bibstyle.bst

# Cohomologically smooth morphisms for
étale $\mathbb{F}_{p}$-sheaves in characteristic $p$

Felix Lotter

###### Abstract.

We scrutinise the notions of cohomologically smooth morphisms and smooth
objects for the six functor formalism of étale $\mathbb{F}_{p}$-sheaves on
schemes in characteristic $p$. We show that only cohomologically étale
morphisms are cohomologically smooth in this setting. This is complemented by
a characterisation of cohomologically étale morphisms in arbitrary
characteristic. In fact, we prove that such a morphism is already étale up to
universal homeomorphism.

###### Contents

  1. 1 Introduction
  2. 2 Abstract 6-functor formalisms
  3. 3 The geometry of cohomologically étale morphisms
  4. 4 Cohomologically smooth morphisms for $\mathbb{F}_{p}$-sheaves in characteristic p
  5. 5 Smooth objects for $\mathbb{F}_{p}$-sheaves in characteristic $p$

##  1\. Introduction

The main goal of this paper is to answer the following question:
Which morphisms admit Poincaré duality for étale $\mathbb{F}_{p}$-sheaves on
schemes in characteristic $p$?
Let us clarify what we mean by this.

###  1.1. The 6-functor formalism

For a scheme $X$, one can consider the category $\mathrm{Sh}(X,\Lambda)$ of
étale sheaves of $\Lambda$-modules on $X$, where $\Lambda$ is some torsion
ring. This category is equipped with a tensor product $\operatorname{\otimes}$
and an internal $\operatorname{Hom}$ object $\operatorname{\underline{Hom}}$;
for a morphism $f:Y\to X$ there is an induced pullback functor
$f^{*}:\mathrm{Sh}(X,\Lambda)\to\mathrm{Sh}(Y,\Lambda)$ which admits a right
adjoint $f_{*}:\mathrm{Sh}(Y,\Lambda)\to\mathrm{Sh}(X,\Lambda)$. Deriving all
of these functors yields four of the six operations: The tensor product
$\operatorname{\otimes}^{L}$, the internal $\operatorname{Hom}$
$R\operatorname{\underline{Hom}}$, the pullback functor
$f^{*}:\mathcal{D}(X,\Lambda)\to\mathcal{D}(Y,\Lambda)$ and its right adjoint
$Rf_{*}:\mathcal{D}(Y,\Lambda)\to\mathcal{D}(X,\Lambda)$. The remaining two
are special in that they do not arise from deriving a functor on the abelian
level and are not even defined for all morphisms: If $f$ is a separated,
finite type morphism of qcqs schemes, one defines a functor
$f_{!}:\mathcal{D}(Y,\Lambda)\to\mathcal{D}(X,\Lambda)$ and again it admits a
right adjoint, $f^{!}:\mathcal{D}(X,\Lambda)\to\mathcal{D}(Y,\Lambda)$, called
the exceptional pullback. These $6$ functors are subject to certain relations,
e.g. the projection formula and base change formula. Roughly, all of this data
is what constitutes an abstract 6-functor formalism; but for a long time this
term was an unprecise heuristic. However, a concise definition was found
recently by L. Mann in [Man22a], vastly streamlining the previous work of Liu-
Zheng ([LZ17]) and Gaitsgory-Rozenblyum ([GR19]).

###  1.2. Cohomologically smooth morphisms

Using Mann’s definition of an abstract six functor formalism, one can give
meaning to the question if a morphism admits Poincaré duality by employing the
notion of cohomologically smooth morphisms. Cohomological smoothness was first
defined in the context of étale cohomology of diamonds by P. Scholze in
[Sch17] to capture the cohomological properties that are expected from
"smooth" morphisms. This notion makes sense in any abstract 6-functor
formalism. Roughly, a cohomologically smooth morphism is a morphism $f$ for
which the functor $f^{!}$ right adjoint to $f_{!}$ is given by twisted
pullback (in a universal way). If one considers the étale $6$-functor
formalism of $\mathbb{F}_{\ell}$-sheaves on the category of separated, finite
type schemes over a field $k=\bar{k}$ with $\ell\in k^{\times}$, all smooth
morphisms are cohomologically smooth and this amounts indeed to Poincaré
duality in étale cohomology (cf. Example 2.33).

Let us now make our question more precise:

######  Main question.

What are the cohomologically smooth morphisms for the 6-functor formalism of
étale $\mathbb{F}_{p}$-sheaves on finite type, separated schemes over a field
of characteristic $p$?

In this setting, it is clear that Poincaré duality can not be expected from
smooth morphisms in a naive way. Consider for example the projective line
$\mathbb{P}^{1}_{k}$ over an algebraically closed field $k$ of characteristic
$p$. Then one computes
$R\Gamma(\mathbb{P}^{1}_{k},\mathbb{F}_{p})\cong\mathbb{F}_{p}[0]$ via the
Artin-Schreier sequence, but Poincaré duality would predict that
$H^{0}(\mathbb{P}^{1}_{k},\mathbb{F}_{p})^{\lor}\cong
H^{2}(\mathbb{P}^{1}_{k},\mathbb{F}_{p})$. In fact, we will see that the
observation that $\mathbb{P}^{1}_{k}$ has no higher
$\mathbb{F}_{p}$-cohomology even implies that
$\mathbb{P}^{1}_{k}\to\operatorname{Spec}k$ can not be cohomologically smooth
(Proposition 4.3). Using the strong stability properties of cohomologically
smooth morphisms, we will deduce from this that the cohomologically smooth
morphisms must be quasi-finite. It turns out that this is enough to give a
full geometric characterisation of them, answering our main question:

######  Theorem (4.9).

Let $k$ be a field of characteristic $p$ and let $\mathcal{D}_{p}$ be the
6-functor formalism of étale $\mathbb{F}_{p}$-sheaves111See Theorem 2.26 for
the precise definition. on the category $\mathcal{C}$ of finite type,
separated schemes over $k$. Let $f:Y\to X$ be a morphism in $\mathcal{C}$.
Then the following are equivalent:

  1. i)

$f$ is $\mathcal{D}_{p}$-cohomologically smooth.

  2. ii)

$f$ is $\mathcal{D}_{p}$-cohomologically étale.

  3. iii)

$f^{\mathrm{perf}}$ is étale.

where $f^{\mathrm{perf}}$ denotes the perfection of $f$ (see Proposition
2.19).

###  1.3. Cohomologically étale morphisms

In the above theorem the notion of a cohomologically étale morphism appeared,
see Definition 2.61. Here, a cohomologically étale morphism is just a morphism
$f$ for which $f_{!}$ is "universally" left adjoint to $f^{*}$; in other
words, it is a cohomologically smooth morphism for which the twisted pullback
$f^{!}$ is just the pullback $f^{*}$ (Lemma 3.26).

We mentioned that it is enough to prove that cohomologically smooth morphisms
are quasi-finite to deduce the characterisation above. This is due to the
following more general description of cohomologically étale morphisms:

######  Theorem (3.27).

Let $\Lambda$ be a finite ring and $\mathcal{D}_{\Lambda}$ the 6-functor
formalism of étale $\Lambda$-sheaves on the category $\operatorname{Sch}$ of
qcqs schemes. Let $f:Y\to X$ be a morphism of finite type in
$\operatorname{Sch}$. Then the following are equivalent:

  1. i)

$f$ is $\mathcal{D}_{\Lambda}$-cohomologically smooth and quasi-finite.

  2. ii)

$f$ is $\mathcal{D}_{\Lambda}$-cohomologically étale.

  3. iii)

$f^{\textrm{awn}}$ is étale.

where $f^{\mathrm{awn}}$ denotes the absolute weak normalisation of $f$ (see
Lemma 2.18).

The notion of absolutely weakly normal schemes was introduced by D. Rydh in
[Ryd10]. In positive characteristic it agrees with the notion of perfect
schemes (i.e. schemes on which the Frobenius endomorphism is an isomorphism)
and the absolute weak normalisation is just the perfection (cf. Proposition
2.19).

While the equivalence of $i)$ and $ii)$ in the above theorem follows rather
formally, we will deduce the equivalence of $ii)$ and $iii)$ using the descent
results proven in [BM21] by B. Bhatt and A. Mathew. More precisely, we will
first observe that cohomologically étale morphisms are universally open and
topologically unramified (i.e. their diagonal is topologically open, see
Definition 3.7) and then prove that each universally open, topologically
unramified morphism in $\mathcal{C}$ is étale after absolute weak
normalisation (Theorem 3.20), using descent for étale morphisms along
universally open covers on the absolutely weakly normal site while doing
induction on the cardinality of geometric fibers.

###  1.4. Smooth objects

Finally, we address a generalisation of our main question:
If $f:Y\to X$ is a morphism in $\mathcal{C}$, for which objects
$A,B\in\mathcal{D}(Y)$ is
$f_{!}(A\operatorname{\otimes}\rule{8.5359pt}{0.42677pt})$ "universally" left
adjoint to $f^{*}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}B$ in the
étale $6$-functor formalism of $\mathbb{F}_{p}$-sheaves?
That is, we want to understand the $f$-smooth objects (see Definition 2.35).
In the absolute case, i.e. for $f:X\to\operatorname{Spec}k$, we will
characterise them as the scyscraper sheaves on $X$ which are dualisable on
their support (Proposition 5.12). We can deduce the following statement about
the relative case from this:

######  Theorem (5.14).

Assume $k=\bar{k}$ and let $f:Y\to X$ be a morphism in $\mathcal{C}$ and $A$
an $f$-smooth object. Let $Z$ be the support of $A$. Then $A|_{Z}$ is
$f|_{Z}$-smooth and $Z\to X$ is quasi-finite.

It is a nice property of quasi-finite morphisms in our setting that they are
cohomologically étale over a stratification of the target (Corollary 3.33). In
particular, the support of smooth objects is cohomologically étale over a
stratification of the target and in particular they are dualisable on the
corresponding stratification of the support (Proposition 5.15). For formal
reasons, we also know that sheaves whose support is cohomologically étale over
the target and which are dualisable on their support are smooth. However,
these are not all smooth objects which can be seen by just taking direct sums
of smooth objects (Example 5.17) and we were not able to find a full
characterisation of smooth objects in the relative case.

######  Question (5.18).

Let $f:Y\to X$ be a morphism in $\mathcal{C}$. Are $f$-smooth objects
generated by smooth objects of the form in Proposition 5.16 under direct sums
and cones?

###  1.5. Relation to the Riemann-Hilbert correspondence

The formalism of $\mathbb{F}_{p}$-sheaves in characteristic $p$ is related to
coherent cohomology in a certain sense by the $\mathrm{mod}\ p$ Riemann-
Hilbert correspondence proven by B. Bhatt and J. Lurie in [BL19]. Also see
[Mat23] for an alternative exposition. They show that $\mathbb{F}_{p}$-sheaves
in characteristic $p$ correspond to algebraic Frobenius modules via an
equivalence of categories which satisfies certain compatibilities with
pullback, pushfoward and $!$-functors. However, coherent cohomology in the
classic sense cannot be upgraded to a 6-functor formalism in a reasonable way.
This problem can be avoided by using the theory of condensed mathematics
developed by P. Scholze and D. Clausen, which might open up a possibility to
turn the $\mathrm{mod}\ p$ Riemann-Hilbert correspondence into a
correspondence of 6-functor formalisms. Cohomological smoothness would
transfer along such a correspondence which adds additional motivation to
Theorem 4.9.

###  1.6. Overview

This paper is organised into four sections.

In section 2 we recall the notion of 6-functor formalisms in the sense of
[Man22a] and the formal groundwork that we will need later on. Following
[Sch23], we construct the étale $6$-functor formalism in a generality which
allows us to pass to the absolute weak normalisation (which we also briefly
recall from [Ryd10] in this context), using the results of P. Hamacher in
[Ham19]. We then discuss the notions of smooth objects and cohomologically
smooth and cohomologically étale morphisms and prove important criteria and
stability properties. Here we essentially follow the arguments given in
[Man22b], [Sch23] and [Zav23]. This is also intended as a small overview over
the different (but mostly equivalent) formal notions and their properties in
these three sources, and the relations between them. While some of the
statements do not seem to be recorded in the literature yet, there is hardly a
claim of originality in this section.

Section 3 is concerned with cohomologically étale morphisms in the étale
6-functor formalism and in particular our geometric characterisation of them
via the absolute weak normalisation. After observing some important properties
of absolutely weakly normal schemes, we first give a description of the
cohomologically unramified morphisms (i.e. morphisms whose diagonal is
cohomologically étale, cf. Definition 2.64); namely, we prove that they are
exactly the topologically unramified ones. We then show that topologically
unramified morphisms are quasi-finite (Corollary 3.13) and that a quasi-finite
cohomologically smooth morphism is open (Proposition 3.14). All in all, we see
that cohomologically étale morphisms are universally open and topologically
unramified as they are both cohomologically smooth and cohomologically
unramified. We then use the descent results for perfect schemes from [BM21] to
prove that a morphism with these properties is étale after absolute weak
normalisation (Theorem 3.20). To finish the proof of Theorem 3.27, we prove
that a quasi-finite morphism is cohomologically étale if and only if it is
cohomologically smooth. We end the section with a digression on quasi-finite
morphisms in positive characteristic. Namely, we show that they are
cohomologically étale over a stratification of the target (Corollary 3.33)
which will have an interesting consequence for the discussion of smooth
objects (Proposition 5.15).

Using the results from section 3, we then answer our main question from above
in section 4: For this, we first show that
$\mathbb{A}_{k}^{n}\to\operatorname{Spec}k$ is not
$\mathcal{D}_{p}$-cohomologically smooth for any $n>0$ by an explicit
calculation and then use this fact to characterise the cohomologically smooth
morphisms to a field. By stability of cohomologically smooth morphisms under
base change, this shows that $\mathcal{D}_{p}$-cohomologically smooth
morphisms are quasi-finite (Proposition 4.8).

Finally, we employ a similar strategy for the description of smooth objects in
section 5: We first deal with the absolute case, i.e. the objects which are
smooth for morphisms to the base field. By an explicit calculation, we prove
that $\mathbb{F}_{p}$ is not smooth for
$\mathbb{A}_{k}^{1}\to\operatorname{Spec}k$ (Proposition 5.6) and can deduce
the characterisation of smooth objects from this (Proposition 5.12), mostly by
formal arguments. By stability of smooth objects under base change, this
readily implies the statement about the relative case (Corollary 5.9).

###  1.7. Acknowledgements

This paper grew out of my master’s thesis, so first and foremost I want to
thank Johannes Anschütz who suggested the main question and was a thoughtful
and caring advisor throughout. This paper owes a huge intellectual debt to his
ideas. I also want to thank Peter Scholze for his insightful lectures on six
functor formalisms in the winter semester 22/23 which introduced me to this
highly fascinating topic, and I am very grateful to Lucas Mann for kindly
answering some intricate questions about abstract six functor formalisms. A
special thanks goes to my friend and fellow student Tim Kuppel for countless
illuminating discussions.

##  2\. Abstract 6-functor formalisms

###  2.1. The definition

We recall the definition of an abstract 6-functor formalism. This requires
some preparation. We adopt most of the terminology and notation from [Man22a,
A.5].

######  Definition 2.1 ([Man22a, A.5.1]).

A geometric setup is a pair $\mathfrak{G}:=(\mathcal{C},E)$, where
$\mathcal{C}$ is an $\infty$-category with all finite limits and $E$ is a
collection of homotopy classes of edges in $\mathcal{C}$ with the following
two properties:

  1. i)

$E$ contains all isomorphisms and is stable under composition.

  2. ii)

$E$ is stable under pullback along arbitrary morphisms in $\mathcal{C}$.

If $E$ satisfies also

  1. iii)

For $f\in E$ and $g$ any morphism, $f\circ g\in E$ implies $g\in E$.

then we call $(\mathcal{C},E)$ a good geometric setup. We write
$\mathcal{C}(\mathfrak{G}):=\mathcal{C}$ and $E(\mathfrak{G}):=E$.

######  Remark 2.2.

The condition that $\mathcal{C}$ has finite limits is not part of L. Mann’s
definition of a geometric setup. However, this assumption simplifies the
discussion of general 6-functor formalisms and is satisfied in the cases
relevant for us in this paper. In general, for Mann’s definition of a
6-functor formalism, one only needs to require existence of pullbacks for
morphisms in $E$. The notion of good geometric setups is non-standard.

######  Example 2.3.

The category $\mathcal{C}$ of qcqs schemes (or a slice thereof) with $E$ the
class of all separated, finite type morphisms is a geometric setup. We will
later extend the class $E$ slightly, cf. Corollary 2.24.

Given a geometric setup $(\mathcal{C},E)$, one can construct the symmetric
monoidal $\infty$-category
$\mathrm{Corr}(\mathcal{C})^{\operatorname{\otimes}}_{E,all}$ of
correspondences associated to $(\mathcal{C},E)$. We refer to [Man22a, A.5] for
the definition and record only some important facts about this category:

  * •

The underlying $\infty$-category $\mathrm{Corr}(\mathcal{C})_{E,all}$ of
$\mathrm{Corr}(\mathcal{C})^{\operatorname{\otimes}}_{E,all}$ has as objects
the objects of $\mathcal{C}$.

  * •

Morphisms $Y\to X$ in $\mathrm{Corr}(\mathcal{C})_{E,all}$ are given by
diagrams

${Y^{\prime}}$${Y}$${X}$$\scriptstyle{\in E}$

where $Y^{\prime}\to X$ is a morphism in $E$. For two morphisms $Z\leftarrow
Z^{\prime}\to Y$ and $Y\leftarrow Y^{\prime}\to X$ there is a 2-simplex

${{Z^{\prime}\times_{Y}Y^{\prime}}}$ ${{Z^{\prime}}}$ ${{Y^{\prime}}}$ ${{Z}}$
${Y}$ ${X}$ $\scriptstyle{\in E}$$\scriptstyle{\in E}$$\scriptstyle{\in E}$

witnessing that $Z\leftarrow Z^{\prime}\times_{Y}Y^{\prime}\to X$ is a
composition of the two morphisms.

  * •

The symmetric monoidal structure $\operatorname{\otimes}$ on
$\mathrm{Corr}(\mathcal{C})_{E,all}$ is on objects just given by the cartesian
product $X\operatorname{\otimes}Y=X\times Y$.222For simplicity, we describe
monoidal structures and (lax) symmetric monoidal functors here in a naive way
as well and refer for the correct definition (as a coCartesian
fibration/$\infty$-operad) to [Man22a, A.5.4]. Given two morphisms
$X\leftarrow X^{\prime}\to S$ and $Z\leftarrow Z^{\prime}\to Y$, the induced
morphism $Z\operatorname{\otimes}X\to Y\operatorname{\otimes}S$ is given by
$Z\times X\leftarrow Z^{\prime}\times X^{\prime}\to Y\times S$.

In particular, we have the following three kinds of morphisms in
$\mathrm{Corr}(\mathcal{C})_{E,all}$:

  1. (1)

For a map $f:Y\to X$ in $\mathcal{C}$ a map $X\to Y$ given by the diagram
$X\leftarrow Y=Y$.

  2. (2)

For a map $f:Y\to X$ in $E$ a map $Y\to X$ given by the diagram $Y=Y\to X$.

  3. (3)

For $X\in\mathcal{C}$, a morphism $X\operatorname{\otimes}X\to X$ induced by
the diagram $X\times X\overset{\Delta}{\leftarrow}X\to X$.

######  Remark 2.4.

Note that in particular there is an embedding of symmetric monoidal
$\infty$-categories

$\mathcal{C}^{\mathrm{op}}\hookrightarrow\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}$

given by the contravariant functor which sends an object to itself and a
morphism $Y\to X$ to the corresponding morphism of type (1) in the above list.
Here, $\mathcal{C}^{\text{op}}$ is equipped with the cocartesian monoidal
structure.

Let us now recall the definition of a 3-functor formalism.

######  Definition 2.5 ([Man22a, Definition A.5.6]).

Let $(\mathcal{C},E)$ be a geometric setup. A 3-functor formalism is a lax
symmetric monoidal functor

$\mathcal{D}:\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}\to\mathrm{Cat}_{\infty}$

where $\mathrm{Cat}_{\infty}$ is equipped with the cartesian symmetric
monoidal structure.

The three functors are then induced by the three maps in
$\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}$ mentioned above:

  1. (1)

The map $X\to Y$ given by the diagram $X\overset{f}{\leftarrow}Y=Y$ induces
the pullback functor $f^{*}:\mathcal{D}(X)\to D(Y)$.

  2. (2)

The map $Y\to X$ given by the diagram $Y=Y\overset{f}{\to}X$ induces the lower
shriek functor $f_{!}:\mathcal{D}(Y)\to\mathcal{D}(X)$.

  3. (3)

The morphism $X\operatorname{\otimes}X\to X$ induces the tensor product
$\operatorname{\otimes}:\mathcal{D}(X)\times\mathcal{D}(X)\to\mathcal{D}(X\times
X)\to\mathcal{D}(X)$ where the first map is induced by the lax monoidality of
$\mathcal{D}$ and the second map is given by $\Delta^{*}$. This equips
$\mathcal{D}(X)$ with a symmetric monoidal structure $\operatorname{\otimes}$.

######  Notation 2.6.

$f^{*},f_{!}$ and $\operatorname{\otimes}$ are sometimes also denoted
$\mathcal{D}^{*}(f)$, $\mathcal{D}_{!}(f)$ and
$\operatorname{\otimes}^{\mathcal{D}}$ to indicate the 6-functor formalism
that induces them.

Definition 2.5 encodes both base change and projection formula into the
functoriality of $\mathcal{D}$:

######  Proposition 2.7.

Let $\mathcal{D}$ be a 3-functor formalism on a geometric setup
$(\mathcal{C},E)$. Then:

  1. i)

(Base Change) For every cartesian diagram

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

with $g\in E$ there is a natural isomorphism of functors
$\mathcal{D}(X^{\prime})\to\mathcal{D}(Y)$

$f^{*}g_{!}\cong g^{\prime}_{!}{f^{\prime}}^{*}$

  2. ii)

(Projection Formula) For every $f:Y\to X$ in $E$, there is a natural
isomorphism of functors $\mathcal{D}(Y)\times\mathcal{D}(X)\to\mathcal{D}(X)$

$f_{!}(\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}f^{*}\rule{8.5359pt}{0.42677pt})\cong
f_{!}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}\rule{8.5359pt}{0.42677pt}$

###### Proof.

We only show $i)$, the argument for $ii)$ is similar (cf. [Man22a, A.5.8]). By
definition, $f^{*}$ is induced by the $1$-simplex
$a:=(X\overset{f}{\leftarrow}Y=Y)$ and $g_{!}$ is induced by the $1$-simplex
$b:=(X^{\prime}=X^{\prime}\overset{g}{\to}X)$ in
$\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}$. By the
definition of $\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}$
there is a 2-simplex witnessing that the morphism $c=(X^{\prime}\leftarrow
Y^{\prime}\to Y)$ is a composition $a\circ b$ (as the diagram is cartesian,
cf. the description of
$\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}$ above); but
another 2-simplex realises $c$ as a composition of $X^{\prime}\leftarrow
Y^{\prime}=Y^{\prime}$ and $Y^{\prime}=Y^{\prime}\to Y$, which correspond
under $\mathcal{D}$ to $g^{\prime*}$ and $f^{\prime}_{!}$. ∎

######  Corollary 2.8 (Künneth Formula).

Let $\mathcal{D}$ be a 3-functor formalism on a geometric setup
$(\mathcal{C},E)$. For every cartesian diagram

${Z\times_{X}Y}$${Z}$${Y}$${X}$$\scriptstyle{f^{\prime}}$$\scriptstyle{g^{\prime}}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f}$

with $f,g\in E$ (and thus $h\in E$) and for all $A\in\mathcal{D}(Y)$ and
$B\in\mathcal{D}(Z)$ there is a natural isomorphism

$h_{!}(g^{\prime*}A\operatorname{\otimes}f^{\prime*}B)\cong
f_{!}A\operatorname{\otimes}g_{!}B$

###### Proof.

We use base change and projection formula (Proposition 2.7) to calculate

$h_{!}(g^{\prime*}A\operatorname{\otimes}f^{\prime*}B)\cong
f_{!}g^{\prime}_{!}(g^{\prime*}A\operatorname{\otimes}f^{\prime*}B)\cong
f_{!}(A\operatorname{\otimes}g^{\prime}_{!}f^{\prime*}B)\cong
f_{!}(A\operatorname{\otimes}f^{*}g_{!}B)\cong
f_{!}A\operatorname{\otimes}g_{!}B$

∎

To get a $6$-functor formalism, it remains to require the existence of
adjoints:

######  Definition 2.9 ([Man22a, Definition A.5.7]).

Let $(\mathcal{C},E)$ be a geometric setup. A 6-functor formalism is a
3-functor formalism

$\mathcal{D}:\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}\to\mathrm{Cat}_{\infty}$

such that for all $X\in\mathcal{C}$ the symmetric monoidal $\infty$-category
$\mathcal{D}(X)$ is closed and all $f^{*}:\mathcal{D}(X)\to\mathcal{D}(Y)$ and
$f_{!}:\mathcal{D}(Y)\to\mathcal{D}(X)$ admit right adjoints. We write
$\operatorname{\underline{Hom}}$ for the internal Hom, $f_{*}$ for the right
adjoint of $f^{*}$ and $f^{!}$ for the right adjoint of $f_{!}$.

Naturally, we expect statements dual to Proposition 2.7 for the adjoints
$f_{*},f^{!}$ and $\operatorname{\underline{Hom}}$.

######  Proposition 2.10.

Let $\mathcal{D}$ be a 6-functor formalism on a geometric setup
$(\mathcal{C},E)$. Then:

  1. i)

(Adjoint Base Change) For every cartesian diagram

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

with $g\in E$ there is a natural isomorphism of functors
$\mathcal{D}(Y)\to\mathcal{D}(X^{\prime})$

$g^{!}f_{*}\cong f^{\prime}_{*}g^{\prime!}$

  2. ii)

Let $f:Y\to X$ be a morphism in $E$.

     * •

(Interior Pullback) There is a natural isomorphism of functors
$\mathcal{D}(X)\times\mathcal{D}(X)\to\mathcal{D}(Y)$

$f^{!}\operatorname{\underline{Hom}}(\rule{8.5359pt}{0.42677pt},\rule{8.5359pt}{0.42677pt})=\operatorname{\underline{Hom}}(f^{*}\rule{8.5359pt}{0.42677pt},f^{!}\rule{8.5359pt}{0.42677pt})$

     * •

(Verdier Duality) There is a natural isomorphism of functors
$\mathcal{D}(Y)\times\mathcal{D}(X)\to\mathcal{D}(X)$

$f_{*}\operatorname{\underline{Hom}}(\rule{8.5359pt}{0.42677pt},f^{!}\rule{8.5359pt}{0.42677pt})\cong\operatorname{\underline{Hom}}(f_{!}\rule{8.5359pt}{0.42677pt},\rule{8.5359pt}{0.42677pt})$

###### Proof.

Using the adjunctions, all assertions are equivalent to the corresponding
statements in Proposition 2.7 by the Yoneda lemma. ∎

######  Remark 2.11.

Let $\mathcal{D}$ be a 3- (resp. 6-)functor formalism on a geometric setup
$\mathfrak{G}=(\mathcal{C},E)$ and $X\in\mathcal{C}$. Then $\mathcal{D}$
induces a 3- (resp. 6-)functor formalism on the slice geometric setup
$\mathfrak{G}_{/X}:=(\mathcal{C}_{/X},E_{/X})$ (where $E_{/X}$ are the edges
in $\mathcal{C}_{X}$ which lie in $E$). Indeed, this is just $\mathcal{D}$
precomposed with the lax symmetric monoidal functor
$\mathrm{Corr}(\mathcal{C}_{/X})_{E_{/X},all}^{\operatorname{\otimes}}\to\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}$
coming from the natural lax symmetric monoidal functor
$(\mathcal{C}_{/X}^{\mathrm{op}})^{\amalg}\to(\mathcal{C}^{\mathrm{op}})^{\amalg}$.

###  2.2. Construction of 6-functor formalisms

In practice, for a morphism $f:Y\to X$, one often wants to define $f_{!}$ as
$p_{!}j_{!}$ where $f=p\circ j$ is a decomposition of $f$ into a morphism $p$
for which $p^{*}$ admits a right adjoint $p_{!}$ and a morphism $j$ for which
$j^{*}$ admits a left adjoint $j_{!}$ (e.g. compactifications in étale
cohomology). This motivates the following strategy to construct 3- or
6-functor formalisms: First construct a functor
$\mathcal{D}_{0}:\mathcal{C}^{op}\to\mathrm{Cat}_{\infty}$, inducing the
functors $f^{*}$. Choose subclasses $I$ and $P$ of morphisms in $E$ for which
$f^{*}$ admits a left resp. right adjoint. Under the right conditions on $I$,
$P$ and $E$, one can then define the functor $f_{!}$ for $f$ in $I$ or $P$ as
the respective adjoint and extend this to all morphisms in $E$, yielding an
extension of $\mathcal{D}_{0}$ to the desired 3- or 6-functor formalism
$\mathcal{D}:\mathrm{Corr}(\mathcal{C})_{E,all}^{\operatorname{\otimes}}\to\mathrm{Cat}_{\infty}$.
This was made precise by L. Mann in [Man22a, Appendix A].

######  Definition 2.12.

[Man22a, Definition A.5.9] Let $(\mathcal{C},E)$ be a geometric setup. A
suitable decomposition of $E$ is a pair $I,P\subseteq E$ of subclasses with
the following properties:

  1. i)

Every $f\in E$ is of the form $f=p\circ j$ for some $j\in I$ and some $p\in
P$.

  2. ii)

Every morphism $f\in I\cap P$ is $n$-truncated for some $n\geq-2$ (which may
depend on $f$).333As $\mathcal{C}$ has finite limits, here the following
inductive definition is available (cf. [Lur09, Lemma 5.5.6.15]): isomorphisms
are $-2$-truncated and a morphism is called $n$-truncated if its diagonal is
$n-1$-truncated. In particular, if $\mathcal{C}$ is (the nerve of) a
1-category, then this is automatic.

  3. iii)

$I$ and $P$ contain all identity morphisms and are stable under pullback along
arbitrary morphisms in $\mathcal{C}$.

  4. iv)

Given $f:Y\to X$ and $g:Z\to Y$ in $\mathcal{C}$ such that $f\in I$ (resp.
$f\in P$), then $g\in I$ (resp. $g\in P$) if and only if $f\circ g\in I$
(resp. $f\circ g\in P$).

######  Example 2.13.

Let $(\mathcal{C},E)$ be the geometric setup of qcqs schemes with all
separated, finite type morphisms as $E$. Let $I$ be the subclass of all open
immersions and $P$ the subclass of all proper morphisms. Then by Nagata’s
theorem on compactifications (cf. [Sta, 0F41]), every $f\in E$ can be factored
as $f=p\circ j$ where $j\in I$ and $p\in P$. Every morphism $f\in\mathcal{C}$
is $0$-truncated and $I$ and $P$ are stable under pullback. Moreover, $iv)$
clearly holds for both $I$ and $P$. So $I$ and $P$ are a suitable
decomposition of $E$.

We make a small definition to gather all the data in one object.

######  Definition 2.14.

We call a quadruple $\mathfrak{G}=(\mathcal{C},E,I,P)$ a construction setup if
$(\mathcal{C},E)$ is a geometric setup and $I,P\subseteq E$ is a suitable
decomposition of $E$. We will sometimes write e.g. $I(\mathfrak{G})$ to denote
the set $I$ associated to $\mathfrak{G}$ and similarly for $\mathcal{C},E$ and
$P$.

######  Remark 2.15.

This is non-standard terminology. Note that if
$\mathfrak{G}=(\mathcal{C},E,I,P)$ is a construction setup, then
$(\mathcal{C},E)$ is a good geometric setup: Indeed, by the usual argument
(cf. e.g. the proof of Proposition 2.25) this is equivalent to the question if
the diagonal $\Delta_{g}$ lies in $E$ if $g$ does. We can write $g=p\circ j$
where $p\in P$ and $j\in I$. Then $\Delta_{p\circ j}$ is the composition of
$\Delta_{j}$ with a base change of $\Delta_{p}$. But since $I$ and $P$ form a
suitable decomposition, $\Delta_{j}\in I$ and $\Delta_{p}\in P$ and thus
$\Delta_{g}=\Delta_{p\circ j}\in E$ as $E$ is stable under pullback and
composition.

######  Proposition 2.16 ([Man22a, Proposition A.5.10]).

Let $\mathfrak{G}=(\mathcal{C},E,I,P)$ be a construction setup. Let
$\mathcal{D}_{0}:(\mathcal{C}^{\mathrm{op}})^{\amalg}\to\mathrm{Cat}_{\infty}$
be a lax symmetric monoidal functor, inducing for every $f:Y\to X$ the
pullback functor $f^{*}:\mathcal{D}(X)\to\mathcal{D}(Y)$. Assume that the
following two conditions are satisfied:

  1. i)

For every $j:U\to X$ in $I$, $j^{*}$ admits a left adjoint $j_{!}$ satisfying
base change and projection formula.444Because of the adjointness of $j_{!}$
and $j^{*}$ this is really a condition, not a datum: for both base change and
projection formula there are natural maps between the respective functors. For
example, here, we have a natural map
$j_{!}(\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}j^{*}\rule{8.5359pt}{0.42677pt})\to
j_{!}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}\rule{8.5359pt}{0.42677pt}$
adjoint to the map
$\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}j^{*}\rule{8.5359pt}{0.42677pt}\to
j^{*}(j_{!}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}\rule{8.5359pt}{0.42677pt})=j^{*}j_{!}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}j^{*}\rule{8.5359pt}{0.42677pt}$
induced by the unit. The construction for morphisms $p\in P$ is similar, with
the difference that the transformations go in the other direction.

  2. ii)

For every $p:Y\to X$ in $P$, $p^{*}$ admits a right adjoint $p_{!}$ satisfying
base change and projection formula.

Moreover, assume the following holds:

  1. iii)

For every cartesian diagram

${U^{\prime}}$${X^{\prime}}$${U}$${X}$$\scriptstyle{j^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p}$$\scriptstyle{j}$

with the horizontal morphisms in $I$ and vertical morphisms in $P$, the
natural map $j_{!}p^{\prime}_{!}\to p_{!}j^{\prime}_{!}$ is an
isomorphism.555This is the map adjoint to $p^{\prime}_{!}\to
p^{\prime}_{!}j^{\prime*}j^{\prime}_{!}\xleftarrow{\sim}j^{*}p_{!}j^{\prime}_{!}$
where the first map is the unit of the $j_{!}$-$j^{*}$-adjunction and the
second comes from the natural morphism $j^{*}p_{!}\to
p^{\prime}_{!}j^{\prime*}$ which is an isomorphism by $ii)$.

Then $\mathcal{D}_{0}$ can be extended to a $3$-functor formalism
$\mathcal{D}$ such that $f_{!}:=\mathcal{D}_{!}(f)$ agrees with the respective
adjoint of $f^{*}$ for morphisms $f$ in $I$ or $P$.
If moreover $\mathcal{D}(X)$ is a closed $\infty$-category for all
$X\in\mathcal{C}$, $f^{*}$ has a right adjoint for all $f$ in $\mathcal{C}$
and $f_{!}$ itself admits a right adjoint $f^{!}$ for $f\in P$, then
$\mathcal{D}$ is already a $6$-functor formalism.

###### Proof.

See [Man22a, Proposition A.5.10]. ∎

We call a 6-functor formalism that arises from the construction in Proposition
2.16 a 6-functor formalism on the construction setup $\mathfrak{G}$.

###  2.3. The 6 functors for étale sheaves

Now we want to define the étale 6-functor formalism on the category
$\operatorname{Sch}$ of qcqs schemes. As in Example 2.13 we could choose all
separated, finite type morphisms as the class $E$ of our geometric setup.
However, we need a slight extension of this setting which enables us to pass
to the absolute weak normalisation in our formalism.

######  Definition 2.17 ([Ryd10, Definition B.1]).

A scheme $X$ is called absolutely weakly normal (a.w.n.) if $X$ is reduced and
every separated universal homeomorphism $\pi:X^{\prime}\to X$ is an
isomorphism if $X^{\prime}$ is reduced.

######  Lemma 2.18.

The inclusion $\mathrm{Awn}\to\operatorname{Sch}$ of the full subcategory of
absolutely weakly normal schemes admits a right adjoint
$(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$, called the absolute weak
normalisation. It is given by sending $X$ to the initial object
$X^{\mathrm{awn}}$ of the category $\mathcal{U}_{X}$ of universal
homeomorphisms $X^{\prime}\to X$ and a morphism $Y\to X$ to the morphism
$Y^{\mathrm{awn}}\to Y\times_{X}X^{\mathrm{awn}}\to X^{\mathrm{awn}}$ (note
that $Y\times_{X}X^{\mathrm{awn}}\to Y$ is a universal homeomorphism).

###### Proof.

By [Sta, 0EUS], $X^{\mathrm{awn}}$ exists indeed for all $X$ and $X\mapsto
X^{\mathrm{awn}}$ yields a functor $\operatorname{Sch}\to\mathrm{Awn}$. In
fact, it is proven there that the initial object is given by
$(\varprojlim_{X^{\prime}\in\mathcal{U}^{\prime}_{X}}X^{\prime}\to X)$ (where
the limit is taken in $\operatorname{Sch}$) with $\mathcal{U}^{\prime}_{X}$
the category of universal homeomorphisms $X^{\prime}\to X$ of finite type.
Thus, for an a.w.n. scheme $Y$ it follows that
$\operatorname{Hom}(Y,X^{\mathrm{awn}})\cong\varprojlim_{X^{\prime}\to
X\in\mathcal{U}_{X}}\operatorname{Hom}(Y,X^{\prime})$ and it remains to show
that

$\varprojlim_{X^{\prime}\to
X\in\mathcal{U}_{X}}\operatorname{Hom}(Y,X^{\prime})\to\operatorname{Hom}(Y,X)$

is an isomorphism. Note that for a morphism $f:Y\to X$, the base change
$g^{\prime}:Y^{\prime}\to Y$ of any universal homeomorphism $g:X^{\prime}\to
X$ is again a universal homeomorphism and thus there is a unique section
$Y\cong Y^{\mathrm{awn}}\to Y^{\prime}$ of $g^{\prime}$ since $Y$ is a.w.n..
In particular, there is a unique $Y\to X^{\prime}$ such that $Y\to
X^{\prime}\to X$ is $f$. This proves both surjectivity and injectivity. ∎

We note that in positive characteristic the absolute weak normalisation has a
particularly nice description:

######  Proposition 2.19.

A scheme $X$ in characteristic $p>0$ is a.w.n. if and only if the Frobenius
endomorphism $\varphi:X\to X$ is an isomorphism, i.e. $X$ is perfect. In
particular, the absolute weak normalisation agrees with the perfection
$(\rule{8.5359pt}{0.42677pt})^{\mathrm{perf}}$, i.e. the functor mapping
$X\mapsto\varprojlim_{\varphi}X$.

###### Proof.

The second statement follows from the first by Lemma 2.18, noting that the
perfection is right adjoint to the inclusion of perfect schemes. Let us prove
the first statement. The "only if" is clear from the definition, as the
Frobenius is a universal homeomorphism. For the converse, let $X$ be perfect.
Then one can use that for a universal homeomorphism $Y\to X$ the induced map
$\varprojlim_{\varphi}Y=Y^{\mathrm{perf}}\to X^{\mathrm{perf}}$ is an
isomorphism (cf. [BS17, Lemma 3.8]); in particular,
$X^{\mathrm{awn}}\cong(X^{\mathrm{awn}})^{\mathrm{perf}}\cong
X^{\mathrm{perf}}\cong X$, where the first isomorphism comes from the proven
direction, the second one from the fact that $X^{\mathrm{awn}}\to X$ is a
universal homeomorphism and the last one is implied by the assumption. Thus
$X$ is a.w.n.. ∎

We want the class $E$ of our geometric setup to contain all separated
morphisms of finite type and their absolute weak normalisations; on the other
hand, we want to apply Proposition 2.16, so compactifications as in Example
2.13 should still be available. As absolute weak normalisations of finite type
morphisms are in general not at all of finite type, we need to weaken the
assumptions in Nagata’s theorem. The following result by P. Hamacher solves
this problem.

######  Definition 2.20.

[Ham19, Definition 1.4] A morphism $f:Y\to X$ is called of finite expansion if
$X$ can be covered by opens $U_{i}$ such that there are $V_{j}$ covering $Y$
with the following property: each $f|_{V_{j}}$ factors as
$V_{j}\to\mathbb{A}^{n}_{U_{i}}\to U_{i}\to X$ for some $i$ and some $n$ such
that $V_{j}\to\mathbb{A}^{n}_{U_{i}}$ is integral.

######  Theorem 2.21.

Let $f:Y\to X$ be a separated morphism of finite expansion between qcqs
schemes. Then $f=p\circ j$ where $j:Y\to\bar{Y}$ is an open immersion and
$p:\bar{Y}\to X$ is separated and universally closed.

###### Proof.

See [Ham19, Theorem 1.17]. ∎

######  Example 2.22.

Clearly, all integral morphisms and all finite type morphisms are of finite
expansion. Using the stability under composition proven below, this already
shows that absolute weak normalisations of finite type morphisms are of finite
expansion, using that for a morphism $f:Y\to X$ $f^{\mathrm{awn}}$ factors as
$Y^{\mathrm{awn}}\cong(Y\times_{X}X^{\mathrm{awn}})^{\mathrm{awn}}\to
Y\times_{X}X^{\mathrm{awn}}\to X^{\mathrm{awn}}$ (note that
$(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$ commutes with all limits by
Lemma 2.18) where the first map is integral (as it is a universal
homeomorphism, cf. [Sta, 04DF]) and the second one is a base change of $f$.

######  Proposition 2.23.

Let $S$ be the set of morphisms in $\operatorname{Sch}$ which are of finite
expansion. Then:

  1. i)

$S$ is stable under composition.

  2. ii)

$S$ is stable under pullback along arbitrary morphisms in
$\operatorname{Sch}$.

###### Proof.

$ii)$ is clear by stability of both parts of the defining factorisations under
base change. $i)$ reduces to the following observation: if we have
factorisations $Z\to\mathbb{A}^{m}_{Y}\to Y$, $Y\to\mathbb{A}^{n}_{X}\to X$
where the first morphism is integral, then
$Z\to\mathbb{A}^{m}_{Y}\cong\mathbb{A}^{m}_{X}\times_{X}Y\to\mathbb{A}^{m}_{X}\times_{X}\mathbb{A}^{n}_{X}\cong\mathbb{A}^{n+m}_{X}$
is integral as well, again by stability of integral morphisms under base
change. ∎

######  Corollary 2.24.

Let $E$ be the set of separated morphisms of finite expansion. Then
$(\operatorname{Sch},E)$ is a geometric setup.

###### Proof.

This follows from the previous proposition as separated morphisms are stable
under composition and pullback. ∎

Now Theorem 2.21 dictates the choice of $I,P\subseteq E$.

######  Proposition 2.25.

Let $E$ be the set of separated morphisms of finite expansion. Let $I$ be the
class of open immersions and $P$ the class of universally closed maps in $E$.
Then $I$ and $P$ form a suitable decomposition of $E$, inducing a construction
setup which we will denote by $\mathfrak{G}$.

###### Proof.

We need to check $i)-iv)$ in Definition 2.12. $i)$ is Theorem 2.21. $ii)$ is
clear as all morphisms in $\mathcal{C}(\mathfrak{G}^{\prime})$ are
$0$-truncated. $iii)$ is obvious as well for both $I$ and $P$. It remains to
show $iv)$. The "only if" part is clear. The "if" part follows as usual from
considering the commutative diagram

${Z}$${{Y\times_{X}Z}}$${Z}$${Y}$${X}$$\scriptstyle{\Gamma}$$\scriptstyle{f}$$\scriptstyle{f\circ
g}$$\scriptstyle{g}$

Since the graph $\Gamma$ is a base change of $\Delta_{f}$, this implies the
statement for morphisms in $I$ as then $\Delta_{f}$ is an isomorphism, and for
morphisms in $P$ as then $\Delta_{f}$ is a closed immersion. ∎

Now we can consider 6-functor formalisms on the construction setup
$\mathfrak{G}$. The central example for us is étale cohomology with
$\mathbb{F}_{p}$-module coefficients. More generally, one can allow modules
over arbitrary torsion rings. For a torsion ring $\Lambda$ and
$X\in\mathcal{C}(\mathfrak{G})$ we consider the $\infty$-derived category
$\mathcal{D}(X_{\mathrm{\acute{e}t}},\Lambda)$ of étale sheaves of
$\Lambda$-modules on $X$666That is, the category obtained from the homotopy
category $\mathcal{K}(X_{\mathrm{\acute{e}t}},\Lambda)$ of the abelian
category of étale sheaves of $\Lambda$-modules by declaring the quasi-
isomorphisms to be weak equivalences. This agrees with the category of
hypersheaves $\mathrm{HypShv}(X_{\mathrm{\acute{e}t}},D(\Lambda))$, cf.
[Lur18, Corollary 2.1.2.3]., which is a stable, presentable $\infty$-category
with the standard $t$-structure. By general topos theory, a morphism $f:Y\to
X$ induces a pullback functor
$f^{*}:\mathcal{D}(X_{\mathrm{\acute{e}t}},\Lambda)\to\mathcal{D}(Y_{\mathrm{\acute{e}t}},\Lambda)$.
We need to construct the corresponding
$\mathcal{D}_{\Lambda,0}:\mathcal{C}(\mathfrak{G})^{\mathrm{op}}\to\mathrm{Cat}_{\infty}$
and check the conditions in Proposition 2.16. For the construction of
$\mathcal{D}_{\Lambda,0}:\mathcal{C}(\mathfrak{G})^{\mathrm{op}}\to\mathrm{Cat}_{\infty}$
we refer to [LZ17, Section 2]. The symmetric monoidal structure
$\operatorname{\otimes}$ on $\mathcal{D}(X_{\mathrm{\acute{e}t}},\Lambda)$
comes from deriving the usual tensor product $\operatorname{\otimes}$ on
$\mathrm{Shv}(X_{\mathrm{\acute{e}t}},\Lambda)$, the category of étale sheaves
of $\Lambda$-modules on $X$. Accordingly, its right adjoint
$\operatorname{\underline{Hom}}$ is obtained by deriving the
$\operatorname{\underline{Hom}}$ in
$\mathrm{Shv}(X_{\mathrm{\acute{e}t}},\Lambda)$. Similarly, for a morphism $f$
in $\mathcal{C}(\mathfrak{G})$, deriving $f_{*}$ yields a right adjoint to
$f^{*}$, which we will also denote by $f_{*}$.

######  Theorem 2.26.

$\mathcal{D}_{\Lambda,0}$ extends to a 6-functor formalism
$\mathcal{D}_{\Lambda}$ on $\mathfrak{G}$.

###### Proof.

We refer to [Sch23, Theorem 7.16]. In loc. cit. the statement is proven for
the assignment
$X\mapsto\mathrm{HypShv}(X_{\mathrm{\acute{e}t}},\hat{D}_{pf}(\mathbb{Z}))$
(where $\hat{D}_{pf}(\mathbb{Z})$ is the full subcategory of torsion complexes
in $D(\mathbb{Z})$), but the same argument applies mutatis mutandis to
$\mathcal{D}_{\Lambda,0}=\mathrm{HypShv}((\rule{8.5359pt}{0.42677pt})_{\mathrm{\acute{e}t}},D(\Lambda))$.

∎

The étale site (and thus étale cohomology) is invariant under universal
homeomorphisms ([Sta, 04DZ, 03SI]). This makes the following observation
particularly relevant:

######  Lemma 2.27.

A morphism $f$ in $\operatorname{Sch}$ is a universal homeomorphism if and
only if $f^{\mathrm{awn}}$ is an isomorphism.

###### Proof.

Clearly, if $f:Y\to X$ is a universal homeomorphism then so is
$f^{\mathrm{awn}}$. But $Y^{\mathrm{awn}},X^{\mathrm{awn}}$ are reduced and
a.w.n., so $f^{\mathrm{awn}}$ must be an isomorphism by definition. The
converse is clear since the natural morphisms $X^{\mathrm{awn}}\to X$ are
universal homeomorphisms. ∎

######  Proposition 2.28.

The functor $(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$ from Lemma 2.18
exhibits the full subcategory $\mathrm{Awn}$ of a.w.n. schemes in
$\operatorname{Sch}$ as the colocalisation of $\operatorname{Sch}$ at the
universal homeomorphisms.

###### Proof.

By Lemma 2.27, a morphism $f$ in $\operatorname{Sch}$ is a universal
homeomorphism if and only if $f^{\mathrm{awn}}$ is an isomorphism. By Lemma
2.18 the inclusion $\mathrm{Awn}\to\operatorname{Sch}$ is left-adjoint to the
absolute weak normalisation and so by [Lur17, Example 1.3.4.3]
$\mathrm{Awn}^{\mathrm{op}}\simeq\operatorname{Sch}^{\mathrm{op}}[(W^{\mathrm{op}})^{-1}]$
where $W$ is the class of universal homeomorphisms in $\operatorname{Sch}$. ∎

######  Proposition 2.29.

Let $\mathcal{D}_{\Lambda}$ be the 6-functor formalism on
$\mathfrak{G}=(\operatorname{Sch},E,I,P)$ from above. Let $\mathrm{Awn}$
denote the full subcategory of a.w.n. schemes in $\operatorname{Sch}$ and
$E^{\mathrm{awn}}$ the classes of morphisms in $\mathrm{Awn}$ which lie in
$E$. Then $\mathcal{D}_{\Lambda}$ factors as
$\mathcal{D}_{\Lambda}^{\mathrm{awn}}\circ\mathcal{P}$ where
$\mathcal{D}_{\Lambda}^{\mathrm{awn}}$ is the restriction of
$\mathcal{D}_{\Lambda}$ to the geometric setup
$\mathfrak{G}^{\mathrm{awn}}=(\mathrm{Awn},E^{\mathrm{awn}})$ and
$\mathcal{P}:\mathrm{Corr}(\operatorname{Sch})_{E,all}\to\mathrm{Corr}(\mathrm{Awn})_{E^{\mathrm{awn}},all}$
is the lax symmetric monoidal functor induced by
$(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}:\operatorname{Sch}_{k}\to\mathrm{Awn}$.

###### Proof.

First note that $(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$ induces a lax
symmetric monoidal functor
$(\operatorname{Sch}^{\mathrm{op}})^{\amalg}\to(\mathrm{Awn}^{\mathrm{op}})^{\amalg}$
(as a right adjoint, the absolute weak normalisation commutes with all
products). Let $\mathcal{D}_{\Lambda,0}$ denote the restriction of
$\mathcal{D}_{\Lambda}$ to $\operatorname{Sch}^{\mathrm{op}}$ and
$\mathcal{D}_{\Lambda,0}^{\mathrm{awn}}$ the restriction of
$\mathcal{D}_{\Lambda}^{\mathrm{awn}}$ to $\mathrm{Awn}^{\mathrm{op}}$. As
pullback along universal homeomorphisms $Y\to X$ induces equivalences of
categories $\mathcal{D}_{\Lambda}(X)\simeq\mathcal{D}_{\Lambda}(Y)$,
$\mathcal{D}_{\Lambda,0}$ factors as
$\mathcal{D}_{\Lambda,0}^{\mathrm{awn}}\circ(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$
by Proposition 2.28. Now by definition $\mathcal{D}_{\Lambda}$ is obtained
from $\mathcal{D}_{\Lambda,0}$ via the construction result Proposition 2.16.
The restricted formalism $\mathcal{D}_{\Lambda}^{\mathrm{awn}}$ can be
obtained from $\mathcal{D}_{\Lambda,0}^{\mathrm{awn}}$ via Proposition 2.16 as
well, using that
$(\mathrm{Awn},E^{\mathrm{awn}},I^{\mathrm{awn}},P^{\mathrm{awn}})$ is still a
construction setup (where $I^{\mathrm{awn}}$, $P^{\mathrm{awn}}$ are the
classes of morphisms in $\mathrm{Awn}$ which lie in $I$ or $P$ respectively;
in other words, they are the image of the classes $I$ and $P$ under
$(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$). We will sketch how this
implies the claim by following the proof of the construction result in
[Man22a, Proposition A.5.10]: It is shown there that the construction of
$\mathcal{D}_{\Lambda}$ resp. $\mathcal{D}_{\Lambda}^{\mathrm{awn}}$ is
equivalent to the construction of a functor777We use the notation from
[Man22a, Proposition A.5.10].

$\delta^{*}_{3,\left\\{3\right\\}}(\mathcal{C}_{\amalg})_{I^{-},P^{-},all}\to\mathrm{Cat}_{\infty}$

where $\mathcal{C}$ is $\operatorname{Sch}$ or $\mathrm{Awn}$, respectively
(and $I^{-}$,$P^{-}$ are the respective classes). To construct this functor,
one starts with the composed functor

$\delta^{*}_{3,\left\\{1,2,3\right\\}}(\mathcal{C}_{\amalg})_{I^{-},P^{-},all}\to(\mathcal{C}^{\mathrm{op}})^{\amalg}\to\mathrm{Cat}_{\infty}$

where the first functor is the diagonal functor and the second one is
$\mathcal{D}_{\Lambda,0}$ resp. $\mathcal{D}_{\Lambda,0}^{\mathrm{awn}}$. The
trisimplical set
$\delta^{*}_{3,\left\\{1,2,3\right\\}}(\mathcal{C}_{\amalg})_{I^{-},P^{-},all}$
can be roughly described as follows: Its objects are the objects of
$\mathcal{C}_{\amalg}:=(\mathcal{C}^{\mathrm{op}})^{\amalg,\mathrm{op}}$ and
its morphisms are cubes made out of cartesian squares, such that all the
morphisms in vertical resp. horizontal direction lie in $I^{-}$ resp. $P^{-}$,
with the morphisms in all directions being inverted. The above functor then
sends such a cube to the composition of pullback functors along all three
dimension. By our observations above we have a factorisation

$\displaystyle\delta^{*}_{3,\left\\{1,2,3\right\\}}((\operatorname{Sch})_{\amalg})_{I^{-},P^{-},all}\to(\operatorname{Sch}^{\mathrm{op}})^{\amalg}\to(\mathrm{Awn}^{\mathrm{op}})^{\amalg}\to\mathrm{Cat}_{\infty},$

writing
$\mathcal{D}_{\Lambda,0}=\mathcal{D}_{\Lambda,0}^{\mathrm{awn}}\circ(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$.
Note that $(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$ preserves morphisms in
$I$ and $P$ and so the functor
$\delta^{*}_{3,\left\\{1,2,3\right\\}}((\operatorname{Sch})_{\amalg})_{I^{-},P^{-},all}\to(\operatorname{Sch}^{\mathrm{op}})^{\amalg}\to(\mathrm{Awn}^{\mathrm{op}})^{\amalg}$
agrees with the functor

$\delta^{*}_{3,\left\\{1,2,3\right\\}}((\operatorname{Sch})_{\amalg})_{I^{-},P^{-},all}\to\delta^{*}_{3,\left\\{1,2,3\right\\}}((\mathrm{Awn})_{\amalg})_{I^{-},P^{-},all}\to\mathrm{Awn}^{\mathrm{op}}$

where the first functor sends an object to its absolute weak normalisation and
a cube to the corresponding cube obtained by applying
$(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$ to all edges; the second functor
is again the diagonal. Now the desired functor

$\delta^{*}_{3,\left\\{3\right\\}}((\mathcal{C})_{\amalg})_{I^{-},P^{-},all}\to\mathrm{Cat}_{\infty}$

is obtained by inverting the morphisms in the vertical and horizontal
direction, which is done by passing to the respective adjoints in
$\mathrm{Cat}_{\infty}$. So we get an induced morphism

$\delta^{*}_{3,\left\\{3\right\\}}((\operatorname{Sch})_{\amalg})_{I^{-},P^{-},all}\to\delta^{*}_{3,\left\\{3\right\\}}((\mathrm{Awn})_{\amalg})_{I^{-},P^{-},all}\to\mathrm{Awn}^{\mathrm{op}}\to\mathrm{Cat}_{\infty}$

which gives, by going backwards through the equivalences of simplicial sets, a
factorisation

$\mathrm{Corr}(\operatorname{Sch})_{E,all}\to\mathrm{Corr}(\mathrm{Awn})_{E^{\mathrm{awn}},all}\to\mathrm{Cat}_{\infty}$

where the first functor $\mathcal{P}$ sends objects to their absolute weak
normalisation and a diagram (corresponding to an $n$-simplex) to the diagram
where all morphisms are replaced by their absolute weak normalisation, and the
second functor is $\mathcal{D}^{\mathrm{awn}}_{\Lambda}$. ∎

To conclude this subsection, let us quickly settle the question what the
dualisable objects in $\mathcal{D}_{\Lambda}(X)$ for
$X\in\mathcal{C}(\mathfrak{G})$ look like. This will be important later.

######  Remark 2.30.

  1. i)

In $\mathcal{D}_{\Lambda}(X)$, the notions of dualisable objects, perfect
objects (in the sense of [Sta, 08G5]) and objects which are étale locally
constant with perfect values agree. The first two are equivalent by [Sta,
0FPP]. It is also clear that objects which are locally constant with perfect
values are perfect in the sense of [Sta], i.e. étale locally strictly perfect
(in the sense of [Sta, 085L]). Finally, the fact that dualisable objects are
locally constant with perfect values is shown in [CD16, Remark 6.3.27].

  2. ii)

It follows from $i)$ that the support of a dualisable object
$A\in\mathcal{D}_{\Lambda}(X)$ is both open and closed. Indeed, as $A$ is
locally constant, it is étale locally supported either everywhere or nowhere
and thus, as étale morphisms are open, the corresponding étale cover yields a
decomposition $X=U\amalg V$ into two open subsets such that $U$ is the support
of $A$.

###  2.4. Cohomologically smooth morphisms

Let us now finally recall the notion of "Poincaré duality" in a 6-functor
formalism.

######  Definition 2.31 ([Sch23, Definition 5.1]).

Let $\mathfrak{G}=(\mathcal{C},E)$ be a geometric setup and $\mathcal{D}$ a
6-functor formalism on $\mathfrak{G}$. Then a morphism $f:Y\to X$ in $E$ is
called ($\mathcal{D}$-)cohomologically smooth if it satisfies the following
conditions:

  1. i)

The natural map
$f^{*}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}f^{!}1_{X}\to
f^{!}\rule{8.5359pt}{0.42677pt}$ adjoint to
$f_{!}(f^{*}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}f^{!}1_{X})=\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}f_{!}f^{!}1_{X}\to\rule{8.5359pt}{0.42677pt}$
is an isomorphism.

  2. ii)

The object $f^{!}1_{X}\in\mathcal{D}(Y)$ (called the dualising complex) is
invertible.

  3. iii)

If $g:X^{\prime}\to X$ is any morphism in $\mathcal{C}$ and $f^{\prime}$ the
pullback of $f$ along $g$, then $f^{\prime}$ satisfies $i)$ and $ii)$.

  4. iv)

In the situation of $ii)$, the natural map
$(g^{\prime})^{*}f^{!}1_{X}\to(f^{\prime})^{!}1_{X^{\prime}}$ is an
isomorphism where $g^{\prime}$ is the pullback of $g$ along $f$.

######  Remark 2.32.

  1. i)

Without condition $ii)$, this is a special case of the notion of $f$-smooth
objects which we will discuss below. Actually, we will see there that
condition $i)$ and $iii)$ are already implied by $iv)$; and it is even enough
to check $iv)$ for $g=f$, cf. Corollary 2.52 below.

  2. ii)

If $f:Y\to X$ is cohomologically smooth, then there is a left adjoint for
$f^{*}$ (given by
$f_{!}(\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}f^{!}1_{X})$) and a
right adjoint for $f^{!}$ (given by
$f_{*}(\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}(f^{!}1_{X})^{-1})$).
The existence of the former again follows from $iv)$ alone.

######  Example 2.33.

  1. i)

If $\mathcal{D}$ is a 6-functor formalism on a construction setup
$\mathfrak{G}$, then all morphisms $f\in I(\mathfrak{G})$ are cohomologically
smooth (as $f^{*}=f^{!}$).

  2. ii)

Any universal homeomorphism $f\in E(\mathfrak{G})$ is
$\mathcal{D}_{\Lambda}$-cohomologically smooth (where $\Lambda$ is any torsion
ring). This follows from the invariance of the étale site under universal
homeomorphisms ([Sta, 03SI]). Indeed, note that $f\in P(\mathfrak{G})$ (a
universal homeomorphism is integral), so $f_{!}=f_{*}$ and since
$(f^{*},f_{*})$ (as underived functors) defines a equivalence of topoi the
right adjoint of $f_{*}$ is $f^{!}=f^{*}$. In fact, this shows that universal
homeomorphisms are even cohomologically étale, see Definition 2.61 below.

  3. iii)

If $\ell\not=p=\mathrm{char}\ k$, then any smooth morphism $f:Y\to X\in
E(\mathfrak{G})$ is $\mathcal{D}_{\mathbb{F}_{\ell}}$-cohomologically smooth,
using Theorem 2.72 below and e.g. [Zav23, Remark 6.1.9]. If $f$ is
additionally of pure dimension $d$ and $Y$ and $X$ are of finite type over
$\operatorname{Spec}k$, then $f^{!}\mathbb{F}_{\ell}$ identifies with the Tate
twist $\mu_{\ell}(d)[2d]$, i.e. $\mu_{\ell}^{\operatorname{\otimes}d}$ for the
étale sheaf $\mu_{\ell}$ of $\ell$-th roots of unity. In particular if
$k=\bar{k}$ and $f:Y\to\operatorname{Spec}k$ is smooth of dimension $d$, then
for any sheaf $A$ on $Y$ we have

$\displaystyle\operatorname{Hom}(H_{c}^{i}(Y,A),\mathbb{F}_{\ell})$
$\displaystyle\cong\operatorname{Hom}(f_{!}A,\mathbb{F}_{\ell}[-i])\cong\operatorname{Hom}(A,f^{!}\mathbb{F}_{\ell}[-i])$
$\displaystyle\cong\operatorname{Hom}(\mathbb{F}_{\ell},\operatorname{\underline{Hom}}(A,f^{*}\mathbb{F}_{\ell}\operatorname{\otimes}\mu_{\ell}(d)[2d-i])$
$\displaystyle\cong
H^{2d-i}(Y,\operatorname{\underline{Hom}}(A,\mathbb{F}_{\ell})\operatorname{\otimes}\mu_{\ell}(d))$

which is usually written as $H^{i}_{c}(Y,A)^{\lor}\cong
H^{2d-i}(Y,A^{\lor}(d))$; this is Poincaré duality.

The properties of cohomologically smooth morphisms are most naturally
discussed in the context of the more general notion of smooth objects for a
morphism $f$.

###  2.5. Smooth and proper objects

Intuitively, smooth and proper objects are the answer to the following
question: Let $\mathcal{D}$ be a 6-functor formalism on a geometric setup
$\mathfrak{G}$. Given a morphism $f:Y\to X$ in $E(\mathfrak{G})$, for which
objects $A,B\in\mathcal{D}(Y)$ is the functor
$f_{!}(A\operatorname{\otimes}\rule{8.5359pt}{0.42677pt})$ "universally" left
resp. right adjoint to the functor
$f^{*}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}B$? It turns out that
these adjunctions can be modelled in another 2-category, called the Lu-Zheng
category $LZ_{\mathcal{D}}$, the category of cohomological correspondences or
simply the magical 2-category. For a precise construction we refer to [Zav23,
Definition 2.2.3]. We assume that $E(\mathfrak{G})$ contains all morphisms in
$\mathcal{C}(\mathfrak{G})$. Then $LZ_{\mathcal{D}}$ can be described as
follows:

  * •

The objects of $LZ_{\mathcal{D}}$ are the objects of
$\mathcal{C}(\mathfrak{G})$.

  * •

Given objects $X,Y$ of $LZ_{\mathcal{D}}$ the category of morphisms $Y\to X$
is $\mathcal{D}(Y\times X)$.

  * •

Given morphisms $\mathcal{D}(Z\times Y)\ni A:Z\to Y$, $\mathcal{D}(Y\times
X)\ni B:Y\to X$ the composition $\mathcal{D}(Z\times X)\ni B\circ A:Z\to X$ is
defined as the convolution

$B\star A:=p_{ZX,!}(p_{YX}^{*}B\operatorname{\otimes}p_{ZY}^{*}A)$

where the $p_{??}$ denote the respective projections from $Z\times Y\times X$.
The identity morphism $id_{X}\in\mathcal{D}(X\times X)$ is given by
$\Delta_{X,!}1_{X}$.

We quickly recall the notions of adjunctions in 2-categories and adjoint
functors of $\infty$-categories.

######  Definition 2.34.

[Lur23, 02CG, 02EK, 02EN]

  1. i)

Let $\mathcal{C}$ be a 2-category. An adjunction between two morphisms
$f\in\mathcal{C}(X,Y)$ and $g\in\mathcal{C}(Y,X)$ is a pair $(\eta,\epsilon)$
of a morphism $\eta:id_{X}\to gf$ in $\mathcal{C}(X,X)$ and a morphism
$\epsilon:fg\to id_{Y}$ in $\mathcal{C}(Y,Y)$ such that the two compositions

$f\overset{f\eta}{\to}fgf\overset{\epsilon f}{\to}f\quad\mathrm{and}\quad
g\overset{\eta g}{\to}gfg\overset{g\epsilon}{\to}g$

are the identities on $f\in\mathcal{C}(X,Y)$ and $g\in\mathcal{C}(Y,X)$
respectively (here we implicitly used the unit and associativity constraints
of the 2-category $\mathcal{C}$).

  2. ii)

Let $\mathcal{C}$ and $\mathcal{D}$ be two $\infty$-categories. An adjunction
of two functors $F:\mathcal{C}\to\mathcal{D}$ and
$G:\mathcal{D}\to\mathcal{C}$ is an adjunction between $[F]$ and $[G]$ in the
homotopy $2$-category $h_{2}(\mathrm{Cat}_{\infty})$.

Now assume $\mathcal{C}(\mathfrak{G})$ has a final object $S$. Then we can
consider the 2-functor
$h_{S}=\operatorname{Hom}(S,\rule{8.5359pt}{0.42677pt})$ from
$LZ_{\mathcal{D}}$ to $\mathrm{Cat}$.888cf. [JY21] for the theory of
representable functors in 2-categories. By definition, this sends an object
$X\in LZ_{\mathcal{D}}$ to $\mathcal{D}(X)$ and a morphism
$\mathcal{D}(Y\times X)\ni A:Y\to X$ to the functor
$FM_{A}:\mathcal{D}(Y)\to\mathcal{D}(X)$ mapping $M$ to $A\star
M=p_{X,!}(A\operatorname{\otimes}p_{Y}^{*}M)$, called the Fourier-Mukai
functor with kernel $A$. In particular, if $X=S$ and $f:Y\to X$, $FM_{A}$ is
exactly the functor
$f_{!}(A\operatorname{\otimes}\rule{8.5359pt}{0.42677pt})$; on the other hand,
if we consider $A$ as morphism $X\to Y$, it is the functor
$A\operatorname{\otimes}f^{*}\rule{8.5359pt}{0.42677pt}$. As the 2-functor
$h_{S}$ preserves adjunctions, an adjunction of morphisms $\mathcal{D}(X)\ni
A:X\to S$ and $\mathcal{D}(X)\ni B:S\to X$ thus yields an adjunction of
functors $f_{!}(A\operatorname{\otimes}\rule{8.5359pt}{0.42677pt})$ and
$B\operatorname{\otimes}f^{*}\rule{8.5359pt}{0.42677pt}$. The following
definition is now sufficiently motivated:

######  Definition 2.35.

Let $\mathcal{D}$ be a 6-functor formalism on a geometric setup
$\mathfrak{G}=(\mathcal{C},E)$ and let $f:Y\to X\in E$. Let
$\mathcal{C}_{E/X}$ be the full subcategory of the slice category
$\mathcal{C}_{/X}$ consisting of objects with the structure morphism in $E$,
and $\mathcal{D}_{E/X}$ the restriction of $\mathcal{D}$ to
$\mathcal{C}_{E/X}$.999See Remark 2.11. Then an object $A\in\mathcal{D}(Y)$ is
called

  1. i)

$f$-smooth (w.r.t. $\mathcal{D}$) if $A$ as a morphism $Y\to X$ in
$LZ_{\mathcal{D}_{E/X}}$ admits a right adjoint $B$.

  2. ii)

$f$-proper (w.r.t. $\mathcal{D}$) if $A$ as a morphism $X\to Y$ in
$LZ_{\mathcal{D}_{E/X}}$ admits a right adjoint $B$.

######  Remark 2.36.

  1. i)

Both properties are stable under base change, as base change defines a
2-functor of the corresponding Lu-Zheng categories, cf. e.g. the proof of
[Man22b, Proposition 7.7].

  2. ii)

Observe that for a morphism $f:Y\to X$ in $E(\mathfrak{G})$, the functor
$f_{!}(A\operatorname{\otimes}\rule{8.5359pt}{0.42677pt})$ always admits the
right adjoint
$\operatorname{\underline{Hom}}(A,f^{!}\rule{8.5359pt}{0.42677pt})$; in
particular, if $A$ is $f$-smooth, there is a natural isomorphism of functors
$\operatorname{\underline{Hom}}(A,f^{!}\rule{8.5359pt}{0.42677pt})\cong
B\operatorname{\otimes}f^{*}\rule{8.5359pt}{0.42677pt}$. Applying this to
$1_{X}$, we see that necessarily
$B\cong\operatorname{\underline{Hom}}(A,f^{!}1_{X})=:\mathbb{D}_{f}(A)$, the
Verdier dual of $A$. One can now wonder if it suffices conversely to supply a
natural isomorphism
$\operatorname{\underline{Hom}}(A,f^{!}\rule{8.5359pt}{0.42677pt})\cong\mathbb{D}_{f}(A)\operatorname{\otimes}f^{*}\rule{8.5359pt}{0.42677pt}$
for $A$ to be $f$-smooth. The problem is that this condition is not
necessarily stable under base change; however, being a bit more careful with
this idea yields Corollary 2.44 below.

In the following we stay in the situation of Definition 2.35 and restrict the
discussion to $f$-smooth objects as the discussion of $f$-proper objects is
dual to this in a certain sense (cf. [Sch23, Remark 6.3]).

We can think about an adjunction in 2-categories in different ways and the
different viewpoints yield useful criteria for $f$-smoothness. The first way
is really just the definition of adjunctions in 2-categories (Definition
2.34): to define an adjunction between morphisms $A:Y\to X$, $B:X\to Y$ in
$LZ_{\mathcal{D}_{E/X}}$ is to provide a unit $\Delta_{Y,!}1_{Y}=id_{Y}\to
B\circ A=p_{1}^{*}B\operatorname{\otimes}p_{2}^{*}A$ and a counit
$f_{!}(A\operatorname{\otimes}B)=A\circ B\to id_{X}=1_{X}$ such that the
triangle identities hold.

######  Definition 2.37.

An $A$-trace-cycle theory on $f$ is a triple $(B,\alpha,\beta)$ of

  1. i)

an object $B\in\mathcal{D}(Y)$,

  2. ii)

an $A$-trace morphism

$\alpha:f_{!}(A\operatorname{\otimes}B)\to 1_{Y},$

  3. iii)

an $A$-cycle map

$\beta:\Delta_{Y,!}1_{Y}\to p_{2}^{*}B\operatorname{\otimes}p_{1}^{*}A,$

such that the corresponding "triangle identity" diagrams commute.101010For an
explicit description of these triangles, see [Sch23, Lecture VI]. Here $p_{i}$
denote the projections $Y\times_{X}Y\to Y$ and $\Delta_{Y}$ the diagonal.

######  Lemma 2.38.

An object $A$ is $f$-smooth if and only if $f$ admits an $A$-trace-cycle
theory.

###### Proof.

This follows by unravelling the definitions. ∎

######  Remark 2.39.

Note the symmetry in this definition: $(B,\alpha,\beta)$ is an $A$-trace-cycle
theory if and only if $(A,\alpha,\beta)$ is a $B$-trace-cycle theory on $f$.
In particular, if $A$ is $f$-smooth the right adjoint of $A$ is $f$-smooth as
well, with right adjoint $A$. In particular, if $A$ is $f$-smooth, then
$\mathbb{D}_{f}(\mathbb{D}_{f}(A))=A$.

######  Corollary 2.40.

$A\in\mathcal{D}(X)$ is $id_{X}$-smooth if and only if it is dualisable.

###### Proof.

This follows from Lemma 2.38 as an $A$-trace cycle theory $(B,\alpha,\beta)$
on $id_{X}$ is precisely the datum of a duality between $A$ and $B$. ∎

Another way to think about the adjunction (see [Lur23, 02D2]) is to ask for a
2-morphism $\epsilon:f_{!}(A\operatorname{\otimes}B)=A\circ B\to id_{X}=1_{X}$
such that for all $T:Z\to Y$, $S:Z\to X$ in $LZ$, the natural map
$\operatorname{Hom}(T,B\circ S)\to\operatorname{Hom}(A\circ T,S)$ is an
isomorphism.111111The map is induced by taking the left adjunct of a given map
$\beta:T\to B\circ S$; this is the composition $A\circ
T\overset{id_{A}\circ\beta}{\Rightarrow}A\circ(B\circ S)\Rightarrow(A\circ
B)\circ S\overset{\epsilon\circ id_{S}}{\Rightarrow}S$. As noted above, for
the adjunction to exist, we must necessarily have $B\cong\mathbb{D}_{f}(A)$.
Now we always have a map
$\epsilon:f_{!}(A\operatorname{\otimes}\mathbb{D}_{f}(A))\to 1_{X}$, namely
the one adjoint to the identity on
$\mathbb{D}_{f}(A)=\operatorname{\underline{Hom}}(A,f^{!}1_{X})$. Moreover,
applying the definitions, we see that

$\operatorname{Hom}(A\circ
T,S)=\operatorname{Hom}(p_{ZX,!}(p_{YX}^{*}A\operatorname{\otimes}p_{ZY}^{*}T),S)\cong\operatorname{Hom}(T,p_{ZY,*}\operatorname{\underline{Hom}}(p_{YX}^{*}A,p_{ZX}^{!}S))$

where we used the adjunctions of the six functors, and

$\operatorname{Hom}(T,B\circ
S)=\operatorname{Hom}(T,p_{ZY,!}(p_{XY}^{*}B\operatorname{\otimes}p_{ZX}^{*}S))$

Also note that $p_{ZY}=id$ as $X$ is final. All these observations culminate
in the following result:

######  Lemma 2.41.

In the situation of Definition 2.35, the following are equivalent:

  1. i)

$A\in\mathcal{D}(Y)$ is $f$-smooth.

  2. ii)

For all morphisms $Z\to X$ and $S\in\mathcal{D}(Z)$, the natural map

$p_{Y}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z}^{*}S\to\operatorname{\underline{Hom}}(p_{Y}^{*}A,p_{Z}^{!}S)$

induced by the identifications above under Yoneda is an
isomorphism.121212Here, $p_{Y},p_{Z}$ denote the projections from
$Z\times_{X}Y$. A calculation yields that this is the map adjoint to the
natural map
$p_{Z,!}(p_{Y}^{*}A\operatorname{\otimes}p_{Y}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z}^{*}S)=p_{Z,!}p_{Y}^{*}(A\operatorname{\otimes}\mathbb{D}_{f}(A))\operatorname{\otimes}S=f^{*}f_{!}(A\operatorname{\otimes}\mathbb{D}_{f}(A))\operatorname{\otimes}S\to
f^{*}f_{!}f^{!}1_{X}\operatorname{\otimes}S\to S$ where we used the projection
formula and base change and the counits of the
$\operatorname{\otimes}$-$\operatorname{\underline{Hom}}$\- and
$f_{!}$-$f^{!}$-adjunction.

  3. iii)

[Sch23, Proposition 6.6] The natural map

$p_{2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{1}^{*}A\to\operatorname{\underline{Hom}}(p_{2}^{*}A,p_{1}^{!}A)$

coming from the map in $ii)$ by choosing $Z=Y$ and $S=A$ is an isomorphism. In
fact, it suffices to check this after applying $\Delta^{!}$ and taking global
sections.

###### Proof.

$i)\Rightarrow ii)$ follows from the discussion above by Yoneda; the only
caveat is that one needs to ensure that the identification
$B\cong\mathbb{D}_{f}(A)$ is compatible with the choice of counits. To see
this, note that by the above discussion for $\mathcal{D}(Y)\ni T:X\to Y$ and
$S=id=1_{X}$ the natural isomorphism $\operatorname{Hom}(T,B\circ
S)\to\operatorname{Hom}(A\circ T,S)$ corresponds to an isomorphism
$\operatorname{Hom}(T,B)\to\operatorname{Hom}(T,\mathbb{D}_{f}(A))$. By
Yoneda, this induces an isomorphism $\varphi:B\to\mathbb{D}_{f}(A)$ and a
quick chase of the identifications reveals that it is given by the morphism
adjoint to the counit $\epsilon:f_{!}(B\operatorname{\otimes}A)\to 1_{X}$. But
then from the commutative diagram

${B}$${\mathbb{D}_{f}(A)}$${\mathbb{D}_{f}(A)}$${\mathbb{D}_{f}(A)}$$\scriptstyle{\varphi}$$\scriptstyle{\varphi}$$\scriptstyle{id}$$\scriptstyle{id}$

we see that the map $\epsilon$ factors through the morphism
$f_{!}(\mathbb{D}_{f}(A)\operatorname{\otimes}A)\to 1$ adjoint to the identity
on $\mathbb{D}_{f}(A)$ via $f_{!}(\varphi\operatorname{\otimes}A)=\varphi\circ
A$ by passing to adjoint morphisms in the horizontal direction.

$iii)$ is a special case of $ii)$, so it remains to show $iii)\Rightarrow i)$.
By the proof of [Lur23, missing02CY], for $A$ to be left adjoint to
$\mathbb{D}_{f}(A)$, it suffices to check that the natural map
$\operatorname{Hom}(T,\mathbb{D}_{f}(A)\circ S)\to\operatorname{Hom}(A\circ
T,S)$ is an isomorphism for the pairs $(T,S)=(\mathbb{D}_{f}(A),id_{X})$ and
$(T,S)=(id_{Y}=\Delta_{!}1_{Y},A)$. But for the first pair this is automatic
from the definition of $\mathbb{D}_{f}(A)$ and $\epsilon$ and for the second
pair this is by construction implied by condition $iii)$. ∎

######  Remark 2.42.

After applying $\Delta^{!}$ to the map in $iii)$ in Lemma 2.41 above, it
becomes a morphism

$\Delta^{!}(p_{2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{1}^{*}A)\to\operatorname{\underline{Hom}}(A,A)$

by the interior pullback formula (Proposition 2.10). The bifunctor
$\Delta^{!}(p_{2}^{*}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}p_{1}^{*}\rule{8.5359pt}{0.42677pt})$
is known as the !-tensor product $\operatorname{\otimes}^{!}$ (and appears
e.g. in the work of Gaitsgory-Rozenblyum, [GR19]). Using this notation, the
criterion takes a more concise form: An object $A$ is $f$-smooth if and only
if the natural map

$\mathbb{D}_{f}(A)\operatorname{\otimes}^{!}A\to\operatorname{\underline{Hom}}(A,A)$

is an isomorphism.

######  Corollary 2.43.

Assume that all $\mathcal{D}(X)$ are stable $\infty$-categories. Then:

  1. i)

If two objects in a distinguished triangle are $f$-smooth, so is the third.

  2. ii)

$f$-smooth objects are stable under retracts.

###### Proof.

The transformation
$p_{Y}^{*}\mathbb{D}_{f}(\rule{8.5359pt}{0.42677pt})\operatorname{\otimes}p_{Z}^{*}S\to\operatorname{\underline{Hom}}(p_{Y}^{*}\rule{8.5359pt}{0.42677pt},p_{Z}^{!}S)$
in condition $ii)$ of Lemma 2.41 is a natural map of exact functors. This
implies both statements. ∎

The following corollary mimics the definition of cohomologically smooth
morphisms:

######  Corollary 2.44.

In the situation of Definition 2.35, an object $A$ is $f$-smooth if and only
if the following hold

  1. i)

The natural map
$\mathbb{D}_{f}(A)\operatorname{\otimes}f^{*}\rule{8.5359pt}{0.42677pt}\to\operatorname{\underline{Hom}}(A,f^{!}\rule{8.5359pt}{0.42677pt})$
adjoint to

$f_{!}(A\operatorname{\otimes}\mathbb{D}_{f}(A)\operatorname{\otimes}f^{*}\rule{8.5359pt}{0.42677pt})\overset{PF}{\cong}f_{!}(A\operatorname{\otimes}\mathbb{D}_{f}(A))\operatorname{\otimes}\rule{8.5359pt}{0.42677pt}\to
f_{!}f^{!}1_{X}\operatorname{\otimes}\rule{8.5359pt}{0.42677pt}\to\rule{8.5359pt}{0.42677pt}$

is an isomorphism.

  2. ii)

If $g:X^{\prime}\to X$ is any morphism in $\mathcal{C}$, $f^{\prime}$ the
pullback of $f$ along $g$, then $f^{\prime}$ satisfies $i)$ with $A$ replaced
by $g^{\prime*}A$ where $g^{\prime}:Y^{\prime}\to Y$ is the pullback of $g$
along $f$.

  3. iii)

In the situation of $ii)$, the natural map
$(g^{\prime})^{*}\mathbb{D}_{f}(A)\to\mathbb{D}_{f^{\prime}}((g^{\prime})^{*}A)$
(the map from Lemma 2.41 for $S=1_{Y^{\prime}}$) is an isomorphism.

###### Proof.

For the "only if" part, we only need to show that $i)$ and $ii)$ hold if $A$
is f-smooth by Lemma 2.41. By stability under base change, it even suffices to
show $i)$. But this follows because

${{\operatorname{Hom}(f_{!}(M\operatorname{\otimes}A),N)}}$${{\operatorname{Hom}(M,\operatorname{\underline{Hom}}(A,f^{!}N))}}$${{\operatorname{Hom}(M,\mathbb{D}_{f}(A)\operatorname{\otimes}f^{*}N)}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

commutes for all $M,N$ by construction.131313More precisely the identity on
$\mathbb{D}_{f}(A)\operatorname{\otimes}f^{*}N$ maps to the map
$f_{!}(\mathbb{D}_{f}(A)\operatorname{\otimes}f^{*}N\operatorname{\otimes}A)\overset{PF}{\cong}f_{!}(\mathbb{D}_{f}(A)\operatorname{\otimes}A)\operatorname{\otimes}N\to
1\operatorname{\otimes}N$ via $\epsilon\operatorname{\otimes}id$ where
$\epsilon$ is the counit of the adjunction in LZ, i.e. given by
$f_{!}(\mathbb{D}_{f}(A)\operatorname{\otimes}A)\to f_{!}f^{!}1_{X}\to 1_{X}$.
But this is precisely how we defined the natural map in $i)$. For the "if"
part, we use $ii)$ and $iii)$ with $g=f$ to see that $iii)$ in Lemma 2.41
holds. ∎

As smooth and proper objects are actually morphisms in a (2-)category, it is a
natural question if and how they can be composed. Indeed, suppose we are given
a commutative triangle

${Z}$${Y}$${X}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{f}$

Then, given a smooth/proper object for $g$ and one for $f$ resp. $h$, one can
wonder if they can be composed to a smooth/proper object for $h$ resp. $f$. By
definition, being $g$-smooth/proper is a statement about adjunctions in
$LZ_{\mathcal{D}_{E/Y}}$ as opposed to $LZ_{\mathcal{D}_{E/X}}$ for $h$ and
$f$. So one needs an appropriate 2-functor $LZ_{\mathcal{D}_{E/Y}}\to
LZ_{\mathcal{D}_{E/X}}$ which allows to compose the given adjoints in
$LZ_{\mathcal{D}_{E/X}}$. Such a functor exists and is given by the functor
$\mathcal{T}$ which is the forgetful functor (along $f$) on objects and sends
a morphism
$A\in\mathcal{D}(S\times_{Y}T)=\operatorname{Hom}_{LZ_{\mathcal{D}_{E/Y}}}(S,T)$
to
$\iota_{!}A\in\mathcal{D}(S\times_{X}T)=\operatorname{Hom}_{LZ_{\mathcal{D}_{E/X}}}(S,T)$
where $\iota$ is the obvious map $S\times_{Y}T\to S\times_{X}T$ (cf. the proof
of [Man22b, Proposition 7.11]). This allows us to prove the following:

######  Lemma 2.45.

Let $\mathcal{D}$ be a 6-functor formalism on a geometric setup
$\mathfrak{G}$. Let $g:Z\to Y$ and $f:Y\to X$ be two morphisms in
$E(\mathfrak{G})$ and $h$ their composition. Let $A\in\mathcal{D}(Y)$ and
$B,C\in\mathcal{D}(Z)$. Then:

  1. i)

If $B$ is $g$-smooth and $A$ is $f$-smooth, $g^{*}A\operatorname{\otimes}B$ is
$h$-smooth.

  2. ii)

If $B$ is $g$-proper and $C$ is $h$-smooth, then
$g_{!}(C\operatorname{\otimes}B)$ is $f$-smooth.

  3. iii)

If $A$ is $f$-proper and $B$ is $g$-proper, then
$B\operatorname{\otimes}g^{*}A$ is $h$-proper.

  4. iv)

If $C$ is $h$-proper and $B$ is $g$-smooth, then
$g_{!}(B\operatorname{\otimes}C)$ is $f$-proper.

More precisely, in the respective cases we have adjunctions

  1. i)

$g^{*}A\operatorname{\otimes}B\dashv\mathbb{D}_{g}(B)\operatorname{\otimes}g^{*}\mathbb{D}_{f}(A)$

  2. ii)

$g_{!}(C\operatorname{\otimes}B)\dashv
g_{!}(\boldsymbol{D}_{g}(B)\operatorname{\otimes}\mathbb{D}_{h}(C))$

  3. iii)

$B\operatorname{\otimes}g^{*}A\dashv
g^{*}\boldsymbol{D}_{f}(A)\operatorname{\otimes}\boldsymbol{D}_{g}(B)$

  4. iv)

$g_{!}(B\operatorname{\otimes}C)\dashv
g_{!}(\boldsymbol{D}_{h}(C)\operatorname{\otimes}\mathbb{D}_{g}(B))$

where $\boldsymbol{D}_{s}(T)$ denotes the right adjoint of an $s$-proper
object $T$.141414Similar as in the case of smooth objects, one can show that
necessarily $\boldsymbol{D}_{s}(T)\cong
p_{1}^{*}\operatorname{\underline{Hom}}(p_{2}^{*}T,\Delta_{!}1)$, cf. [Sch23,
Proposition 6.8].

###### Proof.

We only prove $i)$ and $ii)$ as the arguments for $iii)$ and $iv)$ are dual to
this. In $i)$, $B$ is considered as a morphism $Z\to Y$ and so we can consider
the composition $A\circ\iota_{!}B$ where $\iota$ is the natural morphism $Z\to
Z\times_{X}Y$, i.e. the graph of $g$. As noted above $\mathcal{T}$ defines a
$2$-functor, so $\iota_{!}B$ will be left adjoint to
$\iota_{!}\mathbb{D}_{g}(B)$151515We are a bit sloppy here and do not
distinguish between $Z\times Y$ and $Y\times Z$; but of course we have to be
careful in which direction the morphisms given by the objects go. and thus
$A\circ\iota_{!}B$ will be left adjoint to the composition of right adjoints
$\iota_{!}\mathbb{D}_{g}(B)\circ\mathbb{D}_{f}(A)$. Now we just calculate

$A\circ\iota_{!}B\cong
p_{Z,!}(p_{Y}^{*}A\operatorname{\otimes}\iota_{!}B)\overset{PF}{\cong}p_{Z,!}\iota_{!}(\iota^{*}p_{Y}^{*}A\operatorname{\otimes}B)\cong
g^{*}A\operatorname{\otimes}B$

as claimed. Unravelling the composition of right adjoints in a similar way, we
get
$\iota_{!}\mathbb{D}_{g}(B)\circ\mathbb{D}_{f}(A)\cong\mathbb{D}_{g}(B)\operatorname{\otimes}g^{*}\mathbb{D}_{f}(A)$.

In $ii)$ $B$ is considered as a morphism $Y\to Z$ and so we consider the
composition

$C\circ\iota_{!}B=p_{Y_{!}}(p_{Z}^{*}C\operatorname{\otimes}\iota_{!}B)\overset{PF}{\cong}g_{!}(C\operatorname{\otimes}B)$

and the similar calculation for the right adjoints gives
$\iota_{!}\mathbb{D}_{g}(B)\circ\mathbb{D}_{h}(C)\cong
g_{!}(\boldsymbol{D}_{g}(B)\operatorname{\otimes}\mathbb{D}_{h}(C))$. ∎

######  Corollary 2.46.

In the situation of Definition 2.35, assume that $A$ is $f$-smooth. Then for
every dualisable object $L\in\mathcal{D}(Y)$, $A\operatorname{\otimes}L$ is
$f$-smooth. If $C$ is $f$-proper and $B$ is $f$-smooth, then
$f_{!}(B\operatorname{\otimes}C)$ is dualisable.

###### Proof.

This is a consequence of Corollary 2.40 and Lemma 2.45 by writing $f=id\circ
f$ and $f=f\circ id$. ∎

An interesting property of smooth objects is that they are compact under
(relatively) mild assumptions.

######  Proposition 2.47.

In the situation of Definition 2.35, assume that $\Delta_{f,!}1_{Y}$ is
compact. Then any $f$-smooth object $A\in\mathcal{D}(Y)$ is compact.

###### Proof.

Assume $A$ is $f$-smooth. Then by Lemma 2.41 we have for all
$S\in\mathcal{D}(Y)$ that
$p_{2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{1}^{*}S\cong\operatorname{\underline{Hom}}(p_{2}^{*}A,p_{1}^{!}S)$.
Applying $\Delta_{f}^{!}$ on both sides yields
$\Delta_{f}^{!}(p_{2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{1}^{*}S)\cong\operatorname{\underline{Hom}}(A,S)$
by interior pullback (Proposition 2.10). Thus

$\operatorname{Hom}(A,S)\cong\operatorname{Hom}(1_{Y},\operatorname{\underline{Hom}}(A,S))\cong\operatorname{Hom}(1_{Y},\Delta_{f}^{!}(p_{2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{1}^{*}S))\cong\operatorname{Hom}(\Delta_{f,!}1_{Y},p_{2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{1}^{*}S)$

and the last term commutes with colimits in $S$ as $\Delta_{f,!}1_{Y}$ is
compact by assumption. ∎

######  Example 2.48.

We will see in Corollary 5.3 that if $k$ is an algebraically closed field, the
assumption of Proposition 2.47 holds for all morphisms $f\in E(\mathfrak{G})$
between schemes of finite type over $k$ and 6-functor formalisms
$\mathcal{D}_{\Lambda}$ on $\mathfrak{G}$.

Let us also note the following descent result for smooth objects.

######  Proposition 2.49.

Let $\mathcal{D}$ be a 6-functor formalism on a geometric setup
$\mathfrak{G}$. Let $g:X^{\prime}\to X$ be a map of universal
$\mathcal{D}^{*}$-descent.161616That is, $g$ induces
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arrows are given by the respective pullbacks, and the same holds for any base
change of $g$ If $f:Y\to X\in E(\mathfrak{G})$ and $g^{\prime}:Y^{\prime}\to
X^{\prime}$ is the pullback of $g$ along $f$, then $A\in\mathcal{D}(Y)$ is
$f$-smooth if and only if $g^{\prime*}A\in\mathcal{D}(Y^{\prime})$ is
$f^{\prime}$-smooth, where $f^{\prime}$ denotes the pullback of $f$ along $g$.

###### Proof.

This follows from the argument given in [Man22b, Corollary 7.8]. ∎

Finally, we have the following very useful result on stability of smooth
objects under retracts which generalises Corollary 2.43 $ii)$:

######  Proposition 2.50.

Let $\mathcal{D}$ be a 6-functor formalism on a geometric setup
$\mathfrak{G}$. Consider a diagram

${Z}$${Y}$${X}$$\scriptstyle{i}$$\scriptstyle{h}$$\scriptstyle{f}$

with $f,h\in E(\mathfrak{G})$ such that there is some $r:Y\to Z$ over $X$ with
$r\circ i=id$. Let $A\in\mathcal{D}(Y)$ be $f$-smooth and
$C\in\mathcal{D}(Z)$. If there are maps $i^{*}A\to C$ and $r^{*}C\to A$ such
that $C=i^{*}r^{*}C\to i^{*}A\to C$ is the identity, then $C$ is $h$-smooth
(and its right adjoint is a retract of $r_{!}\mathbb{D}_{f}(A)$).

###### Proof.

See [Sch23, Proposition 6.17]. ∎

###  2.6. Properties of cohomologically smooth morphisms

We can now return to the special case of cohomologically smooth morphisms. We
fix a 6-functor formalism $\mathcal{D}$ on a geometric setup $\mathfrak{G}$.

######  Definition 2.51 ([Zav23, Definition 3.2.4]).

Let $(f:Y\to X)\in E(\mathfrak{G})$. A trace-cycle theory on $f$ is a
$1_{Y}$-trace-cycle theory $(\omega_{f},tr_{f},cl_{f})$ on $f$ such that
$\omega_{f}$ is invertible.

######  Corollary 2.52.

Let $(f:Y\to X)\in E(\mathfrak{G})$. Then the following are equivalent:

  1. i)

$f$ is cohomologically smooth.

  2. ii)

$1_{Y}$ is $f$-smooth and
$\mathbb{D}_{f}(1)=\operatorname{\underline{Hom}}(1_{Y},f^{!}1_{X})=f^{!}1_{X}$
is invertible.

  3. iii)

$f^{!}1_{X}$ is invertible and the natural map $p_{2}^{*}f^{!}1_{X}\to
p_{1}^{!}1_{Y}$ is an isomorphism, where $p_{1},p_{2}$ denote the projections
$Y\times_{X}Y\to Y$.

  4. iv)

$f$ admits a trace-cycle theory.

###### Proof.

This is just Lemma 2.38, Lemma 2.41 and Corollary 2.44 for $A=1_{Y}$ and with
the additional assumption that $\mathbb{D}_{f}(A)=f^{!}1_{X}$ is invertible. ∎

######  Corollary 2.53.

Cohomologically smooth morphisms are stable under composition.

###### Proof.

This follows immediately from Corollary 2.52 and Lemma 2.45 $i)$ as pullback
preserves invertible objects. ∎

######  Corollary 2.54.

Assume $(f:Y\to X)\in E(\mathfrak{G})$ is cohomologically smooth. Then
$f^{!}1_{X}\cong\Delta^{*}p_{1}^{!}1_{Y}$.

###### Proof.

Apply $\Delta^{*}$ to the isomorphism in $iii)$ of Corollary 2.52. ∎

######  Remark 2.55.

This fact is called "Verdier’s diagonal trick" and reduces the computation of
the dualising object to cases where the morphism has a section. This is used
in [Zav23] to deduce a formula for the dualising complex in sufficiently nice
settings.

######  Lemma 2.56.

Let $(f:Y\to X)\in E(\mathfrak{G})$ and $L\in\mathcal{D}(X)$ dualisable. Then
for all $A\in\mathcal{D}(X)$, $f^{!}(L\operatorname{\otimes}A)\cong
f^{*}L\operatorname{\otimes}f^{!}A$.

###### Proof.

Let $K$ be the dual of $L$. As pullback preserves duality, this follows by
Yoneda since

$\displaystyle\operatorname{Hom}(\rule{8.5359pt}{0.42677pt},f^{!}(L\operatorname{\otimes}A))$
$\displaystyle\cong\operatorname{Hom}(f_{!}(\rule{8.5359pt}{0.42677pt})\operatorname{\otimes}K,A)\overset{\text{PF}}{\cong}\operatorname{Hom}(f_{!}(\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}f^{*}K),A)\cong\operatorname{Hom}(\rule{8.5359pt}{0.42677pt},f^{*}L\operatorname{\otimes}f^{!}A)$

∎

If $\mathfrak{G}$ is a good geometric setup, then for a cohomologically smooth
morphism $f$ the inverse of the dualising complex can be described in terms of
the diagonal $\Delta_{f}$.

######  Corollary 2.57.

Assume $\mathfrak{G}$ is a good geometric setup and $(f:Y\to X)\in
E(\mathfrak{G})$ is cohomologically smooth. Then
$(f^{!}1_{X})^{-1}\cong\Delta^{!}1_{Y\times_{X}Y}$. In particular,
$f^{*}=f^{!}\rule{8.5359pt}{0.42677pt}\operatorname{\otimes}\Delta^{!}1_{Y\times_{X}Y}$.

###### Proof.

First note that if $\mathfrak{G}$ is good, $\Delta^{!}$ indeed exists because
$\Delta$ as a section of a morphism in $E$ is contained in $E$. Then note that
by Corollary 2.54
$f^{!}1_{X}\operatorname{\otimes}\Delta^{!}1_{Y\times_{X}Y}\cong\Delta^{*}p_{1}^{!}1_{Y}\operatorname{\otimes}\Delta^{!}1_{Y\times_{X}Y}\cong\Delta^{!}p_{1}^{!}1_{Y}\cong
1_{Y}$ where we used Lemma 2.56 and invertibility of $p_{1}^{!}1_{Y}$. ∎

######  Lemma 2.58.

For a cohomologically smooth morphism $(f:Y\to X)\in E(\mathfrak{G})$ and
$A,B\in\mathcal{D}(X)$ we have
$f^{*}\operatorname{\underline{Hom}}(A,B)\cong\operatorname{\underline{Hom}}(f^{*}A,f^{*}B)$.

###### Proof.

This follows from the interior pullback formula (Proposition 2.10):

$\displaystyle
f^{*}\operatorname{\underline{Hom}}(A,B)\operatorname{\otimes}f^{!}1_{X}\cong
f^{!}\operatorname{\underline{Hom}}(A,B)\cong\operatorname{\underline{Hom}}(f^{*}A,f^{!}B)$
$\displaystyle\cong\operatorname{\underline{Hom}}(f^{*}A,f^{*}B\operatorname{\otimes}f^{!}1_{X})$
$\displaystyle\cong\operatorname{\underline{Hom}}(f^{*}A,f^{*}B)\operatorname{\otimes}f^{!}1_{X}$

where we used that $f^{!}1_{X}$ is invertible and can thus be pulled out of
the internal $\operatorname{Hom}$ functor. ∎

Cohomologically smooth morphisms can be used to check the smoothness of
objects:

######  Theorem 2.59.

Consider a collection of commutative diagram

${Z_{i}}$${Y}$${X}$$\scriptstyle{g_{i}}$$\scriptstyle{h_{i}}$$\scriptstyle{f}$

for $i\in I$ of morphisms in $E(\mathfrak{G})$ with all $g_{i}$
cohomologically smooth and $A\in\mathcal{D}(Y)$, where $I$ is some index set.
If $(g_{i}^{*})_{I}:\mathcal{D}(Y)\to\prod_{I}\mathcal{D}(Z_{i})$ is
conservative, then the following are equivalent:

  1. i)

$A$ is $f$-smooth.

  2. ii)

$g_{i}^{*}A$ is $h_{i}$-smooth for all $i$.

Moreover, if $i)$ or $ii)$ holds then the Verdier-dual
$\mathbb{D}_{h_{i}}(g_{i}^{*}A)$ of $g_{i}^{*}A$ is given by
$g_{i}^{!}\mathbb{D}_{f}(A)$ and $\mathbb{D}_{f}(A)$ is invertible if and only
if $\mathbb{D}_{h_{i}}(g_{i}^{*}A)$ is invertible for all $i$.

###### Proof.

If $A$ is $f$-smooth, then $g_{i}^{*}A$ is $h_{i}$-smooth for all $i$ by Lemma
2.45. To show the converse, we want to use the criterion from Lemma 2.41 iii).
Assume $g_{i}^{*}A$ is $h_{i}$-smooth for all $i$ and consider the natural map

$\Delta_{f}^{!}(p_{Y,2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Y,1}^{*}A)\to\Delta_{f}^{!}\operatorname{\underline{Hom}}(p_{Y,2}^{*}A,p_{Y,1}^{!}A)=\operatorname{\underline{Hom}}(A,A)$

where the identification on the right follows from the interior pullback
formula (Proposition 2.10). We need to show that this map is an isomorphism,
and since $(g_{i}^{*})_{I}$ is conservative, we may check this after pulling
back along each of the $g_{i}^{*}$. Thus, wlog. assume $I=\left\\{0\right\\}$
and write $g:=g_{0}$, $h.=h_{0}$, $Z:=Z_{0}$. Then the right hand side
identifies with $\operatorname{\underline{Hom}}(g^{*}A,g^{*}A)$ by Lemma 2.58.
We claim that the pullback of the left hand side agrees with
$\Delta_{h}^{!}(p_{Z,2}^{*}\mathbb{D}(g^{*}A)\operatorname{\otimes}p_{Z,1}^{*}g^{*}A)$.
Indeed, note that
$\mathbb{D}_{h}(g^{*}A)\cong\operatorname{\underline{Hom}}(g^{*}A,h^{!}1_{X})\cong
g^{!}\operatorname{\underline{Hom}}(A,f^{!}1_{X})\cong g^{!}\mathbb{D}_{f}(A)$
by interior pullback. Then, using Lemma 2.56,

$\displaystyle\Delta_{h}^{!}(p_{Z,2}^{*}g^{!}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z,1}^{*}g^{*}A)$
$\displaystyle\cong\Delta_{h}^{!}(p_{Z,2}^{*}g^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z,1}^{*}g^{*}A\operatorname{\otimes}p_{Z,2}^{*}g^{!}1_{Y})$
$\displaystyle\cong\Delta_{h}^{!}(p_{Z,2}^{*}g^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z,1}^{*}g^{*}A)\operatorname{\otimes}g^{!}1_{Y}$
$\displaystyle\cong\Delta_{h}^{!}(g^{\prime*}p_{Y,2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}g^{\prime*}p_{Z,1}^{*}A)\operatorname{\otimes}g^{!}1_{Y}$

where $g^{\prime}:Z\times_{X}Z\to Y\times_{X}Y$ is the obvious map, given by
the composition $Z\times_{X}Z\to Z\times_{X}Y\to Y\times_{X}Y$ of two base
changes of $g$; in particular it is cohomologically smooth and one calculates
$g^{\prime!}1_{Y\times_{X}Y}=(p_{Z,1}^{*}g^{!}1_{Y})\operatorname{\otimes}(p_{Z,2}^{*}g^{!}1_{Y})$.
So

$\displaystyle\Delta_{h}^{!}(g^{\prime*}(p_{Y,2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z,1}^{*}A))\operatorname{\otimes}g^{!}1_{Y}$
$\displaystyle=\Delta_{h}^{!}g^{\prime!}(p_{Y,2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z,1}^{*}A)\operatorname{\otimes}(\Delta_{h}^{*}g^{\prime!}1_{Y})^{-1}\operatorname{\otimes}g^{!}1_{Y}$
$\displaystyle\cong
g^{!}\Delta_{f}^{!}(p_{Y,2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z,1}^{*}A)\operatorname{\otimes}(g^{!}1_{Y})^{-1}$
$\displaystyle\cong
g^{*}\Delta_{f}^{!}(p_{Y,2}^{*}\mathbb{D}_{f}(A)\operatorname{\otimes}p_{Z,1}^{*}A)$

as claimed. We have shown that pullback along $g$ yields a map

$\Delta_{h}^{!}(p_{Z,2}^{*}\mathbb{D}_{h}(g^{*}A)\operatorname{\otimes}p_{Z,1}^{*}g^{*}A)\to\Delta_{h}^{!}\operatorname{\underline{Hom}}(p_{Z,2}^{*}A,p_{Z,1}^{!}A)$

It remains to show that it agrees (up to homotopy coherence) with the natural
map from Lemma 2.41 which is an isomorphism by assumption. This can be done by
an explicit calculation.

The first statement about the Verdier duals follows from interior pullback
(Proposition 2.10). The "only if" in the second statement is then obvious as
all $g_{i}^{!}1_{Y}$ are invertible by assumption. For the "if" direction we
need to check that the natural map
$\mathbb{D}_{f}(A)\operatorname{\otimes}\operatorname{\underline{Hom}}(\mathbb{D}_{f}(A),1_{Y})\to
1_{Y}$ is an isomorphism which is true as by Lemma 2.58 its pullback along
each $g_{i}$ is an isomorphism (as $g_{i}^{*}\mathbb{D}_{f}(A)\cong
g_{i}^{!}\mathbb{D}_{f}(A)\operatorname{\otimes}(g_{i}^{!}1_{Y})^{-1}$ is
invertible by the first statement). ∎

In particular, cohomological smoothness is cohomologically smooth-local on the
source in the following sense:

######  Corollary 2.60.

Consider a collection of commutative diagrams

${Z_{i}}$${Y}$${X}$$\scriptstyle{g_{i}}$$\scriptstyle{h_{i}}$$\scriptstyle{f}$

for $i\in I$ of morphisms in $E(\mathfrak{G})$ with all $g_{i}$
cohomologically smooth and $A\in\mathcal{D}(Y)$, where $I$ is some index set.
If $(g_{i}^{*})_{I}:\mathcal{D}(Y)\to\prod_{I}\mathcal{D}(Z_{i})$ is
conservative, then the following are equivalent:

  1. i)

All $h_{i}$ are cohomologically smooth.

  2. ii)

$f$ is cohomologically smooth.

###### Proof.

This follows immediately from Corollary 2.52 and Theorem 2.59. ∎

###  2.7. Cohomologically étale and cohomologically proper morphisms

A cohomologically étale morphism is, roughly speaking, a morphism $f$ for
which $f^{!}$ and $f^{*}$ agree "universally" (with the additional property
that this condition is stable under taking the diagonal). For example,
consider $\mathcal{D}_{\mathbb{F}_{\ell}}$ on $\mathfrak{G}$ from Theorem
2.26. We certainly expect that $f^{!}=f^{*}$ for any étale morphism; and
indeed, we will see below that any étale morphism is cohomologically étale in
this setting (whence the name).

######  Definition 2.61.

[Sch23, Definitions 6.10 and 6.12] Let $\mathcal{D}$ be a 6-functor formalism
on a construction setup $\mathfrak{G}$ and let $(f:Y\to X)\in
E(\mathfrak{G})$. Then

  1. i)

$f$ is called ($\mathcal{D}$-)cohomologically étale if it is an isomorphism or
it is $n$-truncated for some $n$ with $\Delta_{f}$ cohomologically étale and
$1_{Y}$ $f$-smooth.

  2. ii)

$f$ is called ($\mathcal{D}$-)cohomologically proper if it is an isomorphism
or it is $n$-truncated for some $n$ with $\Delta_{f}$ cohomologically proper
and $1_{Y}$ $f$-proper.

######  Remark 2.62.

This is an inductive definition which works by [Lur09, Lemma 5.5.6.15]: A
morphism is $n$-truncated if and only if its diagonal is $n-1$-truncated. But
a $-2$-truncated morphism is an isomorphism. Note that $\Delta_{f}\in E$ if
$f\in E$ as the construction setup $\mathfrak{G}$ is in particular a good
geometric setup, cf. Remark 2.15.

######  Example 2.63.

If $\mathcal{D}$ is a 6-functor formalism on a construction setup
$\mathfrak{G}=(\mathcal{C},E,I,P)$ then all $n$-truncated (for some $n\geq-2$)
morphisms in $I$ are cohomologically étale and all $n$-truncated morphisms in
$P$ are cohomologically proper: As $\Delta_{f}\in I$ (resp. $\Delta_{f}\in P$)
if $f\in I$ (resp. $f\in P$) this follows from the fact that $1$ is $f$-smooth
(resp. $f-$proper) if $f\in I$ (resp. $f\in P$).

We will now again restrict the discussion to the case of cohomologically étale
morphisms.
To make the terminology complete, we introduce a non-standard notion.

######  Definition 2.64.

In the situation of Definition 2.61, we call $f$ cohomologically unramified if
$\Delta_{f}$ is cohomologically étale.

######  Lemma 2.65.

[Sch23, Proposition 6.13] In the situation of Definition 2.61, $f$ is
cohomologically étale if and only if it is cohomologically smooth and
cohomologically unramified. More precisely, if $f$ is cohomologically
unramified, then there is a natural transformation $f^{!}\to f^{*}$ and the
following are equivalent:

  1. i)

$f^{!}1_{X}\to 1_{Y}$ is an isomorphism on global sections.

  2. ii)

$f^{!}\to f^{*}$ is an isomorphism.

  3. iii)

$f$ is cohomologically smooth.

  4. iv)

$f$ is cohomologically étale.

###### Proof.

By definition, a cohomologically étale morphism is cohomologically unramified,
so the first statement really follows from the second.

Assume $f$ is cohomologically unramified. Then it is $n$-truncated for some
$n$. We argue by induction on $n$. If $n=-2$ then $f$ is an isomorphism and
all statements hold (as $I(\mathfrak{G})$ contains all isomorphisms). Now
assume by induction that $ii)$ holds for $\Delta_{f}$. We have a natural
transformation

$p_{2}^{*}f^{!}\to p_{1}^{!}f^{*}$

adjoint to $p_{1,!}p_{2}^{*}f^{!}=f^{*}f_{!}f^{!}\to f^{*}$, where we used the
projection formula. In fact, applied to $1_{X}$ this is the map from Corollary
2.52 resp. Lemma 2.41. Applying $\Delta_{f}^{*}=\Delta_{f}^{!}$ on both sides
yields the promised map $f^{!}\to f^{*}$. Thus $i)\Leftrightarrow iv)$ by
Lemma 2.41 and the defintion. The same result also proves $i)\Rightarrow iii)$
as it implies that already $f^{!}1_{X}=1_{Y}$. But then also $i)\Rightarrow
ii)$ by definition of cohomological smoothness and thus $i)\Leftrightarrow
ii)$. It remains to show $iii)\Rightarrow iv)$, but this is obvious from the
definition by Corollary 2.52. ∎

######  Proposition 2.66.

Cohomologically unramified morphisms are stable under base change and
composition.

###### Proof.

Let $f:Y\to X$ be a cohomologically unramified morphism, in particular
$n$-truncated for some $n$. We induct on $n\geq-2$: If $f$ is an isomorphism,
then any base change is an isomorphism and thus cohomologically unramified;
moreover, compositions $g\circ f$ are cohomologically unramified for all
cohomologically unramified $g$. Now assume stability under base change holds
for all $n-1$-truncated $f$. If $f$ is $n$-truncated and $g:X^{\prime}\to X$
is any morphism and $f^{\prime}$ is the pullback of $f$ along $g$, then
$\Delta_{f^{\prime}}$ is the base change of $\Delta_{f}$ (which is
cohomologically étale by assumption and $n-1$-truncated) along
$Y^{\prime}\times_{X^{\prime}}Y^{\prime}\to Y\times_{X}Y$, so it is
cohomologically étale by the induction hypothesis (and Lemma 2.65). This
proves the first assertion. Now assume that the second assertion holds for
compositions $g\circ f$ with $f$ $n-1$-truncated. If $f$ is $n$-truncated and
$g:X\to S$ is cohomologically unramified, then $\Delta_{g\circ f}$ can be
factored as $\varphi\circ\Delta_{f}:Y\to Y\times_{X}Y\to Y\times_{S}Y$, where
$\varphi$ is the base change of $\Delta_{g}$ along $(f,f)$; by the first
assertion $\varphi$ is thus cohomologically unramified, but it is also
cohomologically smooth since $\Delta_{g}$ is and so it is cohomologically
étale by Lemma 2.65. By induction the composition $\varphi\circ\Delta_{f}$ is
cohomologically unramified (since $\Delta_{f}$ is $n-1$-truncated and
cohomologically étale) and cohomologically smooth by Corollary 2.53. But then
it is cohomologically étale. ∎

######  Corollary 2.67.

Cohomologically étale morphisms are stable under base change and composition.

###### Proof.

This is immediate from Corollary 2.53, Proposition 2.66 and Lemma 2.65. ∎

Let us quickly make some additional observations about cohomologically
unramified morphisms:

######  Remark 2.68.

In the situation of Definition 2.61, let $f:Y\to X\in E(\mathfrak{G})$ be a
cohomologically unramified morphism.

  1. i)

Cohomological smoothness can be cancelled along $f$: A morphism $g:Z\to Y$ is
cohomologically smooth (resp. étale) if and only if $f\circ g$ is. This
follows from stability of cohomological smooth morphisms under base change and
composition by the usual argument (cf. the proof of Proposition 2.25).

  2. ii)

Let $A,S\in\mathcal{D}(Z)$. Then the natural map from Lemma 2.41 becomes
$\mathbb{D}_{f}(A)\operatorname{\otimes}S\to\operatorname{\underline{Hom}}(A,S)$,
using that $\Delta_{f}^{!}=\Delta_{f}^{*}$, and so $A$ is $f$-smooth if and
only if this map is an isomorphism for all $S$ (where $S=A$ suffices). In
particular, if $A$ is $f$-smooth, its Verdier dual agrees with the naive dual;
but then the the case $S=A$ says that $A$ is dualisable.171717One checks that
the natural map is indeed just adjoint to evaluation tensored with $A$; it is
an isomorphism if and only if $A$ is dualisable, cf. [Man22b, Lemma 6.2] or
combine Corollary 2.40 and Lemma 2.41. Thus the $f$-smooth objects are exactly
the dualisable $A$ for which
$\mathbb{D}_{f}(A)\to\operatorname{\underline{Hom}}(A,1)$ (induced by the
natural map $f^{!}1\to 1$ from Lemma 2.65) is an isomorphism. In particular,
if $f$ is cohomologically étale, then $f$-smooth objects are exactly the
dualisable ones.

The following result is helpful to find non-trivial examples of
cohomologically étale morphisms.

######  Lemma 2.69.

In the situation of Definition 2.61, consider a cartesian square

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

with $g$ of universal $\mathcal{D}^{*}$-descent. Then $f^{\prime}$ is
cohomologically étale if and only if $f$ is.

###### Proof.

This follows from the definition and Proposition 2.49. ∎

The following is the most important application for us:

######  Proposition 2.70 ([Sch23, Proposition 7.18]).

Let $\Lambda$ be any torsion ring and $\mathcal{D}_{\Lambda}$ the 6-functor
formalism on $\mathfrak{G}$ from Theorem 2.26. Then all étale morphisms
$f:Y\to X$ in $E(\mathfrak{G})$ are $\mathcal{D}_{\Lambda}$-cohomologically
étale.

###### Proof.

As the diagonal of étale morphisms is an open immersion, we know that all
étale morphisms are cohomologically unramified (recall that $I(\mathfrak{G})$
is the set of open immersions). As étale morphisms are open, we can reduce to
the case that $f$ is surjective. Then we make use of Lemma 2.69: As étale
morphisms satisfy universal $\mathcal{D}^{*}$-descent for
$\mathcal{D}_{\Lambda}$, $f$ is cohomologically étale if and only if this
holds for the base change $p:Y\times_{X}Y\to Y$ of $f$ along itself. Now it
suffices to check that $p^{!}1_{X}\to 1_{Y}$ is an isomorphism by Lemma 2.65,
which may be checked on the cover given by $\Delta_{f}$ and its complement
$U$. Both are open subschemes, and on $\Delta_{f}$ this clearly holds. By
induction on the maximum cardinality of a geometric fiber of $f$ we reduce to
the case $U=\emptyset$ and win. ∎

######  Remark 2.71.

We will use the idea of this proof later again to prove our Theorem 3.20.

As a corollary, we can now prove the following criterion due to B. Zavyalov:

######  Theorem 2.72 ([Zav23, Theorem 3.3.3]).

Let $S$ be a qcqs scheme and consider the restricted 6-functor formalism
$\mathcal{D}_{\Lambda}$ on $\mathfrak{G}_{/S}$ (cf. Remark 2.11). In the
situation of Proposition 2.70, smooth morphisms in
$\mathcal{C}(\mathfrak{G}_{/S})$ are $\mathcal{D}_{\Lambda}$-cohomologically
smooth if and only if $\mathbb{P}^{1}_{S}\to S$ is
$\mathcal{D}_{\Lambda}$-cohomologically smooth.

###### Proof.

The "only if" is clear, we need to show the "if"-direction. So assume that
$\mathbb{P}^{1}_{S}\to S$ is cohomologically smooth. Then
$\mathbb{A}^{1}_{S}\to S$ is as well since cohomological smoothness is stable
under composition (Corollary 2.53). Together with stability under base change
this implies that $\mathbb{A}^{n}_{S}\to S$ is cohomologically smooth for all
$n$ and thus $\mathbb{A}^{n}_{X}\to X$ is cohomologically smooth for all
$(X\to S)\in\mathcal{C}(\mathfrak{G}_{/S})$. Now let $Y\to X$ be a smooth
morphism in $\mathcal{C}(\mathfrak{G}_{/S})$. Then it is of finite
presentation, so it is contained in $E(\mathfrak{G}_{/S})$. As cohomological
smoothness can be checked Zariski-locally on the source by Corollary 2.60, we
may assume that $Y\to X$ factors via an étale morphism over some
$\mathbb{A}^{n}_{X}$; and conclude by Proposition 2.70 and Corollary 2.53. ∎

Finally let us note that the property of being cohomologically étale is
cohomologically étale-local on the source in the following sense:

######  Corollary 2.73.

Consider a collection of commutative diagrams

${Z_{i}}$${Y}$${X}$$\scriptstyle{g_{i}}$$\scriptstyle{h_{i}}$$\scriptstyle{f}$

for $i\in I$ of morphisms in $E(\mathfrak{G})$ with all $g_{i}$
cohomologically étale and $A\in\mathcal{D}(Y)$, where $I$ is some index set.
Assume $f$ is $n$-truncated for some $n$. If
$(g_{i}^{*})_{I}:\mathcal{D}(Y)\to\prod_{I}\mathcal{D}(Z_{i})$ is
conservative, then the following are equivalent:

  1. i)

All $h_{i}$ are cohomologically étale.

  2. ii)

$f$ is cohomologically étale.

###### Proof.

$ii)\Rightarrow i)$ is Corollary 2.67.

Let us prove $i)\Rightarrow ii)$. We induct on $n\geq-2$. If $f$ is an
isomorphism, then it is cohomologically étale in any case (by definition). Now
assume the statement holds for $n-1$-truncated $f$. We claim that there are
commutative diagrams

${Z_{i}}$${Y}$${Y\times_{X}Y}$$\scriptstyle{g_{i}}$$\scriptstyle{h^{\prime}_{i}}$$\scriptstyle{\Delta_{f}}$

with all $h^{\prime}_{i}$ cohomologically étale. Indeed, by assumption all
$\Delta_{h_{i}}:Z_{i}\to Z_{i}\times_{X}Z_{i}$ are cohomologically étale; but
then $h_{i}^{\prime}:Z_{i}\to Z_{i}\times_{X}Z_{i}\to Z_{i}\times_{X}Y\to
Y\times_{X}Y$ does the job by stability of cohomologically étale morphisms
under base change and compositions (Corollary 2.67). We conclude by the
induction hypothesis that $\Delta_{f}$ is cohomologically étale, i.e. $f$
cohomologically unramified and the statement follows from Corollary 2.60 and
Lemma 2.65. ∎

For the arguments in this paper we will only need the following instances of
Corollary 2.60 and Corollary 2.73:

######  Lemma 2.74.

Let $f:Y\to X$ be a morphism in $E(\mathfrak{G})$ and $g:\coprod_{I}Y_{i}\to
Y$ an étale cover of $Y$. Then $f$ is cohomologically smooth (resp. étale) if
and only if each $h_{i}:Y_{i}\to X$ is cohomologically smooth (resp. étale).

###### Proof.

By Proposition 2.70, étale morphisms are cohomologically étale which implies
the statement by Corollary 2.60 resp. Corollary 2.73 as $g^{*}$ is indeed
conservative (the étale topos has enough points). ∎

Moreover, in this paper, the only thing we need to know about the class of
$\mathcal{D}_{\Lambda}$-cohomologically proper morphisms is that it contains
the universally closed morphisms in $E(\mathfrak{G})$ which follows from the
construction, cf. Example 2.63. In particular, we have the following result:

######  Corollary 2.75.

If $f$ is a universally closed morphism in $E(\mathfrak{G})$ and $A$ is
$f$-smooth, then $f_{!}A$ is dualisable.

###### Proof.

Combine Example 2.63 and Corollary 2.46. ∎

##  3\. The geometry of cohomologically étale morphisms

######  Notation 3.1.

In this section, we will always work with the 6-functor formalism
$\mathcal{D}_{\Lambda}$ on the geometric setup $\mathfrak{G}$ from Theorem
2.26 for some non-zero finite torsion ring $\Lambda$. The six functors and the
terms cohomologically smooth, cohomologically étale etc. will always refer to
this formalism.

###  3.1. Cohomologically étale morphisms and the absolute weak normalisation

Our goal in this subsection is to prove that a separated morphism of finite
presentation in $\mathcal{C}(\mathfrak{G})$ is cohomologically étale if and
only if its absolute weak normalisation (see Lemma 2.18) is étale. For this,
we first need to take another look at absolutely weakly normal schemes.

We start with some basic properties which are preserved under absolute weak
normalisation:

######  Proposition 3.2.

Let $f:Y\to X$ be a morphism in $\operatorname{Sch}$. Then the following
properties hold for $f^{\mathrm{awn}}$ if they hold for $f$:

  1. i)

a closed immersion

  2. ii)

an open immersion

  3. iii)

an immersion

  4. iv)

étale

  5. v)

any property of universal homeomorphisms which is stable under base change and
composition

###### Proof.

Note that absolute weak normalisation does not change the underlying
topological spaces as $X^{\mathrm{awn}}\to X$ is a universal homeomorphism
(cf. Lemma 2.18). In fact, we observed there that
$X^{\mathrm{awn}}\cong\varprojlim_{X^{\prime}\to
X\in\mathcal{U}^{\prime}_{X}}X^{\prime}$ where $\mathcal{U}^{\prime}_{X}$ is
the category of universal homeomorphisms $X^{\prime}\to X$ of finite type
(note that it is directed). Equivalently, it is the scheme with underlying
topological space $|X|$ and structure sheaf $\varinjlim_{X^{\prime}\to
X\in\mathcal{U}^{\prime}_{X}}\mathcal{O}_{X^{\prime}}$. This implies $i),ii)$
and $iii)$.

For $iv)$, assume that $f$ is étale. We claim that
$Y\times_{X}X^{\mathrm{awn}}$ is already absolutely weakly normal which is of
course sufficient. Indeed, this claim reduces to the affine case which is
proven in [Ryd10, Proposition B.6].

Finally, assume that $f$ satisfies a property of universal homeomorphisms
which is stable under base change and composition. Then in particular the base
change $Y\times_{X}X^{\mathrm{awn}}\to X^{\mathrm{awn}}$ satisfies this
property; but the morphism $Y^{\mathrm{awn}}\to X^{\mathrm{awn}}$ is just the
composition $Y^{\mathrm{awn}}\to Y\times_{X}X^{\mathrm{awn}}\to
X^{\mathrm{awn}}$ where the first morphisms is a universal homeomorphism. ∎

######  Corollary 3.3.

If $f:Y\to X$ is étale and $X$ is a.w.n., then $Y$ is a.w.n..

###### Proof.

This follows from the proof of $iv)$ in Proposition 3.2. ∎

Recall from Proposition 2.29 that the formalism $\mathcal{D}_{\Lambda}$
factors over the full subcategory $\mathrm{Awn}$ of absolutely weakly normal
schemes via the absolute weak normalisation. We can turn $\mathrm{Awn}$ into a
site, using the $v$-topology:

######  Definition 3.4 ([BM21, Definition 1.1], [Sta, 0ETS]).

A map $f:Y\to X$ of qcqs schemes is a $v$-cover if for any valuation ring $V$
and map $\operatorname{Spec}(V)\to X$ there is an extension $V\to W$ of
valuation rings and a map $\operatorname{Spec}(W)\to Y$ lifting the
composition $\operatorname{Spec}(W)\to\operatorname{Spec}(V)\to X$. An
$h$-cover is a $v$-cover of finite presentation.

The v-topology on the category $\mathrm{Awn}$ of qcqs a.w.n. schemes is the
Grothendieck topology where covering families $\left\\{f_{i}:Y_{i}\to
X\right\\}_{i\in I}$ are families with the following property: for any affine
open $V\subseteq X$ there is a map $t:J\to I$ of sets with $J$ finite and
affine opens $U_{j}\subseteq f_{t(k)}^{-1}(V)$ such that $\coprod_{J}U_{j}\to
V$ is a $v$-cover. We call $\mathrm{Awn}$ the absolutely weakly normal site.

The following result will be important for us:

######  Lemma 3.5.

The site $\mathrm{Awn}$ is subcanonical.

###### Proof.

Let $S$ be a qcqs a.w.n. scheme. We need to show that for a $v$-cover $Y\to X$
with $Y$ and $X$ a.w.n.

${{\operatorname{Hom}(X,S)}}$${{\operatorname{Hom}(Y,S)}}$${\operatorname{Hom}(Y\times_{X}Y,S)}$

is exact. By a standard argument (see the paragraph following [F+05, Lemma
2.61]) this reduces to the case that $Y,X$ and $S$ are affine, say
$Y=\operatorname{Spec}B$, $X=\operatorname{Spec}A$, $S=\operatorname{Spec}R$
so that it suffices to show exactness of

${A}$${B}$${B\operatorname{\otimes}_{A}B}$

We can write the $v$-cover $Y\to X$ as a directed limit of $h$-covers
$Y_{i}=\operatorname{Spec}B_{i}\to X$, cf. [Sta, 0EVP]. By [Sta, 0EVT], the
presheaf $\mathcal{O}^{\mathrm{awn}}$ on $\operatorname{Sch}$ which sends a
scheme $X$ to $\Gamma(X^{\mathrm{awn}},\mathcal{O}_{X^{\mathrm{awn}}})$
satisfies the sheaf property for $h$-coverings. Thus every

${A}$${B_{i}^{\mathrm{awn}}}$${B_{i}^{\mathrm{awn}}\operatorname{\otimes}_{A}B_{i}^{\mathrm{awn}}}$

is exact and taking a colimit of these diagrams yields the claim, using that
$(\rule{8.5359pt}{0.42677pt})^{\mathrm{awn}}$ (as a functor on rings) permutes
with colimits. ∎

As the 6-functor formalism $\mathcal{D}_{\Lambda}$ factors over
$\mathrm{Awn}$, we immediately observe the following:

######  Proposition 3.6.

Let $f:Y\to X$ be a morphism in $E(\mathfrak{G})$. If $f^{\mathrm{awn}}$ is
étale, then $f$ is cohomologically étale.

###### Proof.

By Proposition 2.29 $\mathcal{D}_{\Lambda}$ factors over the absolute weak
normalisation which implies the statement as étale morphisms are
cohomologically étale (Proposition 2.70). ∎

Let us now prove the converse for morphisms of finite presentation. As
cohomologically étale morphisms are in particular cohomologically unramified
(recall Definition 2.64), we start with an analysis of the latter class of
morphims.

It is a well-known fact that unramified morphism are (locally) quasi-finite,
cf. [Sta, 02V5]. We claim that cohomologically unramified morphisms of finite
type are as well. In fact, we can even give a full geometric characterisation
of these morphisms. We introduce a non-standard term:

######  Definition 3.7.

We call a morphism $f$ topologically unramified if its diagonal is open (as a
morphism of topological spaces).

We claim that cohomologically unramified morphisms are exactly the
topologically unramified ones. This follows from the next observation:

######  Lemma 3.8.

Let $g:Y\to X$ be a universally closed morphism in $E(\mathfrak{G})$ such that
$\Lambda$ is $g$-smooth. Then $g(Y)$ is open and closed in $X$.

###### Proof.

By Corollary 2.75, $g_{!}\Lambda$ is dualisable. This implies that it has open
and closed support by Remark 2.30. But it is exactly supported on $g(Y)$: By
proper base change $(g_{!}\Lambda)_{x}=(g_{*}\Lambda)_{x}\cong
R\Gamma(Y_{x},\Lambda)$ for any (geometric) point $x$ in $X$ and we have
$Y_{x}=\emptyset\Leftrightarrow R\Gamma(Y_{x},\Lambda)=0$ (since
$\Lambda\not=0$). ∎

Then we only need the following easy fact:

######  Lemma 3.9.

If $f:Z\to X$ is a closed immersion in $E(\mathfrak{G})$, then $f(Z)$ is open
in $X$ if and only if the induced morphism $f^{\mathrm{red}}$ of the
reductions is an open immersion.

###### Proof.

If $f^{\mathrm{red}}$ is an open immersion, then $f$ is open as reduction does
not change the underlying topological space. Conversely, assume $f$ is open,
then we may assume that it is surjective as $f(Z)$ is open. Then we claim that
$Z^{\mathrm{red}}\to X^{\mathrm{red}}$ is an isomorphism. Wlog. assume that
both schemes are affine. Then surjectivity of $Z^{\mathrm{red}}\to
X^{\mathrm{red}}$ implies that the corresponding surjective ring map is
injective and thus an isomorphism. ∎

######  Corollary 3.10.

A morphism $f\in E(\mathfrak{G})$ is cohomologically unramified if and only if
it is topologically unramified.

###### Proof.

If $f$ is topologically unramified then by Lemma 3.9 the absolute weak
normalisation of its diagonal is cohomologically étale (as it is an open
immersion, cf. Proposition 3.2) and thus the diagonal itself is
cohomologically étale by Proposition 3.6. Conversely, assume that $\Delta_{f}$
is cohomologically étale. As morphisms in $E(\mathfrak{G})$ are separated by
definition, Lemma 3.8 applies and we see that the image of $\Delta_{f}$ is
open; but then it is already open as a map of topological spaces because it is
an immersion. ∎

Let us now prove that topologically unramified morphisms of finite type are
quasi-finite.

######  Lemma 3.11.

Let $k$ be a field. Then a morphism $f:Y\to\operatorname{Spec}k$ of finite
type is topologically unramified if and only if
$f^{\mathrm{red}}:Y^{\mathrm{red}}\to\operatorname{Spec}k$ is étale.

###### Proof.

Assume $f$ is topologically unramified and let $\bar{k}$ be the algebraic
closure of $k$. Then $f_{\bar{k}}:Y_{\bar{k}}\to\operatorname{Spec}\bar{k}$ is
topologically unramified as well, and so by fpqc descent for étale morphisms
wlog $f=f_{\bar{k}}$. As $f^{\mathrm{red}}$ is of finite type, we only need to
check that its diagonal is an open immersion; then it is unramified and thus
étale (it is clearly flat and of finite presentation). But the diagonal is
open by assumption and
$Y^{\mathrm{red}}\times_{\operatorname{Spec}k}Y^{\mathrm{red}}$ is still
reduced as $k$ is algebraically closed, so that we conclude by Lemma 3.9. The
converse is obvious. ∎

######  Lemma 3.12.

Topologically unramified morphisms are stable under base change.

###### Proof.

Let $f:Y\to X$ be a topologically unramified morphism and
$f^{\prime}:Y^{\prime}\to X^{\prime}$ its base change along some
$g:X^{\prime}\to X$. Then $\Delta_{f^{\prime}}$ is the base change of
$\Delta_{f}$ along $Y^{\prime}\times_{X^{\prime}}Y^{\prime}\to Y\times_{X}Y$.
As $\Delta_{f}$ is an immersion, it is universally open if it is open, which
proves the claim. ∎

######  Corollary 3.13.

A topologically unramified morphism $f:Y\to X$ of finite type is quasi-finite.
In particular, cohomologically unramified and cohomologically étale morphisms
of finite type are quasi-finite.

###### Proof.

The second statement follows from the first by Corollary 3.10. Assume $f$ is
topologically unramified and of finite type. We claim that the fibers of $f$
are finite. Indeed, this is clear from Lemma 3.11 and the description of
schemes étale over a field ([Sta, 02GL]) since topologically unramified
morphisms are stable under base change by Lemma 3.12. ∎

So we know that cohomologically étale morphisms are quasi-finite and
cohomologically smooth. By the next result, this has a very useful
consequence.

######  Proposition 3.14.

Let $f\in E(\mathfrak{G})$ be a quasi-finite cohomologically smooth morphism
which is of finite presentation. Then $f$ is universally open.

###### Proof.

As all the assumptions on $f$ are stable under base change, we only need to
show that $f$ is open and it suffices to show that $f$ is generalising by the
finiteness condition, cf. [Sta, 01U1]. So take $y\in Y$ and pick a
generalisation $x\rightsquigarrow f(y)$ which is realised by
$\operatorname{Spec}R\to X$ for some discrete valuation ring $R$. Let $R^{h}$
be the henselisation of $R$ and $Z$ the fiber product
$\operatorname{Spec}R^{h}\times_{X}Y$; let $z\in\operatorname{Spec}R^{h}$ be
the point mapping to $f(y)$ and $\eta\in\operatorname{Spec}R^{h}$ the point
mapping to $x$.

${Z_{1}}$${Z}$${Y}$${\operatorname{Spec}R^{h}}$${X}$$\scriptstyle{g}$

$Z\to\operatorname{Spec}R^{h}$ is cohomologically smooth and quasi-finite as
$Y\to X$ is. As $R^{h}$ is henselian, this implies $Z\cong Z_{1}\coprod Z_{2}$
such that $g:Z_{1}\to\operatorname{Spec}R^{h}$ is finite and
$Z_{2}\times_{\operatorname{Spec}R^{h}}z=\emptyset$ (cf. [Sta, 04GG]); in
particular we have $(z,y)\in g^{-1}(z)$. Moreover, $Z_{1}$ is the (finite)
disjoint union of the spectra of its stalks at closed points (as it is finite
over a henselian ring); thus, replacing $Z_{1}$ by its stalk at the point
$(z,y)$, $Z_{1}\to Z$ is still an open immersion and so $g$ remains
cohomologically smooth and finite. But then by Corollary 2.75 $g_{!}\Lambda$
is dualisable and so by Remark 2.30 its support is either
$\operatorname{Spec}R^{h}$ or empty since $\operatorname{Spec}R^{h}$ is
connected. Note that $g_{!}\Lambda$ is non-zero since $(g_{!}\Lambda)_{z}\cong
R\Gamma(g^{-1}(z),\Lambda)\not=0$ (as $g^{-1}(z)\not=\emptyset$) by proper
base change. So we must have $0\not=(Rg_{!}\Lambda)_{\eta}\cong
R\Gamma(g^{-1}(\eta),\Lambda)$ and thus we find a lift of $\eta$ in $Z_{1}$,
yielding a generalisation $y^{\prime}\rightsquigarrow y$ in $Y$ with
$f(y^{\prime})=x$. ∎

######  Remark 3.15.

By arguing as in [Sch17, Proposition 23.11] and using the results from [BS15,
Section 6.4], the finite presentation assumption in Proposition 3.14 can be
dropped whenever the target $X$ satisfies the following property for each $n$:
all affine $U\in(\mathbb{A}^{n}_{X})_{\mathrm{\acute{e}t}}$ have
$\Lambda$-cohomological dimension $\leq d$ for some fixed $d\in\mathbb{N}$
(here we also used [Sta, 0F31]).

######  Corollary 3.16.

Cohomologically étale morphisms of finite presentation are universally open.

###### Proof.

This is immediate from Corollary 3.13 and Proposition 3.14. ∎

To finish the proof, we claim that if $f:Y\to X$ is a separated topologically
unramified and universally open morphism of finite type with $Y$ and $X$ qcqs,
then $f^{\mathrm{awn}}$ is étale. The idea is to use that étale morphisms
satisfy $v$-descent in $\mathrm{Awn}$ to do induction on the maximum
cardinality of a geometric fiber of the morphism. The induction is made
possible by the following lemma:

######  Lemma 3.17.

Let $f:Y\to X$ be a quasi-finite morphism where $Y$ is quasi-compact. Then
there is some minimal $n_{f}\in\mathbb{N}$ such that all fibers of $f$ have at
most $n_{f}$ points.

###### Proof.

This is [Sta, 03JA]. Using that $Y$ is quasi-compact one reduces to the case
that $Y$ and $X$ are affine, then one reduces via Zariski’s main theorem to
the case that $f$ is finite where it becomes a straightforward calculation. ∎

Next, let us state the descent property that we want to use.

######  Lemma 3.18.

Let $f:Y\to X$ be a separated morphism of qcqs a.w.n. schemes and
$g:X^{\prime}\to X$ a cover in the absolutely weakly normal site
$\mathrm{Awn}$. Let $f^{\prime}:Y^{\prime}\to X^{\prime}$ be the pullback of
$f$ along $g$. Then $f^{\prime}$ is étale if and only if $f$ is étale.

###### Proof.

By [BM21, Theorem 5.6] the functor
$\mathcal{G}:\mathrm{Sch}^{\mathrm{op}}_{\mathrm{qcqs},\mathbb{F}_{p}}\to\mathrm{Cat}_{1}$
that sends a qcqs scheme $X$ to the category of separated etale schemes over
$X$ is a $v$-sheaf. By Corollary 3.3, the restriction of $\mathcal{G}$ to
$\mathrm{Awn}$ sends a qcqs scheme $X$ to the category of separated etale
a.w.n. schemes over $X$. As $\mathrm{Awn}$ is subcanonical by Lemma 3.5, the
statement follows by applying the formal Lemma 3.19 below to $\mathrm{Awn}$
and the class of separated étale morphisms. ∎

######  Lemma 3.19.

Let $\mathcal{C}$ be a subcanonical site, $P$ a homotopy class of morphisms in
$\mathcal{C}$ which is stable under base change, and
$\mathcal{F}:\mathcal{C}^{\mathrm{op}}\to\mathrm{Cat}$ the functor sending $X$
to the category of $(Y\to X)\in P$. Assume $\mathcal{F}$ is a sheaf. Then in
any cartesian diagram

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

where $g$ is a cover in $\mathcal{C}$, $f\in P$ if and only if $f^{\prime}\in
P$.

###### Proof.

$\Rightarrow$ holds by assumption. So assume that $f^{\prime}\in P$, i.e.
$f^{\prime}\in\mathcal{F}(X^{\prime})$. Let $p_{1},p_{2}$ denote the
projections $X^{\prime\prime}:=X^{\prime}\times_{X}X^{\prime}\to X^{\prime}$.
By assumption,
$p_{1}^{*}f^{\prime},p_{2}^{*}f^{\prime}\in\mathcal{F}(X^{\prime\prime})$.
Moreover, there is a canonical isomorphism

$p_{1}^{*}Y^{\prime}\cong Y^{\prime}\times_{X}X^{\prime}\cong
Y\times_{X}X^{\prime}\times_{X}X^{\prime}\cong
X^{\prime}\times_{X}Y\times_{X}X^{\prime}\cong
X^{\prime}\times_{X}Y^{\prime}\cong p_{2}^{*}Y^{\prime}$

over $X^{\prime\prime}$. As $\mathcal{F}$ is a sheaf, this implies that there
is some $(h:Z\to X)\in\mathcal{F}(X)$ (in particular $h\in P$) and an
isomorphism $\varphi:g^{\prime*}Z\cong Y^{\prime}$ over $X^{\prime}$ such that

${{Z\times_{X}X^{\prime}\times_{X}X^{\prime}}}$${{X^{\prime}\times_{X}Z\times_{X}X^{\prime}}}$${{Y\times_{X}X^{\prime}\times_{X}X^{\prime}}}$${{X^{\prime}\times_{X}Y\times_{X}X^{\prime}}}$$\scriptstyle{p_{1}^{*}\varphi}$$\scriptstyle{\cong}$$\scriptstyle{p_{2}^{*}\varphi}$$\scriptstyle{\cong}$

commutes. In other words, there is an isomorphism
$Y^{\prime}\times_{Z}Y^{\prime}\cong Y^{\prime}\times_{Y}Y^{\prime}$
compatible with the projections.
Now for any $T\in\mathcal{C}$, we have that

$h_{T}(Y)\to h_{T}(Y^{\prime})\rightrightarrows
h_{T}(Y^{\prime}\times_{Y}Y^{\prime})$

is exact and the same holds with $Y$ replaced by $Z$; by the observations
above, we thus get an induced isomorphism $h_{T}(Y)\cong h_{T}(Z)$ natural in
$T$. So $Y\cong Z$ over $X$ and thus $f\in P$. ∎

Now we are ready to prove our claim.

######  Theorem 3.20.

Let $f:Y\to X$ be a separated, finite type morphism of qcqs schemes. If $f$ is
universally open and topologically unramified, then $f^{\mathrm{awn}}$ is
étale.

###### Proof.

First, we can reduce to the case that $f$ is surjective: $f$ is open and so we
can replace $X$ by the image of $f$ in $X$ (using Proposition 3.2). Note that
$f$ is quasi-finite by Corollary 3.13, so we can do induction on $n_{f}$ from
Lemma 3.17: If $n_{f}=0$, then $Y=\emptyset$ so the statement holds. Now
assume the statement is proven for all feasible $g$ with $n_{g}<n_{f}$. By
Lemma 3.18 it suffices to show that the base change $p$ of $f^{\mathrm{awn}}$
along itself is étale, as $f^{\mathrm{awn}}$ is universally open (by