $Rf_{*}$ carry $D^{(b)}_{lis}(-,\mathbf{Z}/n)$ into
$D^{(b)}_{zc}(-,\mathbf{Z}/n)$.

As explained in Warning 3.2 (1), the functor $Rf_{*}$ does not preserve
$D^{(b)}_{zc}$ in general, even for Zariski-open immersions. Thus, the above
seems to be the best general statement one can expect.

###### Proof.

The statement is local on the target by Theorem 3.5, so we may assume
$Y=\mathrm{Spa}(A)$ is affinoid. By assumption, we can factor $f$ as
$X\xrightarrow{j}\overline{X}\xrightarrow{g}Y$ with $j$ being a Zariski-open
immersion and $g$ being proper. Using Theorem 3.10, we can reduce to the case
$f=j$ is a Zariski-open immersion. The claim for $Rf_{!}$ is clear from the
definition of Zariski-constructible sheaves (using $\\{X,\overline{X}-X\\}$ as
the stratification on $\overline{X}$ witnessing Zariski-constructibility), so
it remains to check the assertion for $Rf_{*}$. As $Y$ is affinoid, the
Zariski-open immersion $f:X\hookrightarrow Y$ is the algebraization of a
unique open immersion $g:\mathcal{X}\to\mathcal{Y}=\mathrm{Spec}(A)$.
Moreover, any object of $D^{b}_{lis}(X,\mathbf{Z}/n)$ is the analytification
of a unique object in $D^{b}_{lis}(\mathcal{X},\mathbf{Z}/n)$ by Proposition
3.7 (3). Using the compatibility of pushforwards with analytification from
Proposition 3.7 (1), the claim follows from Gabber’s constructibility theorem
in [ILO14, Expose XIII, Theorem 1.1.1]. ∎

Secondly, $!$-pullback along finite morphisms preserves Zariski-
constructibility.

######  Corollary 3.12 (Finite $!$-pullback).

Let $f:X\to Y$ be a finite morphism of rigid spaces over $K$. Then the right
adjoint $Rf^{!}:D^{+}(Y,\mathbf{Z}/n)\to D^{+}(X,\mathbf{Z}/n)$ to
$f_{!}=f_{*}$ constructed in [Hub96, §7.1] preserves $D^{(b)}_{zc}(-)$.

###### Proof.

By Theorem 3.5, the assertion is étale-local on $Y$, so we may assume both $X$
and $Y$ are affinoid, corresponding to a finite map $A\to B$ of affinoid
$K$-algebras. Fix $F\in D^{b}_{zc}(Y,\mathbf{Z}/n)$. Let $S_{F}\subset Y$ be
the smallest Zariski-closed subset of $Y$ containing the support of $F$. We
shall prove the claim by induction on $d_{F}=\dim(S_{F})$.

If $d_{F}=0$, then $F$ is supported at finitely many points. As the claim is
étale local on $Y$, we may then assume that $F$ is a finite direct sum of
sheaves of the form $k_{*}\mathbf{Z}/n$, where $k:W\to Y$ is the inclusion of
a Zariski-closed point. Now $f^{!}$ and $k_{*}$ commute: the corresponding
statement for left adjoints is the proper base change theorem for $f$. We are
thus reduced to checking the statement when $Y$ (and thus $X$) are
$0$-dimensional. In this case, up to universal homeomorphisms, the map $f$ is
finite étale, so $Rf^{!}=f^{*}$, so the claim is clear.

Now assume $d_{F}>0$. We may then choose a Zariski open subset
$j:U\hookrightarrow Y$ such that $U\cap S_{F}$ is dense is $S_{F}$, the
restriction $L:=F|_{U\cap S_{F}}$ is lisse, and $f$ is finite étale (up to
universal homeomophisms) over $U$: one can find such an open $U$ as the
algebraization of an open $\mathcal{U}\subset\mathrm{Spec}(A)$ satisfying the
analogous properties for the map $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ and the
algebraization $\mathcal{F}$ of $F$ (in the sense of Proposition 3.7). Let
$i:Z\hookrightarrow Y$ be the closed complement, so we have the standard exact
triangle

$i_{*}Ri^{!}F\to F\to Rj_{*}(F|_{U}).$

Now $Rj_{*}(F|_{U})\simeq k_{*}Rj^{\prime}_{*}L$, where $j^{\prime}:U\cap
S_{F}\to S_{F}$ and $k:S_{F}\to X$ are the natural maps (and thus Zariski open
and Zariski closed immersions respectively). By Corollary 3.11, the third term
in the triangle above is then Zariski-constructible. The remaining term
$i_{*}Ri^{!}F$ in the triangle is then also Zariski-constructible, so
$Ri^{!}F$ is itself Zariski-constructible. Applying $Rf^{!}$ to the above
triangle gives a triangle

$Rf^{!}i_{*}i^{!}F\to f^{!}F\to Rf^{!}k_{*}Rj^{\prime}_{*}L.$

By proper base change for $f$ as in the previous paragraph, the last term
identifies with $k_{X,*}Rj^{\prime}_{X,*}(f|_{U\cap S_{F}})^{!}L$, where
$k_{X}$ and $j^{\prime}_{X}$ are the base changes of $k$ and $j^{\prime}$
along $f$. As $f$ is finite étale up to universal homeomorphisms over $U$, we
have $(f|_{U\cap S_{F}})^{!}L\simeq(f|_{U\cap S_{F}})^{*}L$, so this object is
lisse on $f^{-1}(U\cap S_{F})$. Corollary 3.11 then implies that the third
term in the triangle above is lisse. For the first term, using proper base
change again lets us write it as $i_{X,*}f_{Z}^{!}i^{!}F$, where $i_{X}$ and
$f_{Z}$ are the base changes of $i$ and $f$ against $f$ and $i$. As $i^{!}F$
is known to be Zariski constructible, the induction hypothesis then shows that
the first term is also Zariski constructible. ∎

######  Remark 3.13.

Using results from [Hub96, §7], in the special case $(p,n)=1$, we can extend
Corollary 3.12 to much larger generality. Indeed, if $f:Y\to X$ is any
separated taut morphism of rigid spaces over $K$, then $Rf^{!}$ sends
$D^{(b)}_{zc}(X,\mathbf{Z}/n)$ into $D^{(b)}_{zc}(Y,\mathbf{Z}/n)$. To see
this, by Theorem 3.5, this assertion can be checked locally on $X$ and $Y$, so
we can assume they are affinoid. The map $f$ can be then be factored as the
composition of a Zariski-closed immersion followed by a smooth map of pure
dimension $d$. The claim for Zariski-closed immersions follows from Corollary
3.12, while that for smooth morphisms follows from Huber’s [Hub96, Theorem
7.5.3], which identifies $Rf^{!}$ with $f^{*}(d)[2d]$.

We deduce the existence of $\otimes$ and $\mathop{R\mathscr{H}\mathrm{om}}$.

######  Corollary 3.14 ($\otimes$ and $\mathop{R\mathscr{H}\mathrm{om}}$).

Let $X$ be a rigid space. For any $\mathscr{F},\mathscr{G}\in
D^{(b)}_{zc}(X,\mathbf{Z}/n)$ with $\mathscr{F}$ having finite Tor dimension,
both $\mathscr{F}\otimes\mathscr{G}$ and
$\mathop{R\mathscr{H}\mathrm{om}}(\mathscr{F},\mathscr{G})$ lie in
$D^{(b)}_{zc}(X,\mathbf{Z}/n)$.

###### Proof.

We may work locally on $X$ on Theorem 3.5, so assume $X$ is affinoid. The
claim about tensor products is clear (e.g., by Proposition 3.7 and the
corresponding statement in algebraic geometry). For
$\mathop{R\mathscr{H}\mathrm{om}}$, we proceed by induction on $d=\dim(X)$,
the case $d=0$ being trivial. Choose a dense Zariski-open $j:U\subset X$ such
that both $\mathscr{F}|_{U}$ and $\mathscr{G}|_{U}$ are lisse. Applying
$\mathop{R\mathscr{H}\mathrm{om}}(-,\mathscr{G})$ to the triangle
$j_{!}j^{\ast}\mathscr{F}\to\mathscr{F}\to i_{\ast}i^{\ast}\mathscr{F}\to$, we
get a triangle

$i_{\ast}\mathop{R\mathscr{H}\mathrm{om}}(i^{\ast}\mathscr{F},i^{!}\mathscr{G})\to\mathop{R\mathscr{H}\mathrm{om}}(\mathscr{F},\mathscr{G})\to
Rj_{\ast}(\mathscr{F}|_{U}^{\vee}\otimes\mathscr{G}|_{U})\to,$

where we simplified the first term using the adjunction defining $i^{!}$, and
the last term by using $\mathop{R\mathscr{H}\mathrm{om}}(A,B)=A^{\vee}\otimes
B$ for $A,B\in D(\mathbf{Z}/n)$ with $A\in D_{perf}(\mathbf{Z}/n)$. Now
induction on dimension and Corolary 3.12 ensure that the first term lies in
$D^{b}_{zc}$. The last term lies in $D^{b}_{zc}$ by Corollary 3.11, so we win.
∎

We also deduce the proper base change theorem.

######  Theorem 3.15.

Let

$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\prime}}$$\scriptstyle{f^{\prime}}$$\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{Y}$

be a Cartesian diagram of rigid spaces over $K$, with $f$ proper. Then for any
$\mathscr{F}\in D^{(b)}_{zc}(X,\mathbf{Z}/n)$, the natural base change map
$g^{\ast}Rf_{\ast}\mathscr{F}\to Rf^{\prime}_{\ast}g^{\prime\ast}\mathscr{F}$
is an isomorphism.

###### Proof.

This easily splits into the two disjoint cases where $p\nmid n$ or $n=p^{a}$.
When $p\nmid n$, the result follows from Huber’s much more general base change
results [Hub96, Theorem 4.1.1.(c)]. We may thus assume that $n=p^{a}$. By
Theorem 3.10, we know that $g^{\ast}Rf_{\ast}\mathscr{F}$ and
$Rf^{\prime}_{\ast}g^{\prime\ast}\mathscr{F}$ are Zariski-constructible, hence
overconvergent, so it suffices to show that the base change map induces an
isomorphism on stalks at all rank one geometric points $\overline{y}\to
Y^{\prime}$. By two applications of [Hub96, Example 2.6.2], we compute that
$(g^{\ast}Rf_{\ast}\mathscr{F})_{\overline{y}}\simeq
R\Gamma(X\times_{Y}\overline{g(y)},\mathscr{F})$ and
$(Rf^{\prime}_{\ast}g^{\prime\ast}\mathscr{F})_{\overline{y}}\simeq
R\Gamma(X\times_{Y}\overline{y},\mathscr{F})$. We now conclude by Lemma 3.25.
∎

We end this section by recording that the equivalence in Proposition 3.7 is
compatible with all the operations we have seen so far.

######  Proposition 3.16.

Fix an affinoid $K$-algebra $A$, and write
$\mathcal{S}=\mathop{\mathrm{Spec}}\,A$ and $S=\mathop{\mathrm{Spa}}A$. The
functor $(-)^{an}:D^{b}_{c}(\mathcal{X},\mathbf{Z}/n)\to
D^{b}_{zc}(X,\mathbf{Z}/n)$ for finite type $\mathcal{S}$-schemes
$\mathcal{X}$ with $X=\mathcal{X}^{an}$ is compatible with $\otimes$,
$\mathop{R\mathscr{H}\mathrm{om}}$ when the first argument has finite Tor
dimension, $f^{\ast}$, $Rf_{\ast}$, $Rf_{!}$ for compactifiable $f$, and
$Ri^{!}$ for any finite morphism $i$. If $p\nmid n$, it is compatible with
$Rf^{!}$ for compactifiable $f$.

###### Proof.

The compatibilities for $f^{\ast}$, $\otimes$, and $j_{!}$ for open immersions
$j$ are easy and left to the reader. The compatibility for $Rf_{\ast}$ is a
special case of Proposition 3.7 (1). For proper $f$, the claim for
$Rf_{\ast}=Rf_{!}$ is [Hub96, Theorem 3.7.2] This implies the claim for
$Rf_{!}$ for compactifiable $f$. The result for $Ri^{!}$ in the case of a
closed immersion follows from the triangle $Ri^{!}\to i^{\ast}\to
i^{\ast}Rj_{\ast}j^{\ast}\to$ and its analytic counterpart, where $j$ is the
complementary open immersion, using the known compatibilities for $i^{\ast}$,
$j^{\ast}$, and $Rj_{\ast}$. Given these compatibilities, the claim for
$\mathop{R\mathscr{H}\mathrm{om}}$ now follows by induction on the dimension,
imitating the devissage carried out in Corollary 3.14. The result for $Ri^{!}$
for general finite maps $i:\mathcal{X}\to\mathcal{Y}$ can be proven by
following the argument in Corollary 3.12.

Finally, the claim for $Rf^{!}$ in the case $(n,p)=1$ is local on the source,
so we may factor $f$ as $g\circ i$ where $g$ is smooth of some pure relative
dimension $d$ and $i$ is a closed immersion. Then $Rf^{!}=Ri^{!}\circ
Rg^{!}=Ri^{!}\circ g^{\ast}[2d](d)$ by Poincaré duality for schemes, and
$Rf^{an!}=Ri^{an!}\circ g^{an\ast}[2d](d)$ by Poincaré duality for rigid
spaces as in Huber’s book. The result now follows from the known
compatibilities for $g^{\ast}$ and $Ri^{!}$. ∎

###  3.4. Verdier duality

It remains to discuss Verdier duality. Recall that if $p$ is invertible in the
coefficient ring $\mathbf{Z}/n$, then [Hub96, §7] shows that any separated
taut morphism $f:X\to Y$ of rigid spaces over $K$ induces a well-behaved
functor $Rf_{!}:D(X,\mathbf{Z}/n)\to D(Y,\mathbf{Z}/n)$ with a well-behaved
right adjoint $Rf^{!}$. In particular, for any separated taut rigid space $X$
and any $n$ prime to $p$, we can define the dualizing complex
$\omega_{X}=R\pi_{X}^{!}(\mathbf{Z}/n)$, where
$\pi_{X}:X\to\mathop{\mathrm{Spa}}\,K$ is the structure map. We shall
construct dualizing complexes in a different way using results from [ILO14];
our construction works without restriction on $p$, and is equivalent to
Huber’s if $(p,n)=1$. We will need the following form of unbounded BBDG gluing
[BBD82] for complexes.

######  Lemma 3.17.

Let $(\mathcal{C},\mathcal{O})$ be a ringed site with a final object $X$ and
fiber products, and with enough points. Let $\mathcal{B}$ be a collection of
open subobjects $U\subset X$ such that $X=\cup_{U\in\mathcal{B}}U$ and for all
$U,V\in\mathcal{B}$ we have $U\cap V=\cup_{W\in\mathcal{B},W\subset U\cap
V}W$.

Suppose we are given objects $K_{U}\in D(U,\mathcal{O})$ for all
$U\in\mathcal{B}$ together with isomorphisms $\rho_{V}^{U}:K_{U}|_{V}\to
K_{V}$ for all $V\subset U$ with $V,U\in\mathcal{B}$ which are compatible with
composition. Finally, suppose that $\mathrm{Ext}^{i}(K_{U},K_{U})=0$ for all
$U\in\mathcal{B}$ and all $i<0$.

Then there exists a pair $(K,\\{\rho_{U}\\}_{U\in\mathcal{B}})$ consisting of
an object $K\in D(X,\mathcal{O})$ and isomorphisms $\rho_{U}:K|_{U}\to K_{U}$
such that $\rho_{V}^{U}\circ\rho_{U}=\rho_{V}$ for all $V\subset U$ with
$V,U\in\mathcal{B}$. The pair $(K,\\{\rho_{U}\\})$ is unique up to unique
isomorphism.

Recall that a subobject $U\subset X$ is open if the associated map
$\mathrm{Sh}(\mathcal{C}_{U})\to\mathrm{Sh}(\mathcal{C})$ is an open immersion
of topoi, cf. [Sta18, Tag 08M0] for the latter notion. Note that unlike the
unbounded gluing results proved in [LO08] or [Sta18, Tag 0DCC], we do not
assume here that the underlying site locally has finite cohomological
dimension. The price to pay is that we only prove gluing for open covers.

###### Proof.

This follows from the proof of [Sta18, Tag 0D6C]. ∎

Next, we recall some results from [ILO14] on the existence and uniqueness of
étale dualizing complexes for a fairly general class of noetherian schemes.
For this, we need good dimension functions.

######  Proposition 3.18.

Let $\mathcal{X}$ be a Noetherian universally catenary scheme whose
irreducible components are equicodimensional in the sense of EGA, i.e. such
that $\dim(\mathcal{O}_{\mathcal{Y},y})=\dim\mathcal{Y}$ for every irreducible
component $\mathcal{Y}\subset\mathcal{X}$ and every closed point
$y\in\mathcal{Y}$. Then the function $\delta(x)=\dim\overline{\\{x\\}}$ is a
dimension function on $\mathcal{X}$, which we call the _canonical_ dimension
function.

The hypotheses here are satisfied for any scheme $\mathcal{X}$ of finite type
over $\mathbf{Z}$, over a field, or over $\mathop{\mathrm{Spec}}A$ for some
affinoid $K$-algebra $A$. We will only need the latter case.

###### Proof.

By [ILO14, Corollaire XIV.2.4.4], the map $x\mapsto\dim\overline{\\{x\\}}$
defines a dimension function on each irreducible component of $\mathcal{X}$.
Since these functions agree on overlaps of irreducible components, this
implies the claim. ∎

In the next theorem, we shall use the language introduced in [ILO14, §XVII].
In particular, we shall use the notion of a potential dualizing complex from
[ILO14, §XVII.2]. Recall that such a complex on a scheme $\mathcal{X}$
equipped with a dimension function $\delta$ is an object
$\omega_{\mathcal{X}}\in D^{+}(\mathcal{X},\mathbf{Z}/n)$ equipped with some
additional data: one has specified isomorphisms
$R\Gamma_{\overline{x}}(\omega_{\mathcal{X}})\cong\mathbf{Z}/n[2\delta(x)](\delta(x))$,
called pinnings, for each geometric point $\overline{x}\to\mathcal{X}$ lying
over a point $x\in\mathcal{X}$, and these isomorphisms are required to be
compatible with immediate specializations in the appropriate sense. The
following theorem asserts that such complexes exist in large generality, and
have good properties.

######  Theorem 3.19 (Existence of dualizing complexes in algebraic geometry).

Let $\mathcal{X}$ be an excellent Noetherian $\mathbf{Z}[1/n]$-scheme
satisfying the hypotheses of Proposition 3.18, and set $\Lambda=\mathbf{Z}/n$.
Then $\mathcal{X}$ admits a potential dualizing complex
$\omega_{\mathcal{X}}\in D^{b}_{ctf}(\mathcal{X},\Lambda)$ relative to the
canonical dimension function, which is unique up to unique isomorphism. The
functor
$\mathbf{D}_{\mathcal{X}}(-)=\mathop{R\mathscr{H}\mathrm{om}}(-,\omega_{\mathcal{X}})$
preserves $D^{b}_{c}(\mathcal{X},\Lambda)$ and the biduality map
$\mathscr{F}\to\mathbf{D}_{\mathcal{X}}\mathbf{D}_{\mathcal{X}}\mathscr{F}$ is
an isomorphism for all $\mathscr{F}\in D^{b}_{c}(\mathcal{X},\Lambda)$.

###### Proof.

The existence and uniqueness is [ILO14, Theorem XVII.5.1.1], while the rest
follows from [ILO14, Theoreme XVII.6.1.1]. ∎

######  Remark 3.20.

Choose $\mathcal{X}$ as in Theorem 3.19. Assume additionally than
$\mathcal{X}$ is regular of (locally constant) dimension $d$. Then the twisted
constant sheaf $\mathbf{Z}/n[2d](d)$ comes equipped with the required pinning
data thanks to absolute cohomological purity [ILO14, Theorem XVI.3.1.1], so it
follows that there is a unique isomorphism
$\mathbf{Z}/n[2d](d)\cong\omega_{\mathcal{X}}$ compatible with the pinnings.

Using the preceding results in algebraic geometry and the algebraization
results in Proposition 3.7, we construct dualizing complexes in rigid geometry
using Lemma 3.17 on gluing.

######  Theorem 3.21 (Existence of dualizing complexes in rigid geometry).

Let $X$ be a rigid space over $K$.

  1. (1)

Existence: There exists a natural dualizing complex $\omega_{X}\in
D^{(b)}_{zc}(X,\mathbf{Z}/n)$, characterized up to unique isomorphism by the
requirement that its formation commutes with passage to open subsets and is
given by the algebraization (in the sense of Proposition 3.7) of the potential
dualizing complexes from Theorem 3.19 when $X$ is affinoid. Moreover, one has
$\omega_{X}\cong(\mathbf{Z}/n)[2d](d)$ for $X$ smooth of pure dimension $d$,
and canonical isomorphisms $\omega_{Z}\simeq Ri^{!}\omega_{X}$ for any finite
morphism $i:Z\to X$.

  2. (2)

$!$-compatibility for $(n,p)=1$: If $X$ is separated and taut and $(n,p)=1$,
then $\omega_{X}\cong R\pi_{X}^{!}(\mathbf{Z}/n)$ where
$\pi_{X}:X\to\mathop{\mathrm{Spa}}K$ is the structure map.

  3. (3)

Biduality: The dualizing functor
$\mathbf{D}_{X}(-)=\mathop{R\mathscr{H}\mathrm{om}}(-,\omega_{X})$ induces a
contravariant self-equivalence of $D^{(b)}_{zc}(X,\mathbf{Z}/n)$ satisfying
biduality $\mathrm{id}\cong\mathbf{D}_{X}\circ\mathbf{D}_{X}$ via the natural
map.

  4. (4)

Duality and finite morphisms: For any finite morphism $i:Z\to X$, there are
natural identifications of functors
$Ri^{!}\mathbf{D}_{X}\cong\mathbf{D}_{Z}i^{\ast}$ and
$i_{\ast}\mathbf{D}_{Z}\cong\mathbf{D}_{X}i_{\ast}$.

  5. (5)

Duality and open immersions: For a Zariski-open immersion $j:U\to X$, there
are natural isomorphisms $j^{*}\mathbf{D}_{X}\simeq\mathbf{D}_{U}j^{*}$ and
$Rj_{*}\mathbf{D}_{U}\simeq\mathbf{D}_{X}j_{!}$.

  6. (6)

Base change: If $L/K$ is an extension of nonarchimedean fields, with
$a^{\ast}:D_{zc}(X,\mathbf{Z}/n)\to D_{zc}(X_{L},\mathbf{Z}/n)$ the natural
pullback map, then $a^{\ast}\omega_{X}\cong\omega_{X_{L}}$.

  7. (7)

Compatibility with algebraic geometry: If $X=\mathcal{X}^{an}$ for a finite
type $K$-scheme $\mathcal{X}$, then $\omega_{X}=(\omega_{\mathcal{X}})^{an}$.

###### Proof.

Let us first construct $\omega_{X}$ by glueing together the analytifications
of the dualizing complexes coming from Theorem 3.19. Let
$U=\mathop{\mathrm{Spa}}A$ be any affinoid rigid space, and set
$\mathcal{U}=\mathop{\mathrm{Spec}}A$. Then $\mathcal{U}$ satisfies the
hypotheses of Theorem 3.19. Let $\omega_{\mathcal{U}}\in
D^{b}_{c}(\mathcal{U},\mathbf{Z}/n)$ be the dualizing complex provided by that
theorem.

By Propositions 3.7 and 3.16, we have an equivalence
$(-)^{an}:D^{b}_{c}(\mathcal{U},\mathbf{Z}/n)\to D^{b}_{zc}(U,\mathbf{Z}/n)$
compatible with $\mathop{R\mathscr{H}\mathrm{om}}$s. We now define
$\omega_{U}:=(\omega_{\mathcal{U}})^{an}$, so
$(\mathbf{D}_{\mathcal{U}}\mathscr{F})^{an}\cong\mathbf{D}_{U}(\mathscr{F}^{an})$
for every $\mathscr{F}\in D^{b}_{c}(\mathcal{U},\mathbf{Z}/n)$. Since the
dualizing functor $\mathbf{D}_{\mathcal{U}}$ induces a contravariant self-
equivalence on $D^{b}_{c}(\mathcal{U},\mathbf{Z}/n)$ which satisfies
biduality, and $(-)^{an}$ is an equivalence of categories, we now conclude the
analogous results for $\mathbf{D}_{U}$.

To construct $\omega_{X}$ over an arbitrary rigid space $X$, we apply Lemma
3.17, taking $\mathcal{B}$ to be the collection of affinoid opens $U\subset
X$, and letting $K_{U}=\omega_{U}$ as constructed in the previous paragraph.
Note that the full faithfulness in Proposition 3.7 and Theorem 3.19 show that
$\mathbf{Z}/n\cong\mathop{R\mathscr{H}\mathrm{om}}(\omega_{U},\omega_{U})$ for
all $U\in\mathcal{B}$, so all negative self-exts vanish. Moreover, for any
inclusion of open affinoid subsets $g:V\subset U$, there is a natural
isomorphism $g^{\ast}\omega_{U}\cong\omega_{V}$ compatible with compositions
by Lemma 3.22 below (applied with $d=0$). The gluing lemma now applies, and
gives a unique $\omega_{X}\in D(X,\mathbf{Z}/n)$ equipped with a transitive
system of isomorphisms $\omega_{X}|_{U}\simeq\omega_{U}$ for all open
affinoids $U\subset X$.

We now prove this construction has all the required properties.

  1. (1)

By construction, the complex $\omega_{X}$ is characterized by the properties
demanded in the first sentence of (1); these also characterize $\omega_{X}$
uniquely (up to unique isomorphism) by the uniqueness assertions in Theorem
3.19 and Lemma 3.22 through the equivalence in Proposition 3.7. It is also
clear from the construction and Remark 3.20 that
$\omega_{X}|_{X^{sm}}\cong\mathbf{Z}/n[2d](d)$, where $d$ is the dimension.
For the compatibility with $Ri^{!}$ for finite morphisms, we reduce to the
affinoid case following our construction, and use compatibility of the
analytification functor with $Ri^{!}$ (Proposition 3.16) to reduce to the
corresponding result in algebraic geometry ([ILO14, Proposition XVII.4.1.2]).
The compatibility with restriction to open subsets is clear from our
construction. As these properties guarantee uniqueness of $\omega_{X}$ up to
isomorphism, we are done with (1).

  2. (2)

The identification with $R\pi_{X}^{!}(\mathbf{Z}/n)$ can be checked locally,
where it follows by factoring $\pi_{X}$ as a closed immersion followed by a
smooth map and using the results on Poincare duality proved in Huber’s book.

  3. (3)

This follows from the corresponding assertion in the affinoid case (which was
explained above whilst constructing $\omega_{X}$) as the property of being
Zariski-constructible is local in the analytic topology (Theorem 3.5).

  4. (4)

This is a formal argument given duality and known adjunctions. For
compatibility with $i_{*}$, fix $G\in D^{(b)}_{zc}(Z,\mathbf{Z}/n)$. Then we
have isomorphisms

$\displaystyle i_{*}\mathbf{D}_{Z}(G)=$ $\displaystyle
i_{*}\mathop{R\mathscr{H}\mathrm{om}}(\mathbf{Z}/n,\mathbf{D}_{Z}(G))$
$\displaystyle=$ $\displaystyle
i_{*}\mathop{R\mathscr{H}\mathrm{om}}(G,\omega_{Z})$ $\displaystyle=$
$\displaystyle i_{*}\mathop{R\mathscr{H}\mathrm{om}}(G,i^{!}\omega_{X})$
$\displaystyle=$
$\displaystyle\mathop{R\mathscr{H}\mathrm{om}}(i_{*}G,\omega_{X})$
$\displaystyle=$ $\displaystyle\mathbf{D}_{X}(i_{*}G).$

where the second isomorphism is by biduality, the third by
$\omega_{Z}=i^{!}\omega_{X}$, and the fourth by the defining property of
$i^{!}$. Comparing the first and last term gives
$i_{*}\mathbf{D}_{Z}=\mathbf{D}_{X}i_{*}$.

For compatibility with $i^{!}$, fix additionally $F\in
D^{(b)}_{zc}(X,\mathbf{Z}/n)$. Then we have bifunctorial isomorphisms

$\displaystyle\mathrm{RHom}(i^{*}\mathbf{D}_{X}(F),G)=$
$\displaystyle\mathrm{RHom}(\mathbf{D}_{X}(F),i_{*}G)$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(\mathbf{D}_{X}(i_{*}G),F)$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(i_{*}\mathbf{D}_{Z}(G),F)$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(\mathbf{D}_{Z}(G),Ri^{!}F)$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(\mathbf{D}_{Z}(Ri^{!}F),G),$

where first equality is by adjunction for $(i^{*},i_{*})$, the second and last
by duality, the third by the equality
$i_{*}\mathbf{D}_{Z}=\mathbf{D}_{X}i_{*}$ we just showed, and the fourth by
adjunction for $(i_{*},i^{!})$.

  5. (5)

The compatibility with $j^{*}$ is built into the construction. The rest is
again a formal argument using duality and known adjunctions. For $F\in
D^{(b)}_{zc}(U,\mathbf{Z}/n)$ and $G\in D^{(b)}_{zc}(X,\mathbf{Z}/n)$, we have

$\displaystyle\mathrm{RHom}(F,Rj_{*}\mathbf{D}_{U}(G))=$
$\displaystyle\mathrm{RHom}(j^{*}F,\mathbf{D}_{U}(G))$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(G,\mathbf{D}_{U}(j^{*}F))$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(G,j^{*}\mathbf{D}_{X}(F))$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(j_{!}G,\mathbf{D}_{X}(F))$ $\displaystyle=$
$\displaystyle\mathrm{RHom}(F,\mathbf{D}_{X}(j_{!}G)),$

with first equality by the adjunction $(j^{*},Rj_{*})$, the second and last by
duality, third by $j^{*}\mathbf{D}_{X}=\mathbf{D}_{U}j^{*}$, and the fourth by
the adjunction $(j_{!},j^{*})$.

  6. (6)

By our construction of dualizing complexes, it suffices to construct a natural
system of such isomorphisms over affinoids. Thus, assume $X=\mathrm{Spa}(A)$
is affinoid with base change $X_{L}=\mathrm{Spa}(A_{L})$. Write $\mathcal{X}$
and $\mathcal{X}_{L}$ for the natural algebraization, and let
$f:\mathcal{X}_{L}\to\mathcal{X}$ be the natural map. It is enough to
construct a natural isomorphism
$f^{*}\omega_{\mathcal{X}}\simeq\omega_{\mathcal{X}_{L}}$. We claim this
follows from [ILO14, Proposition 4.1.1] (and uniqueness of potential dualizing
complexes). To apply this lemma, we need to know that $f$ is a regular map,
and that the dimension function
$y\mapsto\delta^{\prime}(y):=\delta(f(y))-\mathrm{codim}_{f^{-1}(f(y))}(y)$ on
$\mathcal{X}_{L}$ agrees with the standard dimension function
$y\mapsto\dim(\overline{\\{y\\}})$. The regularity of $f$ is discussed in the
last paragraph of the proof of Proposition 3.24 below, and essentially comes
from [And74]. To obtain agreement of the dimension functions, it suffices to
know that $\delta^{\prime}(y)=0$ for a closed point $y$ (as any point on
$\mathcal{X}_{L}$ is linked to a closed point by a finite chain of immediate
specializations). Thus, we must check that
$\dim(\overline{\\{f(y)\\}})=\mathrm{codim}_{f^{-1}(f(y))}(y)$. Since $y$ is a
closed point, this follows from [Con99, Lemma 2.1.5] applied to the affinoid
algebra of functions on integral scheme
$\overline{\\{y\\}}\subset\mathcal{X}$.

  7. (7)

To deduce this from our construction, it suffices to show the following: if
$\mathcal{X}=\mathrm{Spec}(A)$ is an affine finite-type $K$-scheme and
$U=\mathrm{Spa}(B)\subset X=\mathcal{X}^{an}$ is an open affinoid with natural
map $\nu:U\to\mathcal{X}$, then there is a natural isomorphism
$\nu^{*}\omega_{\mathcal{X}}\simeq\omega_{U}$. This follows by Lemma 3.22
below applied to the map $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$ with $d=0$.

∎

In the course of the previous proof, we used the following lemma.

######  Lemma 3.22.

Let $g:V=\mathop{\mathrm{Spec}}B\to U=\mathop{\mathrm{Spec}}A$ be a regular
morphism between excellent affine $\mathbf{Q}$-schemes of finite Krull
dimension. Suppose moreover that $U$ and $V$ have equicodimensional
irreducible components, that $g$ maps closed points to closed points, and that
$V\times_{U}\mathop{\mathrm{Spec}}\kappa(g(x))$ is equidimensional of
dimension $d$ for all closed points $x\in V$ for some (constant) $d\geq 0$.
Then there is a unique isomorphism $g^{\ast}\omega_{U}[2d](d)\cong\omega_{V}$
compatible with the pinnings.

This lemma applies if $g$ arises from an étale map of affinoid rigid spaces
$\mathop{\mathrm{Spa}}B\to\mathop{\mathrm{Spa}}A$, with $d=0$. This is the
only case we will need.

###### Proof.

This is a variant of the argument used in Theorem 3.21 (6). By [ILO14, Exp.
XVII, Prop. 4.1.1], $g^{\ast}\omega_{U}$ is a potential dualizing complex for
the dimension function $\tilde{\delta}:|V|\to\mathbf{Z}$ defined by
$\tilde{\delta}(x)=\dim\overline{\\{g(x)\\}}-\mathrm{codim}_{g^{-1}(g(x))}(x)$.
Our assumptions guarantee that $\tilde{\delta}(x)+d=\dim\overline{\\{x\\}}=0$
for all closed points $x$. Since the difference of any two dimension functions
is locally constant, this implies that
$\tilde{\delta}(x)+d=\dim\overline{\\{x\\}}$ for all $x\in V$, and therefore
that $g^{\ast}\omega_{U}[2d](d)$ is a potential dualizing complex for the
canonical dimension function on $V$. We now conclude by the uniqueness of
potential dualizing complexes. ∎

######  Remark 3.23 (Duality and proper maps).

In Theorem 3.21, we have only discussed the functor $Rf^{!}$ when $p$ is
invertible in the coefficient ring, or when $f$ is finite. However, we expect
that for any proper map $f:X\to Y$, the functor
$Rf_{*}:D_{zc}^{(b)}(X,\mathbf{Z}/p^{n})\to D_{zc}^{(b)}(Y,\mathbf{Z}/p^{n})$
admits a right adjoint $Rf^{!}$ naturally isomorphic to
$\mathbf{D}_{X}f^{\ast}\mathbf{D}_{Y}$. One can check that this expectation
holds iff there is a natural isomorphism $\mathbf{D}_{Y}Rf_{\ast}\cong
Rf_{\ast}\mathbf{D}_{X}$. For $f$ proper and smooth, ongoing work of Zavyalov
confirms these expectations.

###  3.5. Miscellany

We collect some auxiliary results.

First, we note that the entire formalism is compatible with changing the
nonarchimedean base field. More precisely, let $K\to L$ be an extension of
characteristic zero nonarchimedean fields. For any rigid space
$X/\mathop{\mathrm{Spa}}K$, there is a natural map of étale sites
$X_{L,\mathrm{\acute{e}t}}\to X_{\mathrm{\acute{e}t}}$ which induces a
pullback functor $D(X)\to D(X_{L})$ sending $D_{zc}$ into $D_{zc}$.

######  Proposition 3.24.

Notation as above, the change of base field functors $D_{zc}(X)\to
D_{zc}(X_{L})$ are compatible (under the appropriate boundedness conditions)
with the operations $f^{\ast}$, $\otimes$, $\mathop{R\mathscr{H}\mathrm{om}}$,
Verdier duality, $Rf_{\ast}$ for proper $f$, $Rf_{!}$ and $Rf_{\ast}$ on lisse
complexes for Zariski-compactifiable morphisms $f$, and $Rf^{!}$ if either $f$
is a finite morphism or $p$ is invertible in the coefficient ring.

###### Proof.

The compatibilities for $f^{\ast}$, $\otimes$ and $j_{!}$ are trivial. The
compatibilities for $Rf_{\ast}$ in the proper case and $Rj_{\ast}$ in the
Zariski-open lisse case are the hardest; the rest follow from these.

For $Rf_{\ast}$ in the case of a proper map $f:Y\to X$, one easily reduces to
the two disjoint cases $p\nmid n$ and $n=p^{a}$. The first case follows from
Huber’s general base change theorem [Hub96, Theorem 4.1.1.b]. The second case
can be reduced, via [Hub96, Theorem 4.1.1.b’], to the situation where $K$ and
$L$ are algebraically closed, and then by Theorem 3.10 it can be checked on
stalks at classical points of $X_{L}$ which map to classical points of $X$. At
these points it reduces to Lemma 3.25 below.

For $Rj_{\ast}$ on lisse sheaves in the case of a Zariski-open immersion
$j:U\to X$, we can assume that $X=\mathop{\mathrm{Spa}}A$ is affinoid. Write
$j_{L}:U_{L}\to X_{L}=\mathop{\mathrm{Spa}}A\widehat{\otimes}_{K}L$ for the
base change, and let
$j^{alg}:\mathcal{U}\to\mathcal{X}=\mathop{\mathrm{Spec}}A$ and
$j^{alg}_{L}:\mathcal{U}_{L}\to\mathcal{X}_{L}=\mathop{\mathrm{Spec}}A_{L}$ be
the evident algebraizations. By (multiple applications of) Proposition 3.7, we
are reduced to proving that $g^{\ast}Rj^{alg}_{\ast}\mathcal{F}\cong
Rj_{L,\ast}^{alg}g^{\prime\ast}\mathcal{F}$ for any lisse sheaf $\mathcal{F}$
on $\mathcal{U}$, where $g:\mathcal{X}_{L}\to\mathcal{X}$ is the natural map
and $g^{\prime}:\mathcal{U}_{L}\to\mathcal{U}$ is its base change to
$\mathcal{U}$. Since $g$ is regular (see next paragraph), this follows from
regular base change [ILO14, Exp. XVII, Prop. 4.2.1]. (The results for
$\mathop{R\mathscr{H}\mathrm{om}}$ and $Ri^{!}$ can be handled in an entirely
analogous way, with the endgame supplied by [ILO14, Exp. XVII, Prop. 4.2.2 and
Cor. 4.2.3].)

The claimed regularity of $A\to A\widehat{\otimes}_{K}L$ can be reduced by
Noether normalization to the regularity of the map $K\left\langle
x_{1},\dots,x_{n}\right\rangle\to L\left\langle
x_{1},\dots,x_{n}\right\rangle$. By standard excellence properties of affinoid
rings, regularity of this map can be verified after $\mathfrak{m}$-adic
completion at all maximal ideals $\mathfrak{m}$ in the source. This finally
reduces us to the fact that for any separable extension of nonarchimedean
fields $L/K$ the ring map $K[[x_{1},\dots,x_{n}]]\to L[[x_{1},\dots,x_{n}]]$
is regular, which follows from the formal smoothness of this map, cf. [And74].
∎

In the previous proof, we used the following lemma.

######  Lemma 3.25.

Let $C^{\prime}/C/\mathbf{Q}_{p}$ be an extension of algebraically closed
nonarchimedean fields. Then for any proper rigid space $X/C$ and any
$\mathscr{F}\in D^{b}_{zc}(X,\mathbf{Z}/p^{a})$, the natural map
$R\Gamma(X,\mathscr{F})\to R\Gamma(X_{C^{\prime}},\mathscr{F}_{C^{\prime}})$
is an isomorphism.

###### Proof.

By an easy induction on $a$ and Proposition 3.6, we can assume that
$\mathscr{F}=f_{\ast}\mathbf{F}_{p}$ for some finite map $f:X^{\prime}\to X$.
Replacing $X$ by $X^{\prime}$, we can assume further that
$\mathscr{F}=\mathbf{F}_{p}$ is constant. By two applications of the primitive
comparison theorem, it’s enough to check that the natural map

$R\Gamma(X,\mathcal{O}^{+}_{X}/p)\otimes_{\mathcal{O}_{C}/p}\mathcal{O}_{C^{\prime}}/p\to
R\Gamma(X_{C^{\prime}},\mathcal{O}^{+}_{X_{C^{\prime}}}/p)$

is an almost isomorphism. This can be deduced from a purely local statement:
if $U=\mathop{\mathrm{Spa}}A/C$ is any affinoid, then the natural map

$R\Gamma(U,\mathcal{O}^{+}_{U}/p)\otimes_{\mathcal{O}_{C}/p}\mathcal{O}_{C^{\prime}}/p\to
R\Gamma(U_{C^{\prime}},\mathcal{O}^{+}_{U_{C^{\prime}}}/p)$

is an almost isomorphism.

To prove the local statement, choose a perfectoid $\mathbf{Z}_{p}^{d}$-torsor
$A\to A_{\infty}$. Then

$R\Gamma(U,\mathcal{O}^{+}_{U}/p)\cong^{a}R\Gamma_{cts}(\mathbf{Z}_{p}^{d},A_{\infty}^{\circ}/p),$

and similarly for $U_{C^{\prime}}$. The result now follows by writing
$R\Gamma_{cts}(\mathbf{Z}_{p}^{d},-)$ as the usual $d+1$-term Koszul complex
and observing that the natural map
$A_{\infty}^{\circ}\widehat{\otimes}_{\mathcal{O}_{C}}\mathcal{O}_{C^{\prime}}\to
A_{C^{\prime},\infty}^{\circ}$ is an almost isomorphism (e.g., because both
sides provide perfectoid rings of definition for the Tate ring
$A_{\infty}\widehat{\otimes}_{C}C^{\prime}$). ∎

Secondly, with biduality in hand, we can somewhat extend our results on
pushfoward.

######  Proposition 3.26.

Let $f:X\to Y$ be a Zariski-compactifiable morphism, and let $\mathscr{F}\in
D_{zc}^{(b)}(X,\mathbf{Z}/n)$ be an object such that one of the following
holds true:

  1. (1)

$\mathscr{F}$ is lisse or $\mathbf{D}_{X}\mathscr{F}$ is lisse, or

  2. (2)

$\mathscr{F}=j^{\ast}\mathscr{F}^{\prime}$ for some compactification
$X\overset{j}{\to}X^{\prime}\overset{\overline{f}}{\to}Y$ and some
$\mathscr{F}^{\prime}\in D_{zc}^{(b)}(X^{\prime},\mathbf{Z}/n)$.

Then $Rf_{\ast}\mathscr{F}$ and $Rf_{!}\mathscr{F}$ lie in
$D^{(b)}_{zc}(Y,\mathbf{Z}/n)$.

It follows from the above proposition that if $f$ is a Zariski locally closed
immersion, any $\mathscr{F}$ satisfying (1) automatically satisfies (2): we
may simply take $\mathscr{F}^{\prime}=Rf_{*}\mathscr{F}$ by the proposition.

###### Proof.

For the first case, the result is already proved for $\mathscr{F}$ lisse.
Suppose now that $\mathbf{D}_{X}\mathscr{F}$ is lisse, and choose a
compactification $X\overset{j}{\to}X^{\prime}\overset{\overline{f}}{\to}Y$. We
first show that $Rj_{\ast}\mathscr{F}$ is Zariski-constructible. To see this,
use Theorem 3.21 to write

$Rj_{\ast}\mathscr{F}\cong
Rj_{\ast}\mathbf{D}_{X}\mathbf{D}_{X}\mathscr{F}\cong\mathbf{D}_{X^{\prime}}j_{!}\mathbf{D}_{X}\mathscr{F}.$

Then $\mathbf{D}_{X}\mathscr{F}$ is lisse by assumption, so
$j_{!}\mathbf{D}_{X}\mathscr{F}$ is Zariski-constructible by Corollary 3.11,
and then duality preserves Zariski-constructibility. Now the triangle
$j_{!}\mathscr{F}\to Rj_{\ast}\mathscr{F}\to
i_{\ast}i^{\ast}Rj_{\ast}\mathscr{F}\to$ shows that $j_{!}\mathscr{F}$ is also
Zariski-constructible. Applying $R\overline{f}_{\ast}$ now gives the claim.

For the second case, let $i:Z\to X^{\prime}$ be the complementary closed
immersion. Then we get a triangle $Rf_{!}\mathscr{F}\to
R\overline{f}_{\ast}\mathscr{F}^{\prime}\to R(\overline{f}\circ
i)_{\ast}i^{\ast}\mathscr{F}^{\prime}\to$, and the second and third terms are
Zariski-constructible by Theorem 3.10. Likewise, we get a triangle
$R(\overline{f}\circ i)_{\ast}Ri^{!}\mathscr{F}^{\prime}\to
R\overline{f}_{\ast}\mathscr{F}^{\prime}\to Rf_{\ast}\mathscr{F}\to$, and the
first two terms are Zariski-constructible by Theorem 3.10 and Proposition
3.12. ∎

######  Remark 3.27 (Unbounded variants).

We briefly discuss without proof how the results discussed above extend to the
unbounded derived category. We shall need the following notion:

######  Definition 3.28.

Given a non-archimedean base field $K$ and the coefficient ring
$\mathbf{Z}/n$, we say $(\dagger)$ holds if $\mathrm{Gal}(\overline{K}/K)$ has
finite $\ell$-cohomological dimension for all primes $\ell|n$.

This condition is very mild, and holds for example if $K$ is separably closed,
or if $K$ is a local field, or if $K$ has separably closed residue field and
$(n,p)=1$. One can also check that $(\dagger)$ is stable under replacing $K$
by $K_{x}$, where $K_{x}$ is the residue field of any rigid space $X/K$ at any
(adic) point $x\in X$. The main reason for introducing this condition is the
following:

######  Proposition 3.29.

Fix $K$ and $\Lambda=\mathbf{Z}/n$ such that $(\dagger)$ holds. Then for any
rigid space $X/K$, the derived category $D(X_{\mathrm{\acute{e}t}},\Lambda)$
is left-complete and compactly generated, and the functor
$R\lim:D(X_{\mathrm{\acute{e}t}}^{\mathbf{N}},\Lambda)\to
D(X_{\mathrm{\acute{e}t}},\Lambda)$ has bounded cohomological amplitude on any
finite-dimensional open subspace of $X$.

###### Proof.

This is well-known; cf. [Roo06] for the final statement. ∎

Let us now formulate the promised unbounded variants of the results discussed
in this paper. Let $f:X\to Y$ be a map of rigid spaces over $K$, and let
$\mathscr{F}\in D_{zc}(X,\mathbf{Z}/n)$.

  1. (1)

Pullback: The pullback $f^{*}$ takes $D_{zc}$ into $D_{zc}$.

  2. (2)

Proper pushforward: If $\mathscr{F}$ is bounded below or $(\dagger)$ holds,
then $Rf_{*}\mathscr{F}\in D_{zc}$.

  3. (3)

General pushforward: Say $\mathscr{F}$ is lisse and $f$ is Zariski-
compactifiable. If $\mathscr{F}$ is bounded below or $(\dagger)$ holds, then
$Rf_{!}\mathscr{F},Rf_{*}\mathscr{F}\in D_{zc}$.

  4. (4)

Duality: The functor
$\mathbf{D}_{X}(-)=\mathop{R\mathscr{H}\mathrm{om}}(-,\omega_{X})$ carries
$D^{(-)}_{zc}$ into $D^{(+)}_{zc}$. Furthermore, if $(\dagger)$ holds, then
$\mathbf{D}_{X}(-)$ also carries $D^{(+)}_{zc}$ into $D^{(-)}_{zc}$, and gives
an autoequivalence of $D_{zc}(X,\mathbf{Z}/n)$ satisfying biduality.

  5. (5)

$!$-pullback: If $f$ is a finite map, then $Rf^{!}$ takes $D_{zc}$ to
$D_{zc}$.

  6. (6)

$!$-pullback for good coefficients: If $(p,n)=1$, $f$ is any taut separated
map and $(\dagger)$ holds true, then $Rf^{!}$ preserves $D_{zc}$.

  7. (7)

Tensor product: $D^{(-)}_{zc}(X,\mathbf{Z}/n)$ is stable under $\otimes$
inside $D(X,\mathbf{Z}/n)$.

  8. (8)

Internal Hom: The bifunctor $\mathop{R\mathscr{H}\mathrm{om}}(-,-)$ carries
$D^{(-)}_{zc}\times D^{(+)}_{zc}$ into $D^{(+)}_{zc}$.

These assertions are all proven by using homological arguments to reduce to
the locally bounded case (using Proposition 3.29 as needed). We omit the
proofs. The most notably difficult case is (4), where the reduction to the
bounded situation is non-formal, and requires the following lemma.

######  Lemma 3.30.

Let $X$ be a $d$-dimensional rigid space. Then for any
$\mathscr{F}\in\mathrm{Sh}_{zc}(X,\mathbf{Z}/n)$, the complex
$\mathop{R\mathscr{H}\mathrm{om}}(\mathscr{F},\omega_{X})$ is concentrated in
degrees $[-2d,0]$.

In particular, for any finite-dimensional $X$, the functor $\mathbf{D}_{X}(-)$
preserves $D^{b}$ without assuming that $X$ is quasi-compact.

###### Proof.

One could deduce this from the last assertion in [ILO14, Theorem XVII.5.1.1]
using [Gab04, §3] as well as our algebraization Proposition 3.7. For the
convenience of the reader, we give a direct proof, essentially mimicing the
argument in Corollary 3.14 when $\mathscr{G}=\omega_{X}$, while controlling
the amplitude of the terms showing up.

Without loss of generality, we may assume $X$ is affinoid and reduced. We
proceed induction on $d$, the $d=0$ case being trivial. Let $j:U\to X$ be a
smooth Zariski-open subset of pure dimension $d$ with complement of dimension
$<d$ such that $\mathscr{F}|U$ is locally constant. Let
$i:Z=\mathop{\mathrm{Spa}}B\to X$ be the inclusion of the closed complement of
$U$. Applying $\mathop{R\mathscr{H}\mathrm{om}}(-,\omega_{X})$ to the triangle
$j_{!}j^{\ast}\mathscr{F}\to\mathscr{F}\to i_{\ast}i^{\ast}\mathscr{F}\to$, we
get a triangle

$i_{\ast}\mathop{R\mathscr{H}\mathrm{om}}(i^{\ast}\mathscr{F},\omega_{Z})\to\mathop{R\mathscr{H}\mathrm{om}}(\mathscr{F},\omega_{X})\to(Rj_{\ast}j^{\ast}\mathscr{F}^{\vee})[2d](d)\to,$

where we used the isomorphism $\omega_{U}\simeq\mathbf{Z}/n[2d](d)$ (coming
from the smoothness of $U$) to simplify the last term, and Theorem 3.21 (4)
for the first term. By induction, the first term here is concentrated in
degrees $[-2d+2,0]$. It thus suffices to show that
$Rj_{\ast}j^{\ast}\mathscr{F}^{\vee}$ is concentrated in degrees $[0,2d-1]$.
Proposition 3.7 identifies $Rj_{\ast}j^{\ast}\mathscr{F}^{\vee}$ as the
analytification of $Rj^{alg}_{\ast}\mathcal{G}$, where
$j^{alg}:\mathcal{U}=\mathop{\mathrm{Spec}}A-\mathop{\mathrm{Spec}}B\to\mathcal{X}=\mathop{\mathrm{Spec}}A$
is the natural algebraization of $j$ and $\mathcal{G}$ is some locally
constant sheaf on $\mathcal{U}$. By [ILO14, Exp. XVIII-A, Theorem 1.1],
$Rj^{alg}_{\ast}\mathcal{G}$ is concentrated in degrees $[0,2d-1]$, so the
result follows. ∎

###  3.6. Adic coefficients

We fix a characteristic zero nonarchimedean base field $K$ of residue
characteristic $p>0$. Fix any prime $\ell$. In this section we explain a
variant of the theory of Zariski-constructible sheaves with
$\mathbf{Z}_{\ell}$-coefficients, using the formalism666The paper [Sch17]
assumes that a prime number is topologically nilpotent in the base field $K$.
This is the reason we assume that the residue characteristic $p$ of $K$ is
$>0$. Since we only use the relatively formal aspects of [Sch17], we expect
that this assumption can be removed once a theory of adic coefficients over
nonarchimedean base fields of residue characteristic $0$ has been developed.
from [Sch17].

For a rigid space $X/K$, let $X_{v}$ denote the $v$-site of $X$ from [Sch17].
Let ${\mathbf{Z}_{\ell}}=\lim_{n}\underline{\mathbf{Z}/\ell^{n}}$ be the
displayed inverse limit of constant sheaves, regarded as a sheaf of (abstract)
rings. Any perfect complex $M\in D(\mathbf{Z}_{\ell})$ yields a “constant”
$\ell$-complete sheaf $\underline{M}:=\lim_{n}\underline{M/\ell^{n}}\in
D(X_{v},\mathbf{Z}_{\ell})$. Our goal is to build a theory of Zariski-
constructible $\mathbf{Z}_{\ell}$-complexes on a rigid space $X$ where the
locally constant objects are twisted forms of $\underline{M}$ for $M\in
D_{perf}(\mathbf{Z}_{\ell})$. We first recall the basic notion of “étale
$\mathbf{Z}_{\ell}$-sheaves” that is introduced in [Sch17] for the purposes of
defining operations.

######  Construction 3.31.

Fix $n\geq 1$. For any strictly totally disconnected perfectoid space $Y$, the
usual derived category
$D(Y_{\mathop{\mathrm{\acute{e}t}}},\mathbf{Z}/\ell^{n})$ is left-complete and
identifies with a full subcategory $D(Y_{v},\mathbf{Z}/\ell^{n})$ via pullback
along $Y_{v}\to Y_{\mathop{\mathrm{\acute{e}t}}}$. Moreover, containment in
this subcategory can be checked $v$-locally (see [Sch17, Proposition 14.10,
Proposition 14.11 (iii), Theorem 14.12 (ii)]).

For any $v$-stack $X$, let
$D_{\mathop{\mathrm{\acute{e}t}}}(X,\mathbf{Z}/\ell^{n})\subset
D(X_{v},\mathbf{Z}/\ell^{n})$ be the full subcategory spanned by objects whose
pullback along any map $Y\to X$ with $Y$ a strictly totally disconnected
perfectoid space lies in the subcategory
$D(Y_{\mathop{\mathrm{\acute{e}t}}},\mathbf{Z}/\ell^{n})\subset
D(Y_{v},\mathbf{Z}/\ell^{n})$; this condition can be checked after pullback
along a $v$-cover ([Sch17, Remark 14.14]. Imposing this condition for a
complex $K$ is equivalent to imposing it for each $v$-cohomology sheaf
$\mathcal{H}^{i}(K)$ (regarded as a complex) ([Sch17, Proposition 14.16]).
Moreover, if $X$ is a locally spatial diamond (e.g., one attached to a rigid
space), then $D_{\mathop{\mathrm{\acute{e}t}}}(X,\mathbf{Z}/\ell^{n})$ admits
a classical description: it agrees with the left-completion
$\widehat{D}(X_{et},\mathbf{Z}/\ell^{n})$ of the usual derived category
$D(X_{\mathop{\mathrm{\acute{e}t}}},\mathbf{Z}/\ell^{n})$ ([Sch17, Proposition
14.15]).

Finally, for any $v$-stack $X$, write
$D_{\mathop{\mathrm{\acute{e}t}}}(X,\mathbf{Z}_{\ell})\subset
D_{\ell-\text{comp}}(X_{v},\mathbf{Z}_{\ell})$ as the full subcategory derived
$\ell$-complete objects in $D(X_{v},\mathbf{Z}_{\ell})$ whose mod
$\ell$-reduction lies in the subcategory
$D_{\mathop{\mathrm{\acute{e}t}}}(X,\mathbf{Z}/\ell)$ mentioned above ([Sch17,
Definition 26]). If $X$ is a locally spatial diamond, then derived
$\ell$-completeness as well as the previous remark on left-completeness give
an equivalence

$\mathcal{D}_{\mathop{\mathrm{\acute{e}t}}}(X,\mathbf{Z}_{\ell})=\lim_{n}\mathcal{D}_{\mathop{\mathrm{\acute{e}t}}}(X,\mathbf{Z}/\ell^{n})\simeq\lim_{n}\widehat{\mathcal{D}}(X_{\mathop{\mathrm{\acute{e}t}}},\mathbf{Z}/\ell^{n})$
(3)

at the level of the corresponding $\infty$-categories, thus giving a classical
description of the left side in the case of rigid spaces, see [Sch17,
Proposition 26.2]. In fact, we could have defined
$D_{\mathop{\mathrm{\acute{e}t}}}(X,\mathbf{Z}_{\ell})$ as the homotopy
category of the right side above, and thus avoided ever mentioning the ambient