# The moduli of sections has a canonical obstruction theory

Rachel Webb  Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720-3840
U.S.A<EMAIL_ADDRESS>
###### Abstract.

We give a detailed proof that locally Noetherian moduli stacks of sections
carry canonical obstruction theories. As part of the argument we construct a
dualizing sheaf and trace map, in the lisse-étale topology, for families of
tame twisted curves, when the base stack is locally Noetherian.

###### Key words and phrases:

obstruction theory, Hom-stack, moduli of sections, duality for twisted curves

###### 2020 Mathematics Subject Classification:

Primary 14D23, 14D15; Secondary 14F06, 18D99

###### Contents

  1. 1 Introduction
  2. 2 A formal framework for dualizing objects and trace maps
  3. 3 Duality for twisted curves
  4. 4 Obstruction theories via the Fundamental Theorem
  5. A Descent theorems for lisse-étale sheaves on algebraic stacks
  6. B Functoriality of the Fundamental Theorem

##  1\. Introduction

###  1.1. Overview

Let $\mathcal{M}$ be a locally Noetherian algebraic stack and let
$\mathcal{C}\rightarrow\mathcal{M}$ be a family of twisted curves as in
[AOV11, Def 2.1]. Let $\mathcal{Z}\rightarrow\mathcal{C}$ be a morphism of
algebraic stacks such that $\mathcal{Z}\to\mathcal{M}$ is locally of finite
presentation, quasi-separated, and has affine stabilizers. By [HR19, Thm 1.3]
there is an algebraic stack
${\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$ over $\mathcal{M}$
whose fiber over a scheme $T\rightarrow\mathcal{M}$ is

${\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})(T):=\operatorname{Hom}_{\mathcal{C}}(\mathcal{C}\times_{\mathcal{M}}T,\mathcal{Z})$

where the right hand side is the groupoid of morphisms of stacks over
$\mathcal{C}$. Recall that an obstruction theory for
${\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$ is a morphism of
complexes $\phi:E\rightarrow\mathbb{L}_{{\mathrm{Sec}}/\mathcal{M}}$ in the
derived category of
${\mathrm{Sec}}:={\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$
whose mapping cone has vanishing cohomology sheaves in degrees $[-1,\infty)$
(see Section 4.3). An implication of our main theorem is the following.

######  Theorem 1.1.1.

The stack ${\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$ carries
a canonical obstruction theory.

We define the canonical obstruction theory in Section 4.4. Theorem 1.1.1 is
generalized and stated more precisely as Theorem 4.4.1 below. An important
feature of the canonical obstruction theory is its functoriality, as explained
in [CJW21, Appendix A].

When the obstruction theory in Theorem 1.1.1 is perfect and
${\mathrm{Sec}}:={\mathrm{Sec}}_{\mathcal{M}}(\mathcal{Z}/\mathcal{C})$ is
Deligne-Mumford, quasi-separated, and locally finite type over a field, the
machinery in [BF97] and [Kre99, Sec 5.2] defines a virtual fundamental class
on ${\mathrm{Sec}}$. This is a key part of the construction of Gromov-Witten
theory and related enumerative theories: see for example [Beh97, AGV08, CCK15,
CL12]. On the other hand, Theorem 1.1.1 is used with a non-Deligne-Mumford
instance of ${\mathrm{Sec}}$ to functorially compare different obstruction
theories on quasimap moduli spaces in [CJW21, Lem A.2.5]. This comparison is
crucial for the application of [CJW21] to quasimap theory and also for the
computations of quasimap $I$-functions in [Web18, Web21].

###  1.2. Discussion of Theorem 1.1.1

The usual argument supporting Theorem 1.1.1 when ${\mathrm{Sec}}$ is Deligne-
Mumford is as follows (this is used, for example, in [BF97, Prop 6.2]). First
reduce to showing that for each affine $f:T\rightarrow{\mathrm{Sec}}$ and
square-zero quasi-coherent ideal sheaf $I$ on $T$, the induced map

(1)
$\operatorname{Ext}^{i}(\mathsf{L}f^{*}\mathbb{L}_{{\mathrm{Sec}}/\mathcal{M}},I)\rightarrow\operatorname{Ext}^{i}(\mathsf{L}f^{*}E,I)$

has the following properties (see [BF97, Thm 4.5]):

(2) $\displaystyle\bullet$ When $i=1$, (1) is injective on obstructions.
$\displaystyle\bullet$ When $i=0$ and there exists a deformation of $f$ by
$I$, (1) is an isomorphism.

Second, use standard deformation theory to relate the groups
$\operatorname{Ext}^{i}(\mathsf{L}f^{*}\mathbb{L}_{{\mathrm{Sec}}/\mathcal{M}},I)$
(resp. $\operatorname{Ext}^{i}(\mathsf{L}f^{*}E,I)$) to deformations of the
morphism $f:T\rightarrow{\mathrm{Sec}}$ (resp.
$\mathcal{C}\times_{\mathcal{M}}T\rightarrow\mathcal{Z}$). Since morphisms
$T\rightarrow{\mathrm{Sec}}$ are equivalent to morphisms
$\mathcal{C}\times_{\mathcal{M}}T\rightarrow\mathcal{Z}$ by definition of
${\mathrm{Sec}}$, one concludes that (1) is an injection (on obstructions)
when $i=1$ and that the groups in (1) are isomorphic (if the obstruction
vanishes) when $i=0$. We note that this falls just shy of the second
requirement in (2), since it is not clear that the morphism in (1) is itself
an isomorphism.

In this paper, we copy the first step above in Lemma 4.3.2. However, in the
second step, we analyze the functoriality of the isomorphism of Picard
categories

(3)
$\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)\simeq\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I[1]))$

due to Illusie and Olsson ([Ill71, Ols06]), for
$\mathcal{X}\rightarrow\mathcal{Y}$ a representable morphism of algebraic
stacks. (See Section 4.1 for the notation and Theorem 4.2.1 for the precise
statement). Our proof shows that when $i=0$, not only are the groups in (1)
isomorphic (in the case of vanishing obstruction), but in fact the morphism in
(1) is an isomorphism, completing the proof of the second requirement in (2).
Our proof also covers the case when ${\mathrm{Sec}}\rightarrow\mathcal{M}$ is
not representable or even relatively Deligne-Mumford.

The correct approach to Theorem 1.1.1 is likely through derived algebraic
geometry, as in [STV15, Sec 2.2]. The functoriality properties of the
obstruction theory proved in [CJW21, Appendix A] would be natural consequences
of such a construction. Unfortunately this author is not equipped to produce
the argument. Though the statement of Theorem 1.1.1 is certainly familiar, we
note that there does not seem to be a reference in the literature for the
generality in which we have stated it here.

###  1.3. Duality for twisted curves

A key ingredient for the construction of the obstruction theory in Theorem
1.1.1 is the following (stated more precisely as Proposition 3.4.1 below).

######  Theorem 1.3.1.

For every family $p:\mathcal{C}\rightarrow\mathcal{M}$ of tame twisted curves
on a locally Noetherian algebraic stack $\mathcal{M}$, there is a functorial
pair $(\omega_{\mathcal{M}},tr_{\mathcal{M}})$ with $\omega_{\mathcal{M}}$ a
quasi-coherent sheaf on $\mathcal{C}$ and
$tr_{\mathcal{M}}:\mathsf{R}\pi_{*}\omega_{\mathcal{M}}\rightarrow\mathscr{O}_{\mathcal{M}}[-1]$.
When $\mathcal{M}$ is a quasi-separated Noetherian algebraic space, the pair
$(\omega_{\mathcal{M}},tr_{\mathcal{M}})$ agrees with the right adjoint to
$\mathsf{R}p_{*}$.

We restate the last sentence of the theorem more precisely: if
$p:\mathcal{C}\to\mathcal{M}$ is a family of twisted curves, a right adjoint
$p^{!}$ to $\mathsf{R}p_{*}$ exists by [HR17, Thm 4.14(1)] (see also Lemma
3.2.1 below). The last sentence of Theorem 1.3.1 says that if $\mathcal{M}$ is
a quasi-separated Noetherian algebraic space, we have that
$\omega_{\mathcal{M}}[1]=p^{!}\mathscr{O}_{\mathcal{M}}$ and
$tr_{\mathcal{M}}$ is the counit of the $(\mathsf{R}p_{*},p^{!})$ adjunction.

The reason we do not have this agreement for arbitrary locally Noetherian
$\mathcal{M}$ is that it seems difficult to show that $p^{!}$ is compatible
with arbitrary base change. Following the exposition of [Lip09] for schemes,
we prove base change for the right adjoint to pushforward for certain
morphisms of algebraic stacks in Lemma 3.2.2 below (see also [Nee17] for a
complementary result). However, in applications, one would like to have base
change for $p^{!}$ for families of curves over arbitrary morphisms of
algebraic stacks.

The base change problem arises for non-twisted prestable curves as well, and
[Stacks, Tag 0E5W] addresses the issue by “gluing” the pairs
$(\omega_{\mathcal{M}},tr_{\mathcal{M}})$ to get a functorial construction of
a dualizing complex and trace map. We adopt the same strategy to prove Theorem
1.3.1. Again, while the statement of Theorem 1.3.1 is well-known, we do not
know a reference for twisted curves, even over the complex numbers.

###  1.4. Contents of the paper

The main goal of Section 2 is to derive a certain commuting diagram (Lemma
2.4.3) which will be used in our proof of Theorem 1.1.1. Because the notation
is simpler and because we can reuse various parts of the argument in other
parts of the paper, we work in the setting of abstract closed symmetric
monoidal categories.

In Section 3 we prove Theorem 1.3.1. The proof requires us to construct a
special kind of hypercover of an algebraic stack and an associated lisse-étale
topos. This is an application of the general results proved in [Stacks] and we
explain the details in Appendix A.

We explain (3) and prove Theorem 1.1.1 in Section 4. The proof itself is
fairly short, granting the existence of the dualizing complex and the
functoriality of (3). We reserve our proof of the functoriality of (3) for
Appendix B. Since the functoriality is critical to our argument we include the
details, but said details are unsurprising.

###  1.5. The locally Noetherian hypothesis

We expect that the locally Noetherian assumption on $\mathcal{M}$ can be
relaxed. It is used only in the proof of Lemma 3.1.3, to show that pushing
forward to the coarse moduli space of a twisted curve preserves pseudo-
coherent objects. See Remark 3.1.4.

###  1.6. Conventions and notation

We collect some conventions and recurring notation. Our list of notation here
is not exhaustive.

Algebraic stacks. We follow the conventions in [Stacks, Tag 0260]; in
particular, an algebraic stack need not be quasicompact or quasi-separated.

Twisted curves. A morphism $p:\mathcal{C}\rightarrow\mathcal{M}$ of algebraic
stacks is a family of twisted curves if smooth-locally on $\mathcal{M}$ it is
a twisted curve in the sense of [AOV11, Def 2.1].

Notation for closed categories and internal hom
---
Notation |  Category | Internal hom
$\mathrm{Mod}(A)$ |  category of $A$-modules for a sheaf of rings $A$ on a site $\mathscr{S}$ | $\mathcal{H}om_{A}$
$\operatorname{D}(A)$ |  unbounded derived category of $\mathrm{Mod}(A)$ | $\mathsf{R}\mathcal{H}om_{A}$
|  derived global hom functor (valued in the derived category of $\Gamma(\mathscr{S},A)$-modules). |  notated $\mathsf{R}\mathrm{Hom}_{A}$
$\mathcal{X}_{\operatorname{lis-et}}$ (resp. $\mathcal{X}_{\mathrm{et}}$) |  category of sheaves on the lisse-étale (rep. étale) site of an algebraic (resp. Deligne-Mumford) stack $\mathcal{X}$ | not needed
$\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}})$ (resp. $\operatorname{QCoh}(\mathcal{X}_{\mathrm{et}})$) |  category of quasi-coherent sheaves on the lisse-étale (rep. étale) site of an algebraic (resp. Deligne-Mumford) stack $\mathcal{X}$ | not needed
$\operatorname{D}(\mathcal{X}_{\operatorname{lis-et}})$ (resp. $\operatorname{D}(\mathcal{X}_{\mathrm{et}})$) |  unbounded derived category of $\mathscr{O}_{\mathcal{X}}$-modules in $\mathcal{X}_{\operatorname{lis-et}}$ (resp. $\mathcal{X}_{\mathrm{et}}$) | $\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}}}$
|  derived global hom functor (valued in the derived category of $\Gamma(\mathcal{X},\mathscr{O}_{\mathcal{X}})$-modules). |  notated $\mathsf{R}\mathrm{Hom}_{\mathscr{O}_{\mathcal{X}}}$
$\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ (resp. $\operatorname{D_{qc}}(\mathcal{X}_{\mathrm{et}})$) |  full subcategory of $\operatorname{D}(\mathcal{X}_{\operatorname{lis-et}})$ (resp. $\operatorname{D}(\mathcal{X}_{\mathrm{et}})$) on objects with quasi-coherent cohomology sheaves | $\mathsf{R}\mathcal{H}om^{\mathrm{qc}}_{\mathscr{O}_{\mathcal{X}}}$
Operations on sheaves on topoi and algebraic stacks
---
Notation |  Meaning
$H^{i}(\mathcal{F})$ |  The $i^{th}$ cohomology sheaf of a complex $\mathcal{F}$
$\mathsf{R}\Gamma(\mathcal{F})$ |  The derived global sections functor applied to a complex $\mathcal{F}$, commonly notated $\mathsf{R}\Gamma(X,\mathcal{F})$ where $X$ is a topos
$(f^{-1},f_{*})$ |  Adjoint functors defined by a morphism of topoi $f:\mathscr{C}\rightarrow\mathscr{D}$
$f^{*}$ |  For $f:(\mathscr{C},\mathscr{O}_{\mathscr{C}})\rightarrow(\mathscr{D},\mathscr{O}_{\mathscr{D}})$ a morphism of ringed topoi set $f^{*}(-):=f^{-1}(-)\otimes_{f^{-1}\mathscr{O}_{\mathscr{D}}}\mathscr{O}_{\mathscr{C}}$
$\mathsf{L}f^{*}$ |  For $f:\mathcal{X}\rightarrow\mathcal{Y}$ a morphism of algebraic stacks, we denote by $\mathsf{L}f^{*}:\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})\rightarrow\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ the functor $\mathsf{L}f^{*}_{\mathrm{qc}}$ in [HR17, Sec 1.3].
$\mathsf{R}f_{*}$ |  For $f:(\mathscr{C},\mathscr{O}_{\mathscr{C}})\rightarrow(\mathscr{D},\mathscr{O}_{\mathscr{D}})$ a morphism of ringed topoi, this is the usual direct image functor $\operatorname{D}(\mathscr{O}_{\mathscr{C}})\to\operatorname{D}(\mathscr{O}_{\mathscr{D}})$.
$\mathsf{R}f_{*}$ |  For $f:\mathcal{X}\rightarrow\mathcal{Y}$ a concentrated morphism of algebraic stacks, this is the functor $\mathsf{R}(f_{\mathrm{qc}})_{*}:\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})\rightarrow\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$ of [HR17, Sec 1.3] that is right adjoint to $\mathsf{L}f^{*}$. By [HR17, Thm 2.6(2)] it agrees with the restriction of the usual direct image functor $\mathsf{R}(f_{\operatorname{lis-et}})_{*}:\operatorname{D}(\mathcal{X}_{\operatorname{lis-et}})\rightarrow\operatorname{D}(\mathcal{Y}_{\operatorname{lis-et}})$.

###  1.7. Acknowledgements

I am grateful to Bhargav Bhatt for explaining the characterization of an
obstruction theory in Lemma 4.3.2. I am also thankful for many helpful
conversations with Martin Olsson. This project was partially supported by an
NSF Postdoctoral Research Fellowship, award number 200213.

##  2\. A formal framework for dualizing objects and trace maps

###  2.1. Closed symmetric monoidal categories

Because notation is simpler in an abstract setting, we work for a moment with
closed symmetric monoidal categories. If $\mathscr{C}$ is such a category, we
will write $\mathscr{O}_{\mathscr{C}}$ for the unit, $\otimes$ for the
product, and $\operatorname{Hom}$ for internal hom, using $\mathscr{C}(X,Y)$
to denote the set of morphisms between two objects $X,Y\in\mathscr{C}$ and
$1_{X}$ to denote the identity morphism on $X$. We will suppress mention of
the associativity, commutativity, and identity isomorphisms that are part of
the definition of $\mathscr{C}$. If $\mathscr{C}$ and $\mathscr{D}$ are any
two categories and $R:\mathscr{C}\rightarrow\mathscr{D}$ is a functor with a
left adjoint $L$, then for $X\in\mathscr{D}$ and $Y\in\mathscr{C}$ we will
denote the unit and counit of the adjunction by

$\eta^{L}_{X}:X\rightarrow
RL(X)\quad\quad\quad\epsilon^{L}_{Y}:LR(Y)\rightarrow Y$

omitting the decorations on $\eta$ and $\epsilon$ when there is no risk of
confusion.

We will use many specific instances of the following abstract situation.

######  Situation 2.1.1.

We are given $\mathscr{C}$, $\mathscr{D}$ be closed symmetric monoidal
categories with $f^{*}:\mathscr{D}\rightarrow\mathscr{C}$ strong monoidal and
$f_{*}$ a right adjoint.111 Moreover, $f_{*}$ is lax monoidal by [FHM03,
(3.2)] and so our setup is consistent with that used in [Lip09, Sec 3.5]. This
means we have natural isomorphisms

(4) $\displaystyle f^{*}(X)\otimes f^{*}(Y)\rightarrow f^{*}(X\otimes Y)$ (5)
$\displaystyle\mathscr{O}_{\mathscr{C}}\rightarrow
f^{*}\mathscr{O}_{\mathscr{D}}.$

When we are in Situation 2.1.1 we have the following three morphisms at our
disposal. The first we recall from [FHM03, (3.4)]: given $Y\in\mathscr{C}$ and
$X\in\mathscr{D}$ there is a functorial isomorphism

(6)
$\operatorname{Hom}(X,f_{*}Y)\xrightarrow{\sim}f_{*}(\operatorname{Hom}(f^{*}(X),Y)).$

The second is the composition

(7) $f_{*}\operatorname{Hom}(X,Y)\rightarrow
f_{*}\operatorname{Hom}(f^{*}f_{*}X,Y)\xleftarrow[\sim]{\text{\eqref{eq:simplicial1}}}\operatorname{Hom}(f_{*}X,f_{*}Y)$

where the first morphism is induced by the counit of the adjunction. The third
is

(8)
$\operatorname{Hom}(X,Y)\rightarrow\operatorname{Hom}(X,f_{*}f^{*}Y)\xrightarrow[\sim]{\text{\eqref{eq:simplicial1}}}f_{*}\operatorname{Hom}(f^{*}X,f^{*}Y)$

where the first morphism is induced by the unit of the adjunction; it is an
isomorphism if $f^{*}$ is fully faithful. One can check that (8) is functorial
in $X$, $Y$, and in the adjoint pair $(f^{*},f_{*})$ (see [Lip09, Exercise
3.7.1.1]).

We present Example 2.1.2 as the first instance of Situation 2.1.1; more
instances can be found in Examples 3.0.1, B.2.2, B.2.3, and B.3.1.

######  Example 2.1.2.

The following is an example of Situation 2.1.1. Let $B^{\prime}\rightarrow B$
be a homomorphism of rings, and set $\mathscr{C}=\mathrm{Mod}(B^{\prime})$ and
$\mathscr{D}=\mathrm{Mod}(B)$. Define $f^{*}$ to be the extension of scalars
functor $-\otimes_{B^{\prime}}B$ and define $f_{*}$ to be restriction of
scalars $(-)_{B^{\prime}}$. The functor $-\otimes_{B^{\prime}}B$ is strong
symmetric monoidal. One can check from the definition in [FHM03, (3.4)] that
(6) sends $X\rightarrow(Y)_{B^{\prime}}$ to its adjoint arrow
$X\otimes_{B^{\prime}}B\rightarrow Y$, that (7) sends a $B$-module
homomorphism $g:X\rightarrow Y$ to
$(g)_{B^{\prime}}:(X)_{B^{\prime}}\rightarrow(Y)_{B^{\prime}}$, and that (8)
sends a $B^{\prime}$-module homomorphism $h:X\rightarrow Y$ to
$h\otimes_{B^{\prime}}B:X\otimes_{B^{\prime}}B\rightarrow
Y\otimes_{B^{\prime}}B.$

We recall a formal framework for basechange. We do not need monoidal
structures here.

######  Situation 2.1.3.

We have a diagram of categories and functors

(9)
${\mathscr{S}}$${\mathscr{C}}$${\mathscr{T}}$${\mathscr{D}}$$\scriptstyle{g_{*}}$$\scriptstyle{m_{*}^{\prime}}$$\scriptstyle{f_{*}}$$\scriptstyle{m_{*}}$

where the functors $f_{*},g_{*},m_{*},m^{\prime}_{*}$ have left adjoints
$f^{*},g^{*},m^{*},m^{\prime*}$, and we are given a natural transformation
$m^{\prime*}f^{*}\simeq g^{*}m^{*}$.

In this situation we get get a unique natural transformation $m_{*}g_{*}\simeq
f_{*}m^{\prime}_{*}$ such that the adjunctions for
$(m^{\prime*}f^{*},f_{*}m^{\prime}_{*})$ and $(m_{*}g_{*},g^{*}m^{*})$ are
compatible (see [Lip09, Sec 3.6]). We define the basechange map

(10) $m^{*}f_{*}X\rightarrow
g_{*}m^{\prime*}X\quad\quad\quad\text{for}\;X\in\mathscr{S}$

as in [Lip09, Prop 3.7.2(i)]. It may not be an isomorphism in general.

######  Lemma 2.1.4.

For $Y\in\mathscr{D}$ and $X\in\mathscr{C}$ there are commuting diagrams

(11)
${m^{\prime*}f^{*}f_{*}X}$${m^{\prime*}X}$${g^{*}m^{*}f_{*}X}$${g^{*}g_{*}m^{\prime*}X}$$\scriptstyle{m^{\prime*}\epsilon}$(10)$\scriptstyle{\epsilon}$
${m^{*}Y}$${m^{*}f_{*}f^{*}Y}$${g_{*}g^{*}m^{*}Y}$${g_{*}m^{\prime*}f^{*}Y}$$\scriptstyle{m^{*}\eta}$$\scriptstyle{\eta}$(10)

###### Proof.

We show that the first diagram commutes; the second one may be checked
similarly. Commutativity of the first follows from the following commuting
diagram.

(12)
${g^{*}m^{*}f_{*}X}$${g^{*}m^{*}f_{*}m^{\prime}_{*}m^{\prime*}X}$${g^{*}m^{*}m_{*}g_{*}m^{\prime*}X}$${g^{*}g_{*}m^{\prime*}X}$${m^{\prime*}f^{*}f_{*}X}$${m^{\prime*}f^{*}f_{*}m^{\prime}_{*}m^{\prime*}X}$${m^{\prime*}X}$${m^{\prime*}m^{\prime}_{*}m^{\prime*}X}$${m^{\prime*}X}$$\scriptstyle{g^{*}m^{*}f_{*}\eta}$$\scriptstyle{g^{*}\epsilon^{m^{*}}}$$\scriptstyle{\epsilon^{g^{*}}}$$\scriptstyle{m^{\prime*}f^{*}f_{*}\eta}$$\scriptstyle{m^{\prime*}\epsilon}$$\scriptstyle{m^{\prime*}\epsilon}$$\scriptstyle{m^{\prime*}\eta}$$\scriptstyle{\epsilon^{m^{\prime*}}}$

The perimeter of the diagram from $m^{\prime*}f^{*}f_{*}X$ to $m^{\prime*}X$
along the bottom is equal to $m^{\prime*}\epsilon$ using a triangle identity,
while the composition along the top is equal to the composition of the other
three arrows in the desired square. The commutativity of the big cell in (12)
is compatibility of the $(m^{\prime*}f^{*},f_{*}m^{\prime}_{*})$ and
$(m_{*}g_{*},g^{*}m^{*})$ adjunctions—see [Lip09, (3.6.2)]. ∎

###  2.2. An ideal setup

We recall the formal framework of [FHM03, Rmk 5.10].

######  Situation 2.2.1.

We are given closed symmetric monoidal categories $\mathscr{C},\mathscr{D}$
and functors $f_{*},f_{!}:\mathscr{C}\rightarrow\mathscr{D}$ and
$f^{*},f^{!}:\mathscr{D}\rightarrow\mathscr{C}$ such that $(f^{*},f_{*})$ and
$(f_{!},f^{!})$ are adjoint pairs. Moreover, these functors satisfy

  * •

$f^{*}$ is strong symmetric monoidal

  * •

$f_{*}=f_{!}$

  * •

The canonical projection formula morphism

(13) $\pi:Y\otimes f_{*}(X)\rightarrow f_{*}(f^{*}Y\otimes X)$

defined in [Hal, Appendix A] is an isomorphism

  * •

The object $C:=f^{!}\mathscr{O}_{\mathscr{D}}$ is invertible, and

  * •

The canonical morphism

(14) $\varphi:f^{*}Y\otimes f^{!}\mathscr{O}_{\mathscr{D}}\rightarrow f^{!}Y$

defined in [FHM03, (5.5)] is an isomorphism.

######  Example 2.2.2.

Let $p:\mathcal{C}\rightarrow T$ be a family of prestable curves on a quasi-
separated Noetherian scheme $T$, in the sense of [Stacks, Tag 0E6T]. Let
$\mathscr{C}=\operatorname{D_{qc}}(\mathcal{C}_{\mathrm{et}})$,
$\mathscr{D}=\operatorname{D_{qc}}(T_{\mathrm{et}})$, $f_{*}=\mathsf{R}p_{*}$,
and $f^{*}=\mathsf{L}p^{*}$. Then $f_{*}$ has a right adjoint $f^{!}$ and
$f^{!}\mathscr{O}_{T}$ is equal to the relative dualizing sheaf
$\omega_{\mathcal{C}/T}[1]$. These data are an example of Situation 2.2.1. We
will extend this example to families $\mathcal{C}\to T$ of twisted curves in
Example 3.3.1.

######  Lemma 2.2.3.

For every $X\in\mathscr{C}$, the $(\otimes,\operatorname{Hom})$-unit

(15) $\eta_{X}:X\rightarrow\operatorname{Hom}(C,X\otimes C)$

is an isomorphism.

###### Proof.

Since $C$ is invertible, it follows from [May01, Lem 2.9] that $C$ is
dualizable and that the coevaluation map defined there is an isomorphism. It
follows from the definition of the coevaluation map that the unit (15) is an
isomorphism when $X=\mathscr{O}_{\mathscr{C}}$. For general $X$, there is a
commuting square

${\mathscr{O}_{\mathscr{C}}\otimes
X}$${\operatorname{Hom}(C,\mathscr{O}_{\mathscr{C}}\otimes C)\otimes
X}$${\operatorname{Hom}(C,C)\otimes X}$${X}$${\operatorname{Hom}(C,X\otimes
C)}$${\operatorname{Hom}(C,C\otimes
X)}$$\scriptstyle{\eta_{\mathscr{O}_{\mathscr{C}}}\otimes
1_{X}}$$\scriptstyle{\nu}$$\scriptstyle{\sim}$$\scriptstyle{\eta_{X}}$

where the map labeled $\nu$ (defined in [Lew+86, 120]) is an isomorphism since
$C$ is dualizable (see [Lew+86, Prop III.1.3(ii)]). This implies that
$\eta_{X}$ is an isomorphism. The commutativity of the square follows
immediately from the definition of $\nu$ and the functoriality of $\eta$. ∎

Following [FHM03, Def 5.6], we define twisted functors
$f^{!}_{C}(X):=\operatorname{Hom}(C,f^{!}(X))$ and
$f_{!}^{C}(X):=f_{!}(X\otimes C)$. We have an isomorphism $f^{*}Y\rightarrow
f^{!}_{C}(Y)$ for $Y\in\mathscr{C}$ equal to the composition

(16)
$f^{*}(Y)\xrightarrow{\text{\eqref{eq:coev}}}\operatorname{Hom}(C,f^{*}Y\otimes
C)\xrightarrow{\varphi}\operatorname{Hom}(C,f^{!}Y)$

where (15) is an isomorphism by Lemma 2.2.3 and $\varphi$ is an isomorphism by
assumption. Since $(f^{C}_{!},f^{!}_{C})$ is an adjoint pair, we’ve realized
$f^{C}_{!}$ as a left adjoint to pullback. Moreover, there is a projection
isomorphism

$\pi_{C}:Y\otimes f_{!}^{C}(X)\xrightarrow{\sim}f_{!}^{C}(f^{*}(Y)\otimes X)$

defined by replacing $X$ with $X\otimes C$ in $\pi$.

In this setting, we prove commutativity of some diagrams which will be useful
to us.

######  Lemma 2.2.4.

There is a commuting diagram

(17) ${X\otimes f^{C}_{!}f^{*}(Y)}$${X\otimes
f_{!}^{C}f^{!}_{C}(Y)}$${X\otimes Y}$${f_{!}^{C}(f^{*}(X)\otimes
f^{*}(Y))}$${f^{C}_{!}f^{*}(X\otimes Y)}$${f_{!}^{C}f^{!}_{C}(X\otimes
Y)}$(16)$\scriptstyle{\sim}$$\scriptstyle{\pi_{C}}$$\scriptstyle{\sim}$$\scriptstyle{\epsilon}$$\scriptstyle{\sim}$(16)$\scriptstyle{\sim}$$\scriptstyle{\epsilon}$

where the arrows labeled $\epsilon$ are counits for the
$(f^{C}_{!},f_{C}^{!})$ adjunction.

###### Proof.

For any $Z\in\mathscr{D}$, the composition

$f^{C}_{!}f^{*}(Z)\xrightarrow{\text{\eqref{eq:sheaves1}}}f^{C}_{!}f^{!}_{C}(Z)\xrightarrow{\epsilon^{f^{C}_{!}}}Z$

is equal to

$f^{C}_{!}f^{*}(Z)=f_{*}(f^{*}(Z)\otimes C)\xleftarrow[\sim]{\pi}Z\otimes
f_{*}(C)\xrightarrow{\epsilon^{f_{!}}_{\mathscr{O}_{\mathscr{D}}}}Z$

To see this, expand the $(f_{!}^{C},f^{C}_{!})$-counit in terms of the
$(\otimes,\operatorname{Hom})$-counit and the $(f_{!},f^{!})$-counit; commute
the morphism $\varphi$ in the definition of (16) with the
$(\otimes,\operatorname{Hom})$-counit; and finally use the triangle identity
$1_{f^{*}Z\otimes C}=\epsilon^{\otimes}_{f^{*}Z\otimes
C}\circ(\eta^{\otimes}_{f^{*}Z}\otimes 1_{C})$, where $\eta$ and $\epsilon$
here denote the unit and counit of the $(\otimes,\operatorname{Hom})$
adjunction. Now (17) is equivalent to the diagram

${X\otimes f_{*}(f^{*}(Y)\otimes C)}$${X\otimes Y\otimes f_{*}(C)}$${X\otimes
Y}$${f_{*}(f^{*}(X)\otimes f^{*}(Y)\otimes C)}$${f_{*}(f^{*}(X\otimes
Y)\otimes
C)}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\epsilon}$

whose commutativity is proved in [Lip09, Lem 3.4.7(iv)]. ∎

######  Lemma 2.2.5.

Suppose we are in Situation 2.2.1. Then there is an isomorphism
$\gamma:f_{*}\operatorname{Hom}(f^{*}X,f^{!}_{C}(Y))\rightarrow\operatorname{Hom}(f^{C}_{!}(f^{*}X),Y)$
making the following diagram commute.

(18)
${\operatorname{Hom}(f^{C}_{!}f^{!}_{C}(X),Y)}$${\operatorname{Hom}(X,Y)}$${f_{*}\operatorname{Hom}(f^{*}X,f^{*}Y)}$$\scriptstyle{\operatorname{Hom}{(\epsilon,1_{Y})}}$(8)$\scriptstyle{\gamma}$

Here, $\epsilon$ is the counit for the $(f_{!}^{C},f^{!}_{C})$-adjunction.

###### Proof.

The definition of $\gamma$ will come out in the course of the proof: it will
be “conjugate” to $\pi$ via various adjoints (see also [FHM03, (4.1)] and
[Stacks, Tag 0A9Q]). For future reference, we summarize it in the final
paragraph of the proof. To simplify notation, when there is no risk of
confusion, if $F$ is a functor between categories and $\alpha$ is a morphism
of the source category, we will notate $F(\alpha)$ by $\alpha$. For example,
we may use $\epsilon$ as the label for the horizontal arrow in (18).

To show the commutativity of (18) we use the Yoneda embedding: for an
arbitrary $T\in\mathscr{D}$, it suffices to show the commutativity of

${{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}3}\mathscr{D}(T,\operatorname{Hom}(f_{!}^{C}f^{!}_{C}(X),Y)}$${{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}1}\mathscr{D}(T,\operatorname{Hom}(X,Y))}$${{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}2}\mathscr{D}(T,f_{*}\operatorname{Hom}(f^{*}X,f^{*}Y)).}$$\scriptstyle{\epsilon^{f^{C}_{!}}}$(8)$\scriptstyle{\gamma}$

We do this by demonstrating that it is equivalent to the commutativity of

${{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}6}\mathscr{D}(T\otimes
f^{C}_{!}f^{!}_{C}(X),Y)}$${{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}4}\mathscr{D}(T\otimes
X,Y)}$${{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}5}\mathscr{D}(f^{C}_{!}f^{!}_{C}(T\otimes
X),Y).}$$\scriptstyle{\epsilon^{f^{C}_{!}}}$$\scriptstyle{\epsilon^{f^{C}_{!}}}$$\scriptstyle{\tilde{\gamma}}$

where $\tilde{\gamma}$ is defined to equal the isomorphisms in (17). This
second diagram commutes by Lemma 2.2.4.

There are isomorphisms
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}1}={\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>and
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}3}={\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>given by $(\otimes,\operatorname{Hom})$ adjunction, and an isomorphism
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}2}={\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>given by

(19)
$\displaystyle{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}2}\mathscr{D}(T,$
$\displaystyle
f_{*}\operatorname{Hom}(f^{*}X,f^{*}Y))=\mathscr{C}(f^{*}T,\operatorname{Hom}(f^{*}X,f^{*}Y))=\mathscr{C}(f^{*}T\otimes
f^{*}X,f^{*}Y)$
$\displaystyle={}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}7}}\mathscr{C}(f^{*}(T\otimes
X),f^{*}Y)=\mathscr{C}(f^{!}_{C}(T\otimes
X),f^{!}_{C}Y)={}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}5}\mathscr{D}(f^{C}_{!}(f^{!}_{C}(T\otimes
X)),Y)$

where the equalities are $(f^{*},f_{*})$-adjunction,
$(\otimes,\operatorname{Hom})$-adjunction, the isomorphism (4), the
isomorphism (16), and $(f^{C}_{!},f^{!}_{C})$-adjunction. The square with
corners
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}1},{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}3},{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}4},$
and
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>commutes by functoriality of the $(\otimes,\operatorname{Hom})$ adjunction. We
take the commutativity of the square with corners
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}2},{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}3},{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}5},$
and
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>as the definition of $\gamma$. The final square commutes as follows. By the
definition of the vertical map in (18) and adjunction, the composition
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}1}\rightarrow{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}2}\rightarrow{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>is equal to

$\displaystyle\mathscr{D}(T,\operatorname{Hom}(X,Y))$
$\displaystyle\xrightarrow{f^{*}}\mathscr{C}(f^{*}T,f^{*}\operatorname{Hom}(X,Y))\xrightarrow{\otimes
f^{*}X}\mathscr{C}(f^{*}T\otimes f^{*}X,f^{*}\operatorname{Hom}(X,Y)\otimes
f^{*}X)$ $\displaystyle\xlongequal{\text{\eqref{eq:strong-
monoidal}}}\mathscr{C}(f^{*}(T\otimes X),f^{*}(\operatorname{Hom}(X,Y)\otimes
X))\xrightarrow{f^{*}\epsilon^{\otimes}}\mathscr{C}(f^{*}(T\otimes X),f^{*}Y)$

By functoriality of (4), this composition is equivalent to

${}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}1}\mathscr{D}(T,\operatorname{Hom}(X,Y))\xrightarrow{\otimes
X}\mathscr{D}(T\otimes X,\operatorname{Hom}(X,Y)\otimes
X)\xrightarrow{\epsilon^{\otimes}}{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}4}\mathscr{D}(T\otimes
X,Y)\xrightarrow{f^{*}}{}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}7}\mathscr{C}(f^{*}(T\otimes
X),f^{*}Y).$

The first two arrows are precisely the $(\otimes,\operatorname{Hom})$
adjunction
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}1}={\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>Finally, the composition

${}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}4}\mathscr{D}(T\otimes
X,Y)\xrightarrow{f^{*}}{}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}7}}\mathscr{C}(f^{*}(T\otimes
X),f^{*}Y)\xlongequal{\text{\eqref{eq:sheaves1}}}\mathscr{C}(f^{!}_{C}(T\otimes
X),f^{!}_{C}Y)={}^{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}5}\mathscr{D}(f_{!}(f^{!}_{C}(T\otimes
X)),Y),$

where the arrow comes from the previous formula and the two equalities come
from (19), is induced by the counit $\epsilon^{f^{C}_{!}}$ as desired.

To conclude, we summarize the definition of $\gamma$ as promised. After
cancelling assorted isomorphisms with their inverses, we see that it is given
under the Yoneda embedding by

$\displaystyle\mathscr{D}(T,$ $\displaystyle
f_{*}\operatorname{Hom}(f^{*}X,f^{*}Y))=\mathscr{C}(f^{*}T,\operatorname{Hom}(f^{*}X,f^{*}Y))=\mathscr{C}(f^{*}T\otimes
f^{*}X,f^{*}Y)$ $\displaystyle=\mathscr{C}(f^{*}T\otimes
f^{*}X,f^{!}_{C}Y)=\mathscr{D}(f_{!}^{C}(f^{*}T\otimes
f^{*}X),Y)\xrightarrow{\sim}\mathscr{D}(T\otimes f_{!}^{C}f^{*}X,Y)$
$\displaystyle=\mathscr{D}(T,\operatorname{Hom}(f_{!}^{C}f^{*}X,Y))=\mathscr{D}(T,\operatorname{Hom}(f_{!}^{C}f^{!}_{C}X,Y)).$

Where the equalities are $(f^{*},f_{*})$ adjunction,
$(\otimes,\operatorname{Hom})$ adjunction, the isomorphism (16),
$(f^{C}_{!},f^{!}_{C})$ adjunction, the projection formula,
$(\otimes,\operatorname{Hom})$ adjunction, and finally the isomorphism (16)
again. In particular, if we follow $\gamma$ with the inverse of the last
equality, we have defined a natural isomorphism

(20)
$f_{*}\operatorname{Hom}(Z,f^{*}Y)\xrightarrow{\sim}\operatorname{Hom}(f_{!}^{C}Z,Y)$

that is functorial in both arguments. (The content of this statement is that
to define (20) it is not necessary for $Z$ to be of the form $f^{*}X$.) ∎

###  2.3. A modification of the ideal situation

When $\mathcal{C}\rightarrow\mathcal{X}$ is a family of twisted curves on an
algebraic stack $\mathcal{X}$, we would like to apply Situation 2.2.1 by
setting $\mathscr{D}=\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-
et}})$ and $\mathscr{C}=\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-
et}})$. Unfortunately we do not know a proof that the right adjoint $f^{!}$ in
Situation 2.2.1 exists in the generality we would like (see Lemma 3.2.1 and
the discussion following it). Instead, we work in the following weaker
situation, replacing $f^{!}$ with a dualizing complex and trace map.

######  Situation 2.3.1.

We are given closed symmetric monoidal categories $\mathscr{C}$,
$\mathscr{D}$, a functor $f_{*}:\mathscr{C}\rightarrow\mathscr{D}$ with a
right adjoint $f^{*}$, and an invertible object $C\in\mathscr{C}$ with a trace
map $tr:f_{*}C\rightarrow\mathscr{O}_{\mathscr{D}}$. These data satisfy

  * •

$f^{*}$ is strong symmetric monoidal

  * •

The canonical morphism $\pi:Y\otimes f_{*}(X)\rightarrow f_{*}(f^{*}Y\otimes
X)$ is an isomorphism

In this situation we define an adjunction-like map
$a:\mathscr{C}(X,f^{*}Y)\rightarrow\mathscr{D}(f_{*}(X\otimes C),Y)$ as
follows (see also [CJW21, Sec A.2.1]).222Informally we think of $a$ as
realizing $f^{C}_{!}=f_{*}(\cdot\otimes C)$ as a left adjoint to
$f^{*}=f^{!}_{C}$. Given $\phi^{\prime}\in\mathscr{C}(X,f^{*}Y)$, define
$a(\phi^{\prime})$ to be the composition

(21) $f_{*}(X\otimes 1_{C})\xrightarrow{f_{*}(\phi^{\prime}\otimes
C)}f_{*}(f^{*}Y\otimes C)\xleftarrow[\sim]{\pi}Y\otimes
f_{*}C\xrightarrow{id\otimes tr_{C}}Y.$

Observe that $a$ is functorial in both arguments, by which we mean the
following:

  1. (1)

Given $X^{\prime}\in\mathscr{C}$ and $\psi\in\mathscr{C}(X^{\prime},X)$, we
have $a(\phi^{\prime}\circ\psi)=a(\phi^{\prime})\circ f_{*}(\psi\otimes
1_{C})$.

  2. (2)

Given $Y^{\prime}\in\mathscr{D}$ and $\psi\in\mathscr{D}(Y,Y^{\prime})$, we
have $a(f^{*}\psi\circ\phi^{\prime})=\psi\circ a(\phi^{\prime}).$

The next example explains why we call $a$ “adjunction-like.”

######  Example 2.3.2.

Suppose we are in Situation 2.2.1, with an adjoint pair $(f^{*},f_{*})$ and
object $C=f^{!}\mathscr{O}_{\mathscr{D}}\in\mathscr{C}$. Then we have the data
of Situation 2.3.1: we can define
$tr_{C}:f_{*}C\rightarrow\mathscr{O}_{\mathscr{D}}$ to be the counit of the
$(f_{!},f^{!})$ adjunction. Under the isomorphism (16), the adjunction-like
map $a:\mathscr{C}(X,f^{*}Y)\rightarrow\mathscr{D}(f_{*}(X\otimes C),Y)$ is
identified with the adjunction
$\mathscr{C}(X,f^{!}_{C}Y)\simeq\mathscr{D}(f^{C}_{!}X,Y)$. To see this, let
$\phi^{\prime}\in\mathscr{C}(X,f^{*}Y)$. The $(f^{C}_{!},f^{!}_{C})$-adjoint
of $\text{\eqref{eq:sheaves1}}\circ\phi^{\prime}$ is equal to the
$(f_{!},f^{!})$-adjoint of

$X\otimes C\xrightarrow{\phi^{\prime}\otimes 1_{C}}f^{*}Y\otimes
C\xrightarrow{\varphi}f^{!}Y.$

Since $f_{*}=f_{!}$, said adjoint is equal to the composition

$f_{*}(X\otimes C)\xrightarrow{f_{*}(\phi^{\prime}\otimes
1_{C})}f_{*}(f^{*}Y\otimes C)\xrightarrow{\hat{\varphi}}Y.$

where $\hat{\varphi}$ is the $(f_{!},f^{!})$-adjoint of $\varphi$. By the
definition of $\varphi$ in [FHM03, (5.5)], this last composition is equal to
$a(\phi^{\prime})$.

###  2.4. Basechange

We introduce a setting where the adjunction-like morphism $a$ is compatible
with pullback.

######  Situation 2.4.1.

We have a diagram of closed symmetric monoidal categories as in (9) such that
the left adjoints $f^{*},g^{*},m^{*},m^{\prime*}$ are strong symmetric
monoidal. We are given objects $S\in\mathscr{S},C\in\mathscr{C}$ and morphisms
$tr_{S}:g_{*}S\rightarrow\mathscr{O}_{\mathscr{T}},tr_{C}:f_{*}C\rightarrow\mathscr{O}_{\mathscr{D}}$
such that the data for each column of (9) are in Situation 2.3.1, and these
data are compatible as follows.

  * •

The basechange map (10) is an isomorphism.

  * •

We are given an isomorphism $\alpha:m^{\prime*}C\rightarrow S$ making this
diagram commute:

(22)
${m^{*}f_{*}C}$${\mathscr{O}_{\mathscr{T}}}$${g_{*}m^{\prime*}C}$${g_{*}S}$(10)$\scriptstyle{m^{*}tr_{C}}$$\scriptstyle{g_{*}\alpha}$$\scriptstyle{tr_{S}}$

In this situation, Lemma 2.4.2 explains a precise sense in which
$m^{*}a(\phi^{\prime})=a(m^{\prime*}\phi^{\prime})$.

######  Lemma 2.4.2.

Suppose we are in Situation 2.4.1. Let $\phi^{\prime}:X\rightarrow f^{*}(Y)$
be an arrow in $\mathscr{C}$. Then we have
$m^{\prime*}\phi^{\prime}:m^{\prime*}X\rightarrow
m^{\prime*}f^{*}Y=g^{*}m^{*}Y$, and the following diagram commutes:

${m^{*}f_{*}(X\otimes C)}$${m^{*}Y}$${g_{*}(m^{\prime*}X\otimes
S)}$$\scriptstyle{m^{*}a(\phi^{\prime})}$$\scriptstyle{\mathfrak{a}_{m^{*}}}$$\scriptstyle{\sim}$$\scriptstyle{a(m^{\prime*}\phi^{\prime})}$

The isomorphism $\mathfrak{a}$ is equal to (10) followed by (4) and finally
$\alpha$, and in particular it is functorial in $X$. Moreover, suppose we have
a diagram

(23)
${\mathscr{C}_{1}}$${\mathscr{C}_{2}}$${\mathscr{C}_{3}}$${\mathscr{D}_{1}}$${\mathscr{D}_{2}}$${\mathscr{D}_{3}}$$\scriptstyle{f_{1*}}$$\scriptstyle{m^{\prime*}_{12}}$$\scriptstyle{f_{2*}}$$\scriptstyle{m^{\prime*}_{23}}$$\scriptstyle{f_{3*}}$$\scriptstyle{m^{*}_{12}}$$\scriptstyle{m^{*}_{23}}$

together with distinguished objects $C_{i}\in\mathscr{C}_{i}$ and trace maps
$tr_{i}:f_{i*}C_{i}\rightarrow\mathscr{O}_{\mathscr{D}_{i}}$ for each $i$, and
isomorphisms $\alpha_{ij}:m^{\prime*}_{ij}C_{j}\rightarrow C_{i}$ for $i<j$.
Suppose that with these data both squares and the outer rectangle of (23) are
in Situation 2.4.1, and that $\alpha_{13}=\alpha_{12}\circ
m^{\prime*}_{12}(\alpha_{23}).$ Then $\mathfrak{a}_{m^{*}_{12}\circ
m^{*}_{23}}=\mathfrak{a}_{m^{*}_{12}}\circ
m^{\prime*}_{12}(\mathfrak{a}_{m^{*}_{23}}).$

###### Proof.

The proof of [CJW21, Lem A.2.1] works in this more general situation. ∎

######  Lemma 2.4.3.

Suppose we are in Situation 2.4.1, but that the left column of (9) is actually
in Situation 2.2.1: this means we are given a right adjoint $g^{!}$ for
$g_{*}=:g_{!}$ with $g^{!}\mathscr{O}_{\mathscr{T}}=S$ and
$tr_{S}:g_{*}S\rightarrow\mathscr{O}_{\mathscr{T}}$ equal to the counit for
the $(g_{!},g^{!})$ adjunction. Let $\phi^{\prime}:X\rightarrow f^{*}(Y)$ be
an arrow in $\mathscr{C}$ and set $\phi:=a(\phi^{\prime})$. Let $I$ be an
object of $\mathscr{T}$. Then the following diagram commutes.

(24) ${\operatorname{Hom}(m^{*}f_{*}(X\otimes
C),I)}$${\operatorname{Hom}(m^{*}Y,I)}$${g_{*}\operatorname{Hom}(m^{\prime*}X,g^{*}I)}$${g_{*}\operatorname{Hom}(m^{\prime*}f^{*}Y,g^{*}I)}$$\scriptstyle{m^{*}\phi}$$\scriptstyle{m^{\prime*}\phi^{\prime}}$

In this diagram, the equality is comprised of (10), $\alpha$, and (20). The
right vertical arrow is (8) followed by (10).

###### Proof.

The desired commuting diagram is derived from the composition of two. On the
left, we have

(25)
${\operatorname{Hom}(g_{!}^{S}(m^{\prime*}X),I)}$${\operatorname{Hom}(g_{!}^{S}(m^{\prime*}f^{*}Y),I)}$${\operatorname{Hom}(g_{!}^{S}g^{*}(m^{*}Y),I)}$${g_{*}\operatorname{Hom}(m^{\prime*}X,g^{*}I)}$${g_{*}\operatorname{Hom}(m^{\prime*}f^{*}Y,g^{*}I)}$${g_{*}\operatorname{Hom}(g^{*}m^{*}Y,g^{*}I)}$(10)(20)(20)(10)(20)

Here, the arrows pointing left are induced by $m^{\prime*}\phi^{\prime}$. On
the right, we have

(26)
${\operatorname{Hom}(g_{!}^{S}g^{*}(m^{*}Y),I)}$${\operatorname{Hom}(m^{*}Y\otimes
g_{!}^{S}(\mathscr{O}_{\mathscr{T}}),I)}$${\operatorname{Hom}(m^{*}Y,I)}$${\operatorname{Hom}(g_{!}^{S}g^{!}_{S}(m^{*}Y),I)}$${g_{*}\operatorname{Hom}(g^{*}m^{*}Y,g^{*}I)}$$\scriptstyle{\pi_{S}}$$\scriptstyle{\sim}$(16)$\scriptstyle{tr_{S}}$$\scriptstyle{\epsilon}$(8)$\scriptstyle{\gamma}$

The commutativity of the top triangle is Lemma 2.2.4 with $X=m^{*}Y$ and
$Y=\mathscr{O}_{\mathscr{T}}$, and the commutativity of the bottom triangle is
Lemma 2.2.5. We note that definition (20) is equal to $\gamma$ followed by the
equality induced by (16). Finally, by Lemma 2.4.2, the top row of (25)
followed by the top row of (26) agrees with the top row of (24) (after
inserting a copy of (10)). ∎

##  3\. Duality for twisted curves

We explain how which the formal discussion of Section 2 will generally be used
in the remainder of this article. In this section and the remainder of the
paper, we define pseudo-coherent and perfect objects of lisse-étale sites as
in [Stacks, Tag 08FT] and [Stacks, Tag 08G5], respectively. Note that if $X$
is an algebraic stack, pseudo-coherent and perfect objects of
$\operatorname{D}({\mathcal{X}_{\operatorname{lis-et}}})$ are always in
$\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$.

######  Example 3.0.1.

If $f:\mathcal{X}\rightarrow\mathcal{Y}$ is a morphism of algebraic stacks, we
have closed symmetric monoidal categories
$\mathscr{C}=\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ and
$\mathscr{D}=\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$ and a
strong monoidal functor $\mathsf{L}f^{*}:\mathscr{D}\rightarrow\mathscr{C}$.
If $f$ is concentrated we also have
$\mathsf{R}f_{*}:\mathscr{C}\rightarrow\mathscr{D}$ that is a right adjoint to
$\mathsf{L}f^{*}$. In this context, the functor notated $\operatorname{Hom}$
in Section 2 translates to internal hom for
$\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$. However, we note
that by [HR17, Lem 4.3(2)], for any algebraic stack $\mathcal{X}$, we have
equality

$\mathsf{R}\mathcal{H}om^{\mathrm{qc}}_{\mathscr{O}_{\mathcal{X}}}(\mathcal{P},\mathcal{F})\simeq\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}}}(\mathcal{P},\mathcal{F})$

for any $\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-
et}})$ and any perfect complex
$\mathcal{P}\in\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$.

###  3.1. Background on twisted curves

Recall from Section 1.6 that a morphism $p:\mathcal{C}\rightarrow\mathcal{M}$
of algebraic stacks is a family of twisted curves if smooth-locally on
$\mathcal{M}$ it is a twisted curve in the sense of [AOV11, Def 2.1]. In
particular, $p$ is flat and proper and the diagonal
$\mathcal{C}\to\mathcal{C}\times_{\mathcal{M}}\mathcal{C}$ is quasi-finite. If
$r:\mathcal{C}\to C$ is the coarse moduli map, and $\bar{c}\to C$ is a
geometric point, the fiber product
$\operatorname{Spec}(\mathscr{O}_{C,\bar{c}})\times_{C}\mathcal{C}$ is
moreover required to have a certain description (see the full definition in
[AOV11, Def 2.1]). We recall some properties of families of twisted curves.

Our first lemma “spreads out” the local quotient description of a twisted
curve at a geometric point to an étale neighborhood of that point.

######  Lemma 3.1.1.

Let $\mathcal{C}\to T$ be a family of twisted curves over an affine scheme $T$
and let $q:C\to T$ be the coarse moduli space. Let $\bar{c}\to C$ be a
geometric point. Then there is an integer $n\geq 1$ and an affine scheme
$V=\operatorname{Spec}(A)$ with an action of $\mu_{n}$ such that, if
$R=A^{\mu_{n}}$ is the ring of invariants and $U:=\operatorname{Spec}(R)$,
there is a commuting diagram

(27)
${V}$${{[V/\mu_{n}]}}$${U}$${\mathcal{C}}$${C}$$\scriptstyle{\sigma}$$\scriptstyle{\tau}$$\scriptstyle{r}$

where the square is fibered and the vertical maps are étale. Moreover, one of
the following holds:

  1. (1)

$A=R[x]/(x^{n}-t)$ for some $t\in R$ and $\mu_{n}$ acts by $\zeta\cdot
p(x)=p(\zeta x)$.

  2. (2)

$A=R[x,y]/(xy-t,x^{r}-u,y^{r}-v)$ for some $t,u,v\in R$ and $\mu_{n}$ acts by
$\zeta\cdot p(x,y)=p(\zeta x,\zeta^{-1}y)$.

###### Proof.

We prove the lemma when $\bar{c}$ maps to a node of $C$; the case when
$\bar{c}$ maps to a smooth point is similar. By definition [AOV11, Def
2.1(v)], there is a fiber square

(28)
${{[(\operatorname{Spec}(\mathscr{O}_{T,q(\bar{c})}[x,y]/(xy-t))/\mu_{n}]}}$${\mathcal{C}}$${\operatorname{Spec}(\mathscr{O}_{C,\bar{c}})}$${C}$

for some $t\in\mathscr{O}_{T,q(\bar{c})}$, where $\zeta\in\mu_{n}$ acts by
$x\mapsto\zeta\cdot x$ and $y\mapsto\zeta^{-1}\cdot y$. Since $\mathcal{C}$ is
tame, formation of the coarse space commutes with arbitrary base change
[AOV08, Cor 3.3], and we have

$\mathscr{O}_{C,\bar{c}}\simeq(\mathscr{O}_{T,q(\bar{c})}[x,y]/(xy-t))^{\mu_{n}}\simeq\mathscr{O}_{T,q(\bar{c})}[x^{n},y^{n}]/(x^{n}y^{n}-t^{n}).$

If we set $u:=x^{n}$ and $v:=y^{n}$ in $\mathscr{O}_{C,\bar{c}}$, then we may
write the top left corner of (28) as the stack

(29)
$[(\operatorname{Spec}(\mathscr{O}_{C,\bar{c}}[x,y]/(x^{n}-u,y^{n}-v,xy-t))/\mu_{n}].$

Now write $\mathscr{O}_{C,\bar{c}}$ as the inverse limit of affine schemes
$\operatorname{Spec}(R_{i})$ with étale maps to $C$. Since
$\operatorname{Spec}(\mathscr{O}_{C,\bar{c}}[x,y]/(x^{n}-u,y^{n}-v,xy-t))\to\operatorname{Spec}(\mathscr{O}_{C,\bar{c}})$
is finitely presented, there is an index $i_{0}$ and elements $u,v,t\in
R_{i_{0}}$ such that
$\operatorname{Spec}(R_{i_{0}}[x,y]/(x^{n}-u,y^{n}-v,xy-t))$ pulls back to the
affine scheme in (29). Define $\mu_{n}$ to act on
$\operatorname{Spec}(R_{i_{0}}[x,y]/(x^{n}-u,y^{n}-v,xy-t))$ by the same rule
$x\mapsto\zeta\cdot x$ and $y\mapsto\zeta^{-1}\cdot y$.

Let $\mathcal{C}_{R_{i}}$ denote the pullback of $\mathcal{C}$ to
$\operatorname{Spec}(R_{i})$. Observe that for $i\geq i_{0}$, we have two
stacks $[(\operatorname{Spec}(R_{i}[x,y]/(x^{n}-u,y^{n}-v,xy-t))/\mu_{n}]$ and
$\mathcal{C}_{R_{i}}$ defined over $\operatorname{Spec}(R_{i})$ and an
isomorphism between their pullbacks to
$\operatorname{Spec}(\mathscr{O}_{C,\bar{c}})$. By [LM00, Prop 4.18(i)] there
is an index $j\geq i_{0}$ and an isomorphism
$[(\operatorname{Spec}(R_{j}[x,y]/(x^{n}-u,y^{n}-v,xy-t))/\mu_{n}]\simeq\mathcal{C}_{R_{j}}$.
We may set $R:=R_{j}$. ∎

We refer the reader to Section 1.6 for definitions of the direct and inverse
image functors in the next lemma.

######  Lemma 3.1.2.

Let $p:\mathcal{C}\rightarrow\mathcal{M}$ be a family of twisted curves on an
algebraic stack $\mathcal{M}$.

  1. (1)

The morphism $p$ has cohomological dimension $\leq 1$ (in the sense of [HR17,
Def 2.1]).

  2. (2)

The morphism $p$ is concentrated (in the sense of [HR17, Def 2.4]).

  3. (3)

For $\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-et}})$
and $\mathcal{G}\in\operatorname{D_{qc}}(\mathcal{M}_{\operatorname{lis-
et}})$, the projection morphism
$\mathcal{G}\otimes\mathsf{R}p_{*}(\mathcal{F})\xrightarrow{\text{\eqref{eq:projection}}}\mathsf{R}p_{*}(p^{*}\mathcal{G}\otimes\mathcal{F})$
is an isomorphism.

  4. (4)

Given a fiber square of algebraic stacks

${\mathcal{C}^{\prime}}$${\mathcal{C}}$${\mathcal{M}^{\prime}}$${\mathcal{M}}$$\scriptstyle{m^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p}$$\scriptstyle{m}$

and $\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-
et}})$, the basechange map
$\mathsf{L}m^{*}\mathsf{R}p_{*}\mathcal{F}\xrightarrow{\text{\eqref{eq:basechange}}}\mathsf{R}p^{\prime}_{*}\mathsf{L}{m^{\prime}}^{*}\mathcal{F}$
is an isomorphism.

  5. (5)

If $\mathcal{M}$ is locally Noetherian, then the functor $\mathsf{R}p_{*}$
sends perfect complexes to perfect complexes.

###### Proof.

For part (1), by flat base change [HR17, Lem 1.2(4)] we may assume
$\mathcal{M}$ is an affine scheme, but this is [AOV11, Prop 2.6]. Now (1)
implies part (2) by definition, part (3) by [HR17, Cor 4.12], and part (4) by
[HR17, Cor 4.13]. For part (5), we recall that perfection is a flat-local
property of complexes in the sense of [HR17, Lem 4.1], so we may use
basechange [HR17, Cor 4.13] to reduce to the case when $\mathcal{M}$ is a
Noetherian affine scheme. Now the result follows from Lemma 3.1.3 below. ∎

######  Lemma 3.1.3.

Let $p:\mathcal{C}\to T$ be a family of twisted curves over a Noetherian
affine scheme $T$ and let $r:\mathcal{C}\to C$ be the map to the coarse moduli
space.

  1. (1)

The exact functor $r_{*}$ sends pseudo-coherent objects in
$\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-et}})$ to pseudo-
coherent objects in $\operatorname{D_{qc}}(C_{\operatorname{lis-et}})$.

  2. (2)

The functor $\mathsf{R}p_{*}$ sends perfect objects in
$\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-et}})$ to perfect
objects in $\operatorname{D_{qc}}(T_{\operatorname{lis-et}})$.

######  Remark 3.1.4.

We expect that the locally Noetherian hypothesis can be removed using absolute
Noetherian approximation for algebraic stacks as in [Stacks, Tag 0CN4] (see
the proof of [Stacks, Tag 01AH]). We do not, however, know a reference that
allows us to assume the approximating morphism has properties (1) and (2) of
Lemma 3.1.3. Since we are not aware of an application of the non-Noetherian
setting we omit this investigation.

###### Proof of Lemma 3.1.3.

We will repeatedly use the fact that if $X$ is a scheme, there are
equivalences of categories $\operatorname{QCoh}(X_{\operatorname{lis-
et}})\simeq\operatorname{QCoh}(X_{\mathrm{zar}})$ and
$\operatorname{D_{qc}}(X_{\operatorname{lis-
et}})\simeq\operatorname{D_{qc}}(X_{\mathrm{zar}})$, where $X_{\mathrm{zar}}$
is the category of sheaves on the small Zariski site of $X$, and that these
equivalences preserve coherence, pseudo-coherence, and perfection.

To prove (1), let
$\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-et}})$ be
pseudo-coherent. Let $f:U\to\mathcal{C}$ be a smooth cover by a scheme. Since
$f$ defines a morphism of lisse-étale sites and $f^{*}$ is exact,
$f^{*}\mathcal{F}$ is pseudo-coherent by [Stacks, Tag 08H4]. By [Stacks, Tag
08E8], the sheaves $H^{i}(f^{*}\mathcal{F})$ are coherent and vanish for $i\gg
0$. It follows from [Ols07, Rmk 6.10, Prop 6.12] that the sheaves
$H^{i}(\mathcal{F})$ are coherent and vanish for $i\gg 0$. By [Alp13, Thm
4.16(x)] the sheaves $r_{*}H^{i}(\mathcal{F})$ have these same properties, but
since $r_{*}$ is exact we know
$r_{*}H^{i}(\mathcal{F})=H^{i}(r_{*}\mathcal{F}).$ Hence by [Stacks, Tag 08E8]
again, the object $r_{*}\mathcal{F}$ is pseudo-coherent.

To prove (2), let
$\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-et}})$ be
perfect—by [Stacks, Tag 08G8] this is equivalent to pseudo-coherent and
locally of finite tor dimension. By part (1) of this lemma and [Stacks, Tag
0CTL], the pushforward $\mathsf{R}p_{*}\mathcal{F}$ is pseudo-coherent. To see
that $\mathsf{R}p_{*}\mathcal{F}$ locally has finite tor dimension, by
[Stacks, Tag 08EA] it suffices to show that for
$\mathcal{G}\in\operatorname{QCoh}(T)$ the sheaves
$H^{i}(\mathsf{R}p_{*}\mathcal{F}\overset{\mathsf{L}}{\otimes}\mathcal{G})$
vanish for $i$ outside a finite range. By the projection formula [HR17, 4.12]
and flatness of $p$ these are equal to
$H^{i}(\mathsf{R}p_{*}(\mathcal{F}\overset{\mathsf{L}}{\otimes}p^{*}\mathcal{G}))$.
Since $\mathcal{F}$ is a perfect complex on a quasi-compact space, it has
finite tor amplitude, so the spectral sequence

$\mathsf{R}^{m}p_{*}H^{n}(\mathcal{F})\implies\mathsf{R}^{m+n}p_{*}\mathcal{F}$

of [Stacks, Tag 015J] and the fact that $p$ is concentrated finish the proof.
∎

###  3.2. Background on right adjoint to pushforward

We recall some statements about right adjoint to pushforward that hold for
purely formal reasons.

######  Lemma 3.2.1.

Let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a concentrated morphism of
algebraic stacks. Then a right adjoint $f^{!}$ to
$\mathsf{R}f_{*}:\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-
et}})\to\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$ exists,
and for dualizable
$\mathcal{G}\in\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$ the
canonical morphism $f^{*}\mathcal{G}\otimes f^{!}\mathscr{O}_{\mathcal{Y}}\to
f^{!}\mathcal{G}$ defined in (14) is an isomorphism. Moreover, for
$\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ and
$\mathcal{G}\in\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$
there is a functorial isomorphism

(30)
$\mathsf{R}f_{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}}}^{\mathrm{qc}}(\mathcal{F},f^{!}\mathcal{G})\rightarrow\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{Y}}}^{\mathrm{qc}}(\mathsf{R}f_{*}\mathcal{F},\mathcal{G}).$

###### Proof.

Existence of $f^{!}$ is [HR17, Thm 4.14(1)] and that (14) is an isomorphism
follows from [FHM03, Prop 5.4]. The isomorphism (30) is [FHM03, Prop 4.3] (see
also [Stacks, Tag 0A9Q]). ∎

We now explain what it means for $f^{!}$ to be compatible with basechange.
While lemma 3.2.1 applies to arbitrary families of twisted curves
$\mathcal{C}\to\mathcal{M}$, we will see that we need additional assumptions
for the basechange property to hold.

Suppose we have a fiber square of algebraic stacks as below with $m$ and $f$
tor-indpendent (see [HR17, Sec 4.5]) and $f$ concentrated.

(31)
${\mathcal{X}^{\prime}}$${\mathcal{X}}$${\mathcal{Y}^{\prime}}$${\mathcal{Y}}$$\scriptstyle{m^{\prime}}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{m}$

Then $f^{!}$ and $g^{!}$ exist as recalled in Lemma 3.2.1. By [HR17, Cor
4.13], the base change map (10) is an isomorphism (we take the closed
symmetric monoidal categories in (9) to be
$\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$, etc). This lets
us define the functorial base change map
$\mathsf{L}{m^{\prime}}^{*}f^{!}\rightarrow g^{!}\mathsf{L}m^{*}$ to be the
composition

(32) $\mathsf{L}{m^{\prime}}^{*}f^{!}\rightarrow
g^{!}\mathsf{R}g_{*}\mathsf{L}{m^{\prime}}^{*}f^{!}\xrightarrow{\text{\eqref{eq:basechange}}}g^{!}\mathsf{L}m^{*}\mathsf{R}f_{*}f^{!}\rightarrow
g^{!}\mathsf{L}m^{*}.$

We are interested in when (32) is an isomorphism.

######  Lemma 3.2.2.

Suppose we have a tor-independent fiber square (31) of quasi-compact algebraic
stacks with quasi-finite and separated diagonals, and suppose that
$\mathcal{Y}^{\prime}$ and $\mathcal{Y}$ are concentrated with quasi-affine
diagonals. If $f$ is concentrated and $\mathsf{R}f_{*}$ sends perfect
complexes to perfect complexes, then (32) is an isomorphism.

######  Remark 3.2.3.

The hypotheses of the lemma are satisfied if all the stacks in (31) are quasi-
compact tame Deligne-Mumford with separated diagonals, with additional
conditions on $f$ as above.

######  Remark 3.2.4.

The preprint [Nee17] proves that (32) is an isomorphism under very general
conditions. Compared with [Nee17], our Lemma 3.2.2 imposes stricter conditions
on the stacks $\mathcal{X}$, $\mathcal{X}^{\prime}$, $\mathcal{Y}$,
$\mathcal{Y}^{\prime}$, and on the morphism $f$, but we allow $m$ to be
arbitrary, whereas [Nee17] requires $m$ to be flat.

######  Remark 3.2.5.

The proof of Lemma 3.2.2 relies on our ability to find a compact generator for
the algebraic stack $\mathcal{X}$. By [HR17, Thm A], our assumptions that
$\mathcal{X}$ is quasi-compact with quasi-finite and separated diagonal imply
that $\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ is compactly
generated by a single perfect complex $\mathcal{P}$. This means that for any
$\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$, we
have $\mathcal{F}=0$ if and only if
$\operatorname{Hom}_{\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-
et}})}(\mathcal{P}[n],\mathcal{F})=0$ for every $n\in\mathbb{Z}$ (here,
$\operatorname{Hom}_{\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-
et}})}$ denotes the hom-set in the (additive) category
${\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})}$). Since
$\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ is a full
subcategory of $\operatorname{D}(\mathcal{X}_{\operatorname{lis-et}})$, we may
compute the hom set in the larger category. But these hom sets are computed by
the cohomology of the derived global hom functor. We conclude that for any
morphism $f:\mathcal{F}\rightarrow\mathcal{G}$, we have that $f$ is an
isomorphism if and only if
$\mathsf{R}\mathrm{Hom}_{\mathscr{O}_{\mathcal{X}}}(\mathcal{P},f)$ is an
isomorphism.

###### Proof of Lemma 3.2.2.

We explain why the proof of [Lip09, Cor 4.4.3] also works in this setting.

The first step is to reduce to the case where $m$ is quasi-affine. Indeed, by
[Lip09, Prop 4.6.8] the morphism (32) satisfies a cocycle condition for
squares stacked horizontally. This implies that it is enough to prove the
Lemma when $\mathcal{Y}^{\prime}$ is an affine scheme and $m$ is smooth, or
when $\mathcal{Y}^{\prime}$ and $\mathcal{Y}$ are both affine (see [Lip09,
182-4] for more details). By assumption $\mathcal{Y}$ has quasi-affine
diagonal, so in either case the morphism $m$ is quasi-affine.

Now we assume $m$ is quasi-affine. Let
$\mathcal{F}\in\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$ and
let $\mathcal{P}$ be a perfect, compact generator for
$\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-et}})$ (see Remark
3.2.5). Since $\mathcal{Y}^{\prime}$ and $\mathcal{Y}$ are concentrated by
assumption, the morphisms $m$ and $m^{\prime}$ are also concentrated [HR17,
Lem 2.5] and we have functors $\mathsf{R}m_{*}$ and
$\mathsf{R}m^{\prime}_{*}$. To show that (32) is an isomorphism, we claim that
it suffices to show the induced map

(33)
$\mathsf{R}f_{*}\mathsf{R}m^{\prime}_{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}^{\prime}}}^{\mathrm{qc}}(\mathsf{L}m^{\prime*}\mathcal{P},\mathsf{L}m^{\prime*}f^{!}\mathcal{F})\rightarrow\mathsf{R}f_{*}\mathsf{R}m^{\prime}_{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}^{\prime}}}^{\mathrm{qc}}(\mathsf{L}m^{\prime*}\mathcal{P},g^{!}\mathsf{L}m^{*}\mathcal{F})$

is an isomorphism. First, $\mathsf{L}{m^{\prime}}^{*}\mathcal{P}$ is
perfect,333 To see this, note that the equivalence in Proposition A.2.3 sends
perfect objects to perfect objects—one reason is because this is an
equivalence of symmetric monoidal categories and the perfect objects are the
dualizables [Stacks, Tag 0FPP]. Now apply [Stacks, Tag 08H6] to the morphism
of strictly simplicial étale topoi.  so we may replace the functors
$\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}^{\prime}}}^{\mathrm{qc}}$
with $\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}^{\prime}}}$ (see
Example 3.0.1). Next, if (33) is an isomorphism we get an isomorphism of
global derived homs by applying the global sections functor. But
$\mathsf{L}m^{\prime*}\mathcal{P}$ is a perfect generator for
$\operatorname{D_{qc}}(\mathcal{X}^{\prime}_{\operatorname{lis-et}})$ by
[HR17, Cor 2.8]—this is where we use that $m$ (hence $m^{\prime}$) is quasi-
affine. We conclude that (32) is an isomorphism (see Remark 3.2.5).

To show that (33) is an isomorphism, we cite the bottom two cells of the
commuting diagram on [Lip09, 182] to reduce to proving a certain morphism

$\mathsf{R}m_{*}\mathsf{R}g_{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}^{\prime}}}^{\mathrm{qc}}(\mathsf{L}m^{\prime*}\mathcal{P},\mathsf{L}m^{\prime*}f^{!}\mathcal{F})\xrightarrow{\mathsf{R}m_{*}(4.4.1)^{*}_{\mathrm{pc}}}\mathsf{R}m_{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{Y}^{\prime}}}^{\mathrm{qc}}(\mathsf{R}g_{*}\mathsf{L}m^{\prime*}\mathcal{P},\mathsf{L}m^{*}\mathcal{F})$

is an isomorphism.444One may check that Lipman’s discussion of the relevant
commuting diagram here and in (34) uses only formal properties of adjoint
symmetric functors as discussed in [Lip09, Sec 3.5]. The setup in [Lip09, Sec
3.5] is compatible with our Situation 2.1.1 by footnote 1. (In the cited
diagram, the map notated $u^{*}\delta$ is, in our notation, equal to
$\mathsf{L}m^{*}$ applied to the isomorphism (30).) We will not bother to
write the definition of $\mathsf{R}m_{*}(4.4.1)^{*}_{\mathrm{pc}}$ because
[Lip09, Lem 4.6.4] gives a commuting diagram

(34)
${\mathsf{R}f^{\prime}_{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}^{\prime}}}^{\mathrm{qc}}(\mathsf{L}m^{\prime*}\mathcal{P},\mathsf{L}m^{\prime*}f^{!}cF)}$${\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{Y}^{\prime}}}^{\mathrm{qc}}(\mathsf{R}f^{\prime}_{*}\mathsf{L}m^{\prime*}\mathcal{P},\mathsf{L}m^{*}\mathcal{F})}$${\mathsf{R}f^{\prime}_{*}\mathsf{L}m^{\prime*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}}}^{\mathrm{qc}}(\mathcal{P},f^{!}\mathcal{F})}$${\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{Y}^{\prime}}}^{\mathrm{qc}}(\mathsf{L}m^{*}\mathsf{R}f_{*}\mathcal{P},\mathsf{L}m^{*}\mathcal{F})}$${\mathsf{L}m^{\prime*}\mathsf{R}f_{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{X}}}^{\mathrm{qc}}(\mathcal{P},f^{!}\mathcal{F})}$${\mathsf{L}m^{*}\mathsf{R}\mathcal{H}om_{\mathscr{O}_{\mathcal{Y}}}^{\mathrm{qc}}(\mathsf{R}f_{*}\mathcal{P},\mathcal{F})}$$\scriptstyle{(4.4.1)^{*}_{\mathrm{pc}}}$$\scriptstyle{\rho}$(10)(10)$\scriptstyle{\mathsf{L}m^{*}\text{\eqref{eq:sheaf-
adjii}}}$$\scriptstyle{\rho}$

The arrows labeled (10) are isomorphisms by [HR17, Cor 4.13]. The arrows
labeled $\rho$ are defined in [FHM03, (3.3)], and by [FHM03, Prop 3.2] all
instances in this diagram are isomorphisms since $\mathcal{P}$ and
$\mathsf{R}f_{*}\mathcal{P}$ are perfect complexes by assumption. We note that
our definition of $\rho$ agrees with the definition in [Lip09, (3.5.4.5)] by
[Lip09, Exercise 3.5.6(a)]. Finally, in diagram (34), we know that (30) is an
isomorphism; this concludes the proof.

∎

###  3.3. Example of Situation 2.2.1

We realize Situation 2.2.1 as duality for families of twisted curves on
Noetherian algebraic spaces.

######  Example 3.3.1.

Let $p:\mathcal{C}\rightarrow T$ be a family of twisted curves on a quasi-
separated Noetherian algebraic space $T$. Let
$\mathscr{C}=\operatorname{D_{qc}}(\mathcal{C}_{\mathrm{et}})$,
$\mathscr{D}=\operatorname{D_{qc}}(T_{\mathrm{et}})$, $f_{*}=\mathsf{R}p_{*}$,
and $f^{*}=\mathsf{L}p^{*}$. We will write $p^{*}$ for $\mathsf{L}p^{*}$ since
$p$ is flat (this is justified by [HR17, (1.9)]). The projection map (13) is
an isomorphism by Lemma 3.1.2.

The right adjoint $p^{!}$ exists by Lemma 3.2.1. Moreover, by [HR17, Thm A],
the category $\operatorname{D_{qc}}(\mathcal{C}_{\operatorname{lis-et}})$ is
compactly detected by a single perfect complex. Since $\mathsf{R}p_{*}$
preserves perfect complexes (by Lemma 3.1.2) and perfect objects in
$\operatorname{D_{qc}}(T_{\mathrm{et}})$ are also compact, it follows from
[FHM03, Thm 8.4] and [FHM03, Lem 7.4] that (14) is an isomorphism for all $Y$.

It remains to show that $p^{!}\mathscr{O}_{T}$ is invertible. This follows
from Lemma 3.2.2 and Lemma 3.3.2 below.

######  Lemma 3.3.2.

Let $p:\mathcal{C}\rightarrow T$ be a family of twisted curves on a Noetherian
affine scheme $T$. Then $p^{!}\mathscr{O}_{T}$ is represented by a rank one
locally free sheaf in degree -1.

###### Proof.

Let $q:C\rightarrow T$ be the coarse moduli space of $\mathcal{C}$ and let
$r:\mathcal{C}\rightarrow C$ be the coarse moduli map. By [Stacks, Tag 0E6P,
0E6R] we know $q^{!}\mathscr{O}_{T}$ is invertible and supported in degree -1.
In particular it is dualizable, so we have

$p^{!}\mathscr{O}_{T}=r^{!}q^{!}\mathscr{O}_{T}=r^{*}q^{!}\mathscr{O}_{T}\otimes
r^{!}\mathscr{O}_{C}$

where the second equality uses [FHM03, Thm 8.4] and the fact that
$q^{!}\mathscr{O}_{T}$ is dualizable. Hence to prove the lemma it suffices to
show that $r^{!}\mathscr{O}_{C}$ is invertible and supported in degree 0.

Let $\bar{c}\to C$ be a geometric point. By Lemma 3.1.1 we have a local
description of $\mathcal{C}\to C$ near $\bar{c}$ given by the diagram (27).
Note that a right adjoint to pushforward exists for every horizontal map in
(27). It follows from [Nee17, Lem 0.1] that the pullback of
$r^{!}\mathscr{O}_{C}$ to $[V/\mu_{n}]$ is equal to $\tau^{!}\mathscr{O}_{U}$
(note that [Nee17, Lem 0.1] applies since $U\to C$ is étale and $r_{*}$
preserves pseudo-coherent objects by Lemma 3.1.3). Since
$[V/\mu_{n}]\to\mathcal{C}$ is flat, the complex $r^{!}\mathscr{O}_{C}$ is
represented by a quasi-coherent sheaf if and only if $\tau^{!}\mathscr{O}_{U}$
is, and by [Stacks, Tag05B2] (applied on strictly simplicial étale sites as in
Proposition A.2.3) $r^{!}\mathscr{O}_{C}$ is invertible if and only if
$\tau^{!}\mathscr{O}_{U}$ is.

To compute $\tau^{!}\mathscr{O}_{U}$, set $\rho=\tau\circ\sigma$ and we
observe that we have an equality

$\rho^{!}\mathscr{O}_{U}=\sigma^{*}\tau^{!}\mathscr{O}_{U}\otimes\sigma^{!}\mathscr{O}_{[V/\mu_{n}]}$

so it suffices to show that $\rho^{!}\mathscr{O}_{U}$ and
$\sigma^{!}\mathscr{O}_{[V/\mu_{n}]}$ are both invertible and supported in
degree 0. In Lemma 3.3.3 below, we prove the statement about
$\rho^{!}\mathscr{O}_{U}$, as well as the statement that
$pr^{!}\mathscr{O}_{V}$ is a line bundle in degree 0, where $pr:\mu_{n}\times
V\to V$ is the projection. The statement about $pr^{!}\mathscr{O}_{V}$ is
equivalent to the statement about $\sigma^{!}\mathscr{O}_{[V/\mu_{n}]}$ by an
arument identical to the one used in the previous paragraph.

∎

######  Lemma 3.3.3.

The complexes $\rho^{!}\mathscr{O}_{U}$ and $pr^{!}\mathscr{O}_{V}$ are
represented by line bundles supported in degree 0.

###### Proof.

We use the statement of finite duality in [Stacks, Tag 0AX2], which we
translate to a statement about rings using [Stacks, Tag 06Z0]. These results
imply that for a morphism of affine schemes
$\operatorname{Spec}(B)\rightarrow\operatorname{Spec}(A)$, the image of
$\mathscr{O}_{\operatorname{Spec}(A)}$ under the right adjoint to pushforward
is induced by the complex of $B$-modules

(35) $\mathsf{R}\mathcal{H}om_{A}(B,A).$

For $pr$, the relevant ring map is the diagonal $A\rightarrow\prod_{g\in G}A$,
and $B=\prod_{g\in G}A$ is a free $A$-module so (35) is supported in degree 0.
One checks that there is an isomorphism
$B\rightarrow\mathsf{R}\mathcal{H}om_{A}(B,A)$ given by sending $1_{B}$ to the
projection to the identity factor.

For $\rho$, let $R=A^{G}$, the ring of $G$-invariants. Lemma 3.1.1 lists two
possibilities for $A$. In case (1) the computation of (35) is similar to that
for $pr$ since in this case, $A$ is a free $R$-module with basis
$1,x,\ldots,x^{r-1}$ and $\mathsf{R}\mathcal{H}om_{R}(A,R)$ is generated as an
$A$-module by projection to the $x^{r-1}$-factor.

The computation in case (2) is more involved since we have to take a free
resolution of $A$. One may use the resolution

$\ldots\xrightarrow{d_{3}}R^{\oplus 2r-2}\xrightarrow{d_{2}}R^{\oplus
2r-2}\xrightarrow{d_{1}}R^{\oplus 2r-1}\xrightarrow{d_{0}}A\rightarrow 0$

with maps given as follows. If $\\{f_{i},g_{i}\\}_{i=1}^{r-1}$ denotes a free
basis for $R^{\oplus 2r-2}$ and $e$ is the additional basis element of
$R^{\oplus 2r-1}$, then $d_{i}$ is defined by

$\begin{array}[]{llllll}d_{0}:&e\mapsto 1&\quad
d_{i},\;i\;\text{odd}:&f_{i}\mapsto vf_{i}-t^{i}g_{r-i}&\quad
d_{i},\;i>0\;\text{even:}&f_{i}\mapsto uf_{i}+t^{i}g_{r-i}\\\ &f_{i}\mapsto
x^{i}&&g_{r-i}\mapsto ug_{r-i}-t^{r-i}f_{i}&&g_{r-i}\mapsto
t^{r-i}f_{i}+vg_{r-i}\\\ &g_{i}\mapsto y^{i}&&&&\end{array}$

For details, see [Web20, 25-28]. ∎

###  3.4. Example of Situation 2.4.1

We realize Situation 2.4.1 for families of twisted curves on algebraic stacks.
We use the dualizing sheaf and trace map (as in Situation 2.3.1) as a
substitute for the full duality in Example 3.3.1 because we are unable to show
that the basechange morphism (32) is an isomorphism in general.

######  Proposition 3.4.1.

For every family $\mathcal{C}\rightarrow\mathcal{M}$ of twisted curves on a
locally Noetherian algebraic stack $\mathcal{M}$, there is a pair
$(\omega^{\bullet}_{\mathcal{M}},tr_{\mathcal{M}})$ with
$\omega^{\bullet}_{\mathcal{M}}=\omega_{\mathcal{M}}[1]$ where
$\omega_{\mathcal{M}}\in\operatorname{QCoh}(\mathcal{C}_{\operatorname{lis-
et}})$ is locally free and
$tr_{\mathcal{M}}:\mathsf{R}\pi_{*}\omega^{\bullet}_{\mathcal{M}}\rightarrow\mathscr{O}_{\mathcal{M}}$,
such that the following hold:

  1. (1)

The pair is functorial in the following sense. Given a fiber square

(36)
${\mathcal{C}_{\mathcal{N}}}$${\mathcal{C}_{\mathcal{M}}}$${\mathcal{N}}$${\mathcal{M}}$$\scriptstyle{m^{\prime}}$$\scriptstyle{m}$

there is a canonical isomorphism

(37)
${m^{\prime}}^{*}\omega^{\bullet}_{\mathcal{M}}\xrightarrow{\sim}\omega^{\bullet}_{\mathcal{N}}$

such that the following square commutes:

(38)
${\mathsf{L}m^{*}\mathsf{R}p_{*}\omega^{\bullet}_{\mathcal{M}}}$${\mathscr{O}_{\mathcal{N}}}$${\mathsf{R}p_{*}\mathsf{L}{m^{\prime}}^{*}\omega^{\bullet}_{\mathcal{M}}}$${\mathsf{R}p_{*}\omega^{\bullet}_{\mathcal{N}}}$$\scriptstyle{\mathsf{L}m^{*}tr_{\mathcal{M}}}$(10)$\scriptstyle{\sim}$(37)$\scriptstyle{\sim}$$\scriptstyle{tr_{\mathcal{N}}}$

Moreover, if $n:\mathcal{K}\rightarrow\mathcal{N}$ is a morphism of algebraic
stacks and
$\mathcal{C}_{\mathcal{K}}=\mathcal{C}_{\mathcal{N}}\times_{\mathcal{N}}\mathcal{K}$
is the pullback and
$n^{\prime}:\mathcal{C}_{\mathcal{K}}\rightarrow\mathcal{C}_{\mathcal{N}}$ the
projection, then the isomorphism $(m^{\prime}\circ
n^{\prime})^{*}\omega^{\bullet}_{\mathcal{M}}\rightarrow\omega_{\mathcal{N}}$
is equal to the composition
$n^{\prime*}m^{\prime*}\omega^{\bullet}_{\mathcal{M}}\rightarrow
n^{\prime*}\omega^{\bullet}_{\mathcal{N}}\rightarrow\omega^{\bullet}_{\mathcal{K}}.$

  2. (2)

If $\mathcal{M}$ is a quasi-separated Noetherian algebraic space, then
$\omega^{\bullet}_{\mathcal{M}}=p^{!}\mathscr{O}_{\mathcal{M}}$ and
$tr_{\mathcal{M}}$ is the counit of the $(\mathsf{R}p_{*},p^{!})$ adjunction.

For a general base $\mathcal{M}$ we do not know if our construction of
$(\omega^{\bullet}_{\mathcal{M}},tr_{\mathcal{M}})$ agrees with the right
adjoint to pushforward.

######  Remark 3.4.2.

To see that Proposition 3.4.1 gives an example of Situation 2.4.1 compatible
with Example 3.3.1, we use the fact that
$\mathsf{L}m^{*}:\operatorname{D_{qc}}(\mathcal{M}_{\operatorname{lis-
et}})\to\operatorname{D_{qc}}(\mathcal{N}_{\operatorname{lis-et}})$ has a
right adjoint even when $m$ is not concentrated; see [HR17, Sec 1.3].

###### Proof of Proposition 3.4.1.

The idea as follows. We will define the pair
$(\omega^{\bullet}_{\mathcal{M}},tr_{\mathcal{M}})$ when $\mathcal{M}$ is an
algebraic space as required by part (2) of the proposition. When $\mathcal{M}$
is an algebraic stack, we will take this as the smooth-local definition of
$(\omega^{\bullet}_{\mathcal{M}},tr_{\mathcal{M}})$, and using the notion of a
very smooth hypercover explained in Appendix A we will show that these local
pairs “glue” to a global one with the correct properties.

We now proceed with the proof. When $\mathcal{M}$ is a quasi-separated
Noetherian algebraic space, we define $\omega^{\bullet}_{\mathcal{M}}$ and
$tr_{\mathcal{M}}$ as required in part (2) of the proposition (see Example
3.3.1). When both $\mathcal{N}$ and $\mathcal{M}$ are both quasi-separated
Noetherian algebraic spaces, we define (37) to be the base change map (32) (it
is an isomorphism by Lemma 3.2.2). The commuting diagram (38) follows from the
definition of (32), see [Lip09, Rmk 4.4(d)]. The cocycle condition on (37) is
[Lip09, Prop 4.6.8].

Let $\mathcal{M}$ be a locally Noetherian algebraic stack. In this paragraph
we define $\omega_{\mathcal{M}}$. Let $M_{\bullet}\rightarrow\mathcal{M}$ be a
very smooth hypercover (see Definition A.3.6) and let
$\mathcal{C}_{M,\bullet}$ be its pullback to $\mathcal{C}_{\mathcal{M}}$ (see
Remark A.3.7). We have associated categories of quasi-coherent sheaves
$\operatorname{QCoh}(M_{\bullet,\operatorname{lis-et}})$ and
$\operatorname{QCoh}(\mathcal{C}_{M,\bullet,\operatorname{lis-et}})$ as in
Section A.3.2. By Remark A.3.8 we may assume that each $M_{i}$ is a disjoint
union of affine schemes (each Noetherian by [Stacks, Tag 06R6]). In
particular, each $M_{i}$ is a disjoint union of qcqs Noetherian schemes. For
each $n\in\mathbb{Z}_{\geq 0}$ we have families of twisted curves
$\mathcal{C}_{M,n}\rightarrow M_{n}$, and hence the system of locally free
sheaves $\omega_{M_{n}}$ (defined by applying the construction in the previous
paragraph to the Noetherian components of $M_{n}$) together with the
isomorphisms (37) defines an object $\omega_{M,\bullet}$ of
$\operatorname{QCoh}(\mathcal{C}_{M,\bullet,\operatorname{lis-et}})$. By
Proposition A.3.12 the sheaf $\omega_{M,\bullet}$ corresponds to a unique
quasi-coherent sheaf $\omega_{\mathcal{M}}$ in
$\operatorname{QCoh}(\mathcal{C}_{\mathcal{M},\operatorname{lis-et}})$ whose
restriction to $\mathcal{C}_{M_{i}}$ is $\omega_{M_{i}}$. Let
$\omega^{\bullet}_{\mathcal{M}}=\omega_{\mathcal{M}}[1].$

In this paragraph we define $tr_{\mathcal{M}}$. By Remark A.3.14 the complex
$\mathsf{R}p_{*}\omega^{\bullet}_{\mathcal{M}}$ is represented by the element
of $\operatorname{D_{qc}}(M_{\bullet,\operatorname{lis-et}})$ whose $n^{th}$
component is $\mathsf{R}p_{*}\omega^{\bullet}_{M_{n}}$ (see also [Stacks, Tag
0D9P]). We have trace maps
$tr_{M_{n}}:\mathsf{R}p_{*}\omega^{\bullet}_{M_{n}}\to\mathscr{O}_{M_{n}}$ for
each $n$, and these are compatible with the transition maps of $M_{\bullet}$
by (38). Now from Proposition A.3.12 combined with the argument in [Stacks,
Tag 0DL9] we obtain
$tr_{\mathcal{M}}:\mathsf{R}p_{*}\omega^{\bullet}_{\mathcal{M}}\rightarrow\mathscr{O}_{\mathcal{M}}$
(the required Ext groups vanish since
$\mathsf{R}p_{*}\omega^{\bullet}_{\mathcal{M}}$ is a complex in degrees [-1,0]
by Lemma 3.1.2).

Now we check that the pair $(\omega_{\mathcal{M}},tr_{\mathcal{M}})$ has the
properties required in part (1) of the proposition. Suppose we have a fiber
square (36) where $\mathcal{N}$ and $\mathcal{M}$ are algebraic stacks. Let
$N_{\bullet}\to\mathcal{N}$ and $M_{\bullet}\to\mathcal{M}$ be very smooth
hypercovers with $M_{i}$ and $N_{i}$ disjoint unions of affine schemes, with a
morphism $N_{\bullet}\to M_{\bullet}$ commuting with the augmentations and
$m:\mathcal{N}\to\mathcal{M}$ (see Remark A.3.9). Let
$\mathcal{C}_{M,\bullet}$ and $\mathcal{C}_{N,\bullet}$ be the pullbacks of
$M_{\bullet}$ and $N_{\bullet}$ to $\mathcal{C}_{\mathcal{M}}$ and
$\mathcal{C}_{\mathcal{N}}$, respectively. For each $n\in\mathbb{Z}_{\geq 0}$,
the twisted curve $\mathcal{C}_{N,n}\rightarrow N_{n}$ is the pullback of
$\mathcal{C}_{M,n}\rightarrow M_{n}$, and we have isomorphisms
$m^{\prime*}_{n}\omega_{M_{n}}\xrightarrow{\text{\eqref{eq:omega-
iso}}}\omega_{N_{n}}$. Under the identifications
$(a^{*}m^{\prime*}\omega_{\mathcal{M}})|_{M_{n}}\simeq
m^{\prime*}_{n}\omega_{M_{n}}$ of Remark A.3.14, these isomorphisms are
compatible with the transition maps for the sheaves
$a^{*}{m^{\prime}}^{*}\omega_{\mathcal{M}}$ and $a^{*}\omega_{\mathcal{N}}$ in
$\operatorname{QCoh}(\mathcal{C}_{N,\bullet,\operatorname{lis-et}})$ because
(37) satisfies the cocycle condition. By descent we get an isomorphism
$m^{\prime*}\omega^{\bullet}_{\mathcal{M}}\rightarrow\omega^{\bullet}_{\mathcal{N}}$.
To check that this definition makes (38) commute, apply the equivalences
$a^{*}$ and use Remark A.3.14 to get a collection of commuting diagrams
indexed by $n\in\mathbb{Z}_{\geq 0}$.

∎

##  4\. Obstruction theories via the Fundamental Theorem

###  4.1. Some Picard categories

Let $\mathscr{S}$ be a site. We recall the notion of Picard stacks from [73,
Sec XVIII.1.4.5], and observe that a Picard category is just a Picard stack on
the punctual site (see also [73, Def XVIII.1.4.2]). If
$f:\mathcal{P}\rightarrow\mathcal{Q}$ is a morphism of Picard stacks on
$\mathscr{S}$, we define the kernel to be the fiber product
$\mathcal{K}=\bullet\times_{e,\mathcal{Q},f}\mathcal{P}$ where $\bullet$ is
the trivial Picard stack (a constant sheaf with all its fibers equal to a
single point), and $e:\bullet\rightarrow\mathcal{Q}$ is the identity.

######  Example 4.1.1.

Let $\operatorname{D}(\mathscr{S})$ be the unbounded derived category of
abelian sheaves on $\mathscr{S}$. As in [73, Sec XVIII.1.4.11] we have a
functor $ch$ from the subcategory $\operatorname{D}^{[-1,0]}(\mathscr{S})$ to
the category of Picard stacks on $\mathscr{S}$ (in the latter category, arrows
are isomorphism classes of morphisms of stacks). Suppose $A$ is a sheaf of
rings on $\mathscr{S}$ and $\operatorname{D}(A)$ is the unbounded derived
category of sheaves of $A$-modules. For two complexes
$F\in\operatorname{D}^{[-\infty,a]}(A)$ and
$G\in\operatorname{D}^{[a-1,\infty]}(A)$, we define

(39) $\mathrm{Ext}^{0/-1}_{A}(F,G):=ch(\tau_{\leq
0}\mathsf{R}\mathrm{Hom}_{A}(F,G))=ch(\tau_{\leq
0}\mathsf{R}\Gamma\mathsf{R}\mathcal{H}om_{A}(F,G))$

where $\mathsf{R}\mathrm{Hom}_{A}$ is derived global hom for
$\operatorname{D}(A)$ and we have omitted the pushforward from the derived
category of $\Gamma(\mathscr{S},A)$-modules to the category of abelian groups.
Observe that $ch$ is applied here over the site with one object and one
morphism, so $\mathrm{Ext}^{0/-1}_{A}(F,G)$ is actually a Picard category (and
the prestack $pch(\tau_{\leq
0}\mathsf{R}\Gamma\mathsf{R}\mathcal{H}om_{A}(F,G)$ of [73, Sec XVIII.1.4.11]
is actually a stack). If the ring $A$ is clear we will omit it from the
notation. It follows from [73, (XVIII.1.4.11.1)] that isomorphism classes of
objects of $\mathrm{Ext}^{0/-1}_{A}(F,G)$ are equal to
$\operatorname{Ext}^{0}_{A}(F,G)$ and from [73, (XVIII.1.4.11.2)] that
automorphisms of the identity element are $\operatorname{Ext}^{-1}_{A}(F,G)$.

######  Example 4.1.2.

Let $\mathcal{X}\rightarrow\mathcal{Y}$ be a representable morphism of
algebraic stacks and let $I$ be a quasi-coherent sheaf on $\mathcal{X}$. We
recall from [Ols06, Sec 2.2, 2.12] the Picard category
$\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)$ on
$\mathcal{X}_{\mathrm{et}}$: objects are square-zero extensions
$\mathcal{X}\hookrightarrow\mathcal{X}^{\prime}$ of stacks over $\mathcal{Y}$,
together with an isomorphism
$I\rightarrow\ker(\mathscr{O}_{\mathcal{X}^{\prime}}\rightarrow\mathscr{O}_{\mathcal{X}})$
(see [Ols06, Sec 2.2] for details, e.g. arrows).

Now suppose we have the following commuting diagram of algebraic stacks where
$q:\mathcal{X}\hookrightarrow\mathcal{X}^{\prime}$ is a square-zero extension
by a quasi-coherent sheaf $I$, the maps $f$ and $g$ are representable, and we
have fixed 2-morphism $\gamma:r\circ f\rightarrow g\circ q$.

(40)
${\mathcal{X}}$${\mathcal{Y}}$${\mathcal{X}^{\prime}}$${\mathcal{Z}}$$\scriptstyle{f}$$\scriptstyle{q}$$\scriptstyle{r}$$\scriptstyle{g}$

The morphism $r$ induces a morphism
${\underline{R}}:\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)\to\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{X},I)$,
and the perimeter of (40) defines an element of
$\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{X},I)$ (i.e. a functor
$\bullet\to\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{X},I)$, where
$\bullet$ is the groupoid with one object and one arrow). We define the Picard
category $\underline{\mathrm{Def}}(f)$ to be the fiber product

(41)
${\underline{\mathrm{Def}}(f)}$${\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)}$${\bullet}$${\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{X},I)}$$\scriptstyle{{\underline{R}}}$

where the bottom arrow
$\bullet\to\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{X},I)$ is the
section induced by (40). We use $\mathrm{Def}(f)$ to denote the set of
isomorphism classes of $\underline{\mathrm{Def}}(f)$. Explicitly, objects of
$\underline{\mathrm{Def}}(f)$ are triples $(k,\epsilon,\delta)$ such that
$k:\mathcal{X}^{\prime}\rightarrow\mathcal{Y}$ is a 1-morphism, and
$\epsilon:f\rightarrow k\circ q$ and $\delta:r\circ k\rightarrow g$ are
2-morphisms satisfying $q^{*}(\delta)\circ r(\epsilon)=\gamma$. A morphism
from $(k_{1},\epsilon_{1},\delta_{1})$ to $(k_{2},\epsilon_{2},\delta_{2})$ is
a natural transformation $\tau:k_{1}\rightarrow k_{2}$ such that
$q^{*}(\tau)\circ\epsilon_{1}=\epsilon_{2}$ and $\delta_{1}=\delta_{2}\circ
r(\tau)$ (for details see [Web20, Lem 2.4.3]).

######  Example 4.1.3.

As an example of the diagram (40), let
$\mathcal{X}\xrightarrow{f}\mathcal{Y}\xrightarrow{r}\mathcal{Z}$ be morphisms
of algebraic stacks with $\mathcal{X}$ an algebraic space, and let
$I\in\operatorname{QCoh}(\mathcal{X}_{\mathrm{et}})$. Define
$q:\mathcal{X}\rightarrow\mathcal{X}^{\prime}$ to be the trivial extension by
$I$, so we have $q^{\prime}:\mathcal{X}^{\prime}\rightarrow\mathcal{X}$ such
that $q^{\prime}\circ q=1_{\mathcal{X}}$. Now $g:=r\circ f\circ q^{\prime}$ is
representable, and $k=f\circ q^{\prime}$ defines an element of
$\underline{\mathrm{Def}}(f)$.

###  4.2. The Fundamental Theorem

The fundamental property of the cotangent complex is that it provides a
description of the Picard category in Example 4.1.2 in terms of the
construction in Example 4.1.1.

######  Theorem 4.2.1 ([Ols06]).

Let $\mathcal{X}\rightarrow\mathcal{Y}$ be a representable morphism of
algebraic stacks. Then there is an isomorphism of Picard categories:

(42)
$\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)\xrightarrow{\sim}\mathrm{Ext}^{0/-1}_{\mathscr{O}_{\mathcal{X}}}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I[1])$

The definition of (42) is technical and we defer it to Section B.3. For us,
the key property of (42) is that it is functorial under pullback and
basechange as stated in the next two lemmas.

######  Lemma 4.2.2.

Suppose we have maps
$\mathcal{Z}\xrightarrow{f}\mathcal{W}\xrightarrow{g}\mathcal{Y}$ with $f$ and
$g\circ f$ representable. Then given
$I\in\operatorname{QCoh}(\mathcal{Z}_{\operatorname{lis-et}})$, there is a
commuting diagram of Picard categories:

(43)
${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{Z}/\mathcal{W}},I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{Z}/\mathcal{Y}},I[1])}$${\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{Z},I)}$${\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{Z},I)}$$\scriptstyle{A}$$\scriptstyle{B}$(42)(42)

Here $A$ is induced by the canonical map
$\mathbb{L}_{\mathcal{Z}/\mathcal{Y}}\rightarrow\mathbb{L}_{\mathcal{Z}/\mathcal{W}}$
and $B$ is induced by composition with $g$.

Lemma 4.2.2 is a special case of [Ols06, (2.33.3)], but that result is stated
only for isomorphism classes of objects. We will prove Lemma 4.2.2 in Appendix
B. For the second functoriality lemma, suppose we have a fiber square of
algebraic stacks

(44)
${\mathcal{Z}}$${\mathcal{X}}$${\mathcal{W}}$${\mathcal{Y}}$$\scriptstyle{p}$

where the map $\mathcal{W}\rightarrow\mathcal{Y}$ is flat and
$\mathcal{X}\rightarrow\mathcal{Y}$ is representable. Then given a quasi-
coherent sheaf $I\in\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}})$,
there is a morphism of Picard categories

(45)
$\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)\rightarrow\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{Z},p^{*}I)$

sending $\mathcal{X}^{\prime}\rightarrow\mathcal{Y}$ to the pullback
$\mathcal{Z}^{\prime}:=\mathcal{X}^{\prime}\times_{\mathcal{Y}}\mathcal{W}\rightarrow\mathcal{W}$
(observe that, since (44) is fibered, we have an induced map
$\mathcal{Z}\hookrightarrow\mathcal{Z}^{\prime}$ with the desired kernel).

######  Lemma 4.2.3.

Given the fiber square (44) and
$I\in\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}})$, there is a
commuting diagram of Picard categories:

(46)
${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I[1])}$${\mathrm{Ext}^{0/-1}(p^{*}\mathbb{L}_{\mathcal{X}/\mathcal{Y}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{Z}/\mathcal{W}},p^{*}I[1])}$${\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)}$${\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{Z},p^{*}I)}$$\scriptstyle{C}$$\scriptstyle{\sim}$$\scriptstyle{D}$(42)$\scriptstyle{E}$(42)

Here $C$ is induced by (8) in the context of Example B.2.3555Example B.2.3
differs from Example 3.0.1 because it uses general sheaves of
$\mathscr{O}$-modules and hence the $\mathsf{R}\mathcal{H}om$ functor instead
of $\mathsf{R}\mathcal{H}om^{\mathrm{qc}}$., the arrow $D$ is induced by the
canonical map of cotangent complexes (an isomorphism in this case), and $E$ is
(45).

We will prove Lemma 4.2.3 in Appendix B. We conclude this section with a
corollary to Theorem 4.2.1 that may be read as a relative version of the same
theorem.

######  Corollary 4.2.4.

Consider a diagram (40) of algebraic stacks where
$\mathcal{X}\rightarrow\mathcal{X}^{\prime}$ is a square-zero extension with
ideal sheaf $I$, and $f$ and $g$ are representable.

  1. (1)

There is an obstruction
$o(f)\in\operatorname{Ext}^{1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)$
whose vanishing is necessary and sufficient for the set $\mathrm{Def}(f)$ to
be nonempty.

  2. (2)

If $o(f)=0$, then there is an isomorphism
$\underline{\mathrm{Def}}(f)\simeq\mathrm{Ext}^{0/-1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)$.

######  Remark 4.2.5.

It follows from the Corollary that if $o(f)=0$, we get an isomorphism of
groups between
$\operatorname{Ext}^{-1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}})$
and the automorphism group of any element of $\underline{\mathrm{Def}}(f)$.
One can extract from the proof of the Corollary that $\mathrm{Def}(f)$ is a
torsor for
$\operatorname{Ext}^{0}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I).$

###### Proof.

Applying Lemma 4.2.2 to the maps
$\mathcal{X}\rightarrow\mathcal{Y}\xrightarrow{r}\mathcal{Z}$ we get a
commuting diagram

(47)
${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{X}/\mathcal{Z}},I[1])}$${\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)}$${\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{X},I)}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{{\underline{R}}}$

where ${\underline{R}}$ is the same as the map $B$ in the Lemma. When we
restrict (47) to isomorphism classes of objects, we get the commuting square
in the diagram below.

(48)
${\operatorname{Ext}^{1}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I)}$${\operatorname{Ext}^{1}(\mathbb{L}_{\mathcal{X}/\mathcal{Z}},I)}$${\operatorname{Ext}^{1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)}$${\mathrm{Exal}_{\mathcal{Y}}(\mathcal{X},I)}$${\mathrm{Exal}_{\mathcal{Z}}(\mathcal{X},I)}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$$\scriptstyle{\alpha}$$\scriptstyle{ob}$$\scriptstyle{R}$

The top row of the diagram comes from applying $\operatorname{Ext}^{1}(-,I)$
to the distinguished triangle

(49)
$\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}}\rightarrow\mathbb{L}_{\mathcal{X}/\mathcal{Z}}\rightarrow\mathbb{L}_{\mathcal{X}/\mathcal{Y}}.$

The set $\mathrm{Def}(f)$ is nonempty if and only if, in (48), the fiber of
$R$ over the element $[g]\in\mathrm{Exal}_{\mathcal{Z}}(\mathcal{X},I)$
defined by (40) is nonempty. From the long exact sequence for
$\operatorname{Ext}^{i}(-,I)$ applied to (49), we see that this happens if and
only if the image of $[g]$ in
$\operatorname{Ext}^{1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)$
(under the maps given in (48)) is 0. We define

(50) $o(f)=ob(\alpha^{-1}([g])).$

If $\mathrm{Def}(f)$ is not empty, then by Lemma 4.2.6 below
$\underline{\mathrm{Def}}(f)$ is isomorphic to the kernel of the morphism of
Picard categories

${\underline{R}}:\underline{\mathrm{Exal}}_{\mathcal{Y}}(\mathcal{X},I)\rightarrow\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{X},I).$

It follows from (47) and [Ols06, Lem 2.29] applied to the distinguished
triangle

$\mathsf{R}\mathrm{Hom}(\mathbb{L}_{\mathcal{X}/\mathcal{Z}},I[1])\xrightarrow{\beta}\mathsf{R}\mathrm{Hom}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I[1])\rightarrow\mathsf{R}\mathrm{Hom}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I[1])\rightarrow$

induced from (49) that this kernel is canonically isomorphic to
$ch((\tau_{\leq-1}Cone(\tau_{\leq 0}\beta))[-1])$, where $Cone$ denotes the
mapping cone of a morphism. But we compute

$(\tau_{\leq-1}Cone(\tau_{\leq
0}\beta)))[-1]=(\tau_{\leq-1}Cone(\beta))[-1]=\tau_{\leq 0}Cone(\beta[-1])$

so we get that this kernel is isomorphic to
$\mathrm{Ext}^{0/-1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)$.
∎

######  Lemma 4.2.6.

Let $f:\mathcal{P}\rightarrow\mathcal{Q}$ be a morphism of Picard stacks on a
stack $\mathcal{X}$, and let $\mathcal{K}$ denote the kernel. Let
$q:\mathcal{X}\rightarrow\mathcal{Q}$ be a section and
$\mathcal{F}=\mathcal{P}\times_{\mathcal{Q},q}\mathcal{X}$ the fiber product.
If the set of global objects of $\mathcal{F}$ is not empty, then $\mathcal{F}$
is non-canonically isomorphic to $\mathcal{K}$.

###### Proof.

A global object of $\mathcal{F}$ defines a section
$\sigma:\mathcal{X}\rightarrow\mathcal{F}$. One can check that the composition
$\mathcal{K}\times_{\mathcal{X}}\mathcal{F}\rightarrow\mathcal{P}\times_{\mathcal{X}}\mathcal{P}\xrightarrow{\mu}\mathcal{P}$,
where $\mu$ is the group operation, factors through $\mathcal{F}$. We obtain a
morphism

(51)
$\mathcal{K}\xrightarrow{(1_{\mathcal{K}},\sigma)}\mathcal{K}\times_{\mathcal{X}}\mathcal{F}\rightarrow\mathcal{F}.$

On the other hand, we have the composition

(52)
$\mathcal{F}\xrightarrow{(pr_{1},-\sigma)}\mathcal{P}\times_{\mathcal{X}}\mathcal{P}\xrightarrow{\mu}\mathcal{P}\xrightarrow{f}\mathcal{Q}$

where $pr_{1}:\mathcal{F}\rightarrow\mathcal{P}$ is the canonical morphism and
$-\sigma$ is $\sigma$ followed by the inverse morphism. The composition (52)
factors through the identity $e:\mathcal{X}\rightarrow\mathcal{Q}$, so we get
an induced map $\mathcal{F}\rightarrow\mathcal{K}$. One may check that this is
inverse to (51). ∎

###  4.3. Equivalent definitions of an obstruction theory

Let $\mathcal{Y}\rightarrow\mathcal{Z}$ be a morphism of algebraic stacks. If
$\phi:E\rightarrow F$ is a morphism in
$\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$, let
$H^{i}(\phi):H^{i}(E)\rightarrow H^{i}(F)$ denote the induced morphism on
cohomology sheaves. The following definition generalizes [BF97, Def 4.4].

######  Definition 4.3.1.

A morphism $\phi:E\rightarrow\mathbb{L}_{\mathcal{Y}/\mathcal{Z}}$ in
$\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$ is an obstruction
theory if $H^{-1}(\phi)$ is a surjection and $H^{0}(\phi)$, $H^{1}(\phi)$ are
isomorphisms.

Given a morphism $\phi:E\rightarrow\mathbb{L}_{\mathcal{Y}/\mathcal{Z}}$ in
$\operatorname{D_{qc}}(\mathcal{Y}_{\operatorname{lis-et}})$, for every
diagram (40) we have induced homomorphisms of groups (computed a priori in the
lisse-étale topology)

(53)
$\Phi_{i}:\operatorname{Ext}^{i}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)\rightarrow\operatorname{Ext}^{i}(\mathsf{L}f^{*}E,I).$

We now present a well-known local criterion for a morphism $\phi$ to be an
obstruction theory. Similar criteria have appeared in [BF97, Thm 4.5], [AP19,
Cor 8.5] and [Pom15, Thm 3.5]. However, we found the wording in these criteria
to be vague in that they do not explicitly require compatibility between
various morphisms. Since proving said compatibility is a major part of the
paper (it comprises the functoriality computations in Appendix B), we give the
precise statement of the local criterion and a fully detailed proof.

######  Lemma 4.3.2.

The following conditions are equivalent.

  1. (1)

The morphism $\phi$ is an obstruction theory.

  2. (2)

For every diagram (40) with $\mathcal{X}$ a scheme, the following hold:

    1. (a)

the element $\Phi_{1}(o(f))\in\operatorname{Ext}^{1}(\mathsf{L}f^{*}E,I)$
vanishes if and only if $\underline{\mathrm{Def}}(f)$ is nonempty

    2. (b)

if $\Phi_{1}(o(f))=0$ then $\Phi_{0}$ and $\Phi_{-1}$ are isomorphisms

  3. (3)

For every affine scheme $\mathcal{X}$, and smooth map
$\mathcal{X}\rightarrow\mathcal{Z}$, the following hold:

    1. (a)

for every ambient diagram (40) using $\mathcal{X}$, the element
$\Phi_{1}(o(f))\in\operatorname{Ext}^{1}(\mathsf{L}f^{*}E,I)$ vanishes if and
only if $\underline{\mathrm{Def}}(f)$ is nonempty

    2. (b)

for every $I\in\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}})$, the
maps $\Phi_{0}$ and $\Phi_{-1}$ are isomorphisms

######  Remark 4.3.3.

In Lemma 4.3.2, conditions (2) and (3) may be computed in
$\mathcal{X}_{\mathrm{et}}$—so in (3b), one checks every
$I\in\operatorname{QCoh}(\mathcal{X}_{\mathrm{et}})$ (see e.g. [Ols16, Prop
9.2.16]).

###### Proof of Lemma 4.3.2.

The proof of this lemma seems to be well-known; many parts were explained to
me by Bhargav Bhatt. Let $C$ be the mapping cone of
$\phi:E\rightarrow\mathbb{L}_{\mathcal{Y}/\mathcal{Z}}$. Then condition (1) is
equivalent to

$\text{(1')}\;\;\;H^{i}(C)=0\;\text{for}\;i\geq-1.$

Assume (1’). Then $H^{i}(\mathsf{L}f^{*}C)$ also vanish for $i\geq-1$, so a
spectral sequence [Stacks, Tag 07AA] for $\operatorname{Ext}^{i}(-,I)$ implies
$\operatorname{Ext}^{i}(\mathsf{L}f^{*}C,I)=0$ for $i\leq 1$ and any $I$. Now
the long exact sequence of Ext groups arising from the distinguished triangle

(54)
$\mathsf{L}f^{*}E\rightarrow\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}}\rightarrow\mathsf{L}f^{*}C\rightarrow$

implies that $\Phi_{1}$ is injective and $\Phi_{0}$ and $\Phi_{-1}$ are
isomorphisms. Combined with Corollary 4.2.4, this proves (2) (with
$\mathcal{X}$ an arbitrary scheme). Now (2) implies (3) using Example 4.1.3.

Assume (3). Condition (1’) may be checked smooth-locally on $\mathcal{Y}$, so
let $f:\mathcal{X}\rightarrow\mathcal{Y}$ be a smooth morphism from an affine
scheme and let $I\in\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}})$
be arbitrary. We will show that if $i\geq-1$ then
$\operatorname{Ext}^{0}(H^{i}(\mathsf{L}f^{*}C),I)=0$, which implies
$f^{*}H^{i}(C)=H^{i}(\mathsf{L}f^{*}C)=0$ (the first equality is [HR17, (1.9)]
and uses flatness of $f$).

By assumption (3b), the morphisms $\Phi_{0}$ and $\Phi_{-1}$ are isomorphisms.
We show that $\Phi_{1}$ is injective. It follows from Corollary 4.2.4 and
assumption (3a) that if $\Phi_{1}(o(f))=0$ then $o(f)=0$, so it suffices to
show that every element of
$\operatorname{Ext}^{1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)$
is equal to $o(f)$ for some diagram (40), or equivalently that the map $ob$ in
(50) is surjective. This follows from the long exact sequence

$\rightarrow\operatorname{Ext}^{1}(\mathbb{L}_{\mathcal{X}/\mathcal{Z}},I)\xrightarrow{ob}\operatorname{Ext}^{1}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Y}/\mathcal{Z}},I)\rightarrow\operatorname{Ext}^{2}(\mathbb{L}_{\mathcal{X}/\mathcal{Y}},I)\rightarrow$

since
$\mathbb{L}_{\mathcal{X}/\mathcal{Y}}=\Omega^{1}_{\mathcal{X}/\mathcal{Y}}[0]$
is a locally free sheaf in degree 0.

Since $\Phi_{1}$ is injective and $\Phi_{0}$ and $\Phi_{-1}$ are isomorphisms,
the long exact sequence of Ext groups for (54) shows that
$\operatorname{Ext}^{i}(\mathsf{L}f^{*}C,I)=0$ for every $i\leq 1$. By
[Stacks, Tag 07AA] there is a spectral sequence whose second page is

$\operatorname{Ext}^{i}(H^{-j}(\mathsf{L}f^{*}C),I)\implies\operatorname{Ext}^{i+j}(\mathsf{L}f^{*}(C),I).$

A priori we know $H^{i}(\mathsf{L}f^{*}C)=0$ for $i\geq 2$. By the above
spectral sequence, the group
$\operatorname{Ext}^{0}(H^{1}(\mathsf{L}f^{*}C),I)$ is equal to
$\operatorname{Ext}^{-1}(\mathsf{L}f^{*}C,I)$ which vanishes for every $I$.
This forces $H^{1}(\mathsf{L}f^{*}C)$ to vanish. Inductively applying the same
argument to $\operatorname{Ext}^{0}(\mathsf{L}f^{*}C,I)$ and then
$\operatorname{Ext}^{1}(\mathsf{L}f^{*}C,I)$ shows that
$H^{0}(\mathsf{L}f^{*}C)$ and $H^{-1}(\mathsf{L}f^{*}C)$ vanish as well.

∎

###  4.4. Moduli of sections

Consider a tower of algebraic stacks

$\mathcal{Z}\rightarrow\mathcal{C}\xrightarrow{p}\mathcal{M}$

as in Section 1. There we defined the moduli of sections
${\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$. By [HR19, Thm
1.3] and our assumption that $\mathcal{M}$ is locally Noetherian, the stack
${\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$ is also locally
Noetherian. The stack
${\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$ has a universal
curve $\mathcal{C}_{{\mathrm{Sec}}_{\mathcal{M}}(\mathcal{Z}/\mathcal{C})}$
and a universal section
$f_{{\mathrm{Sec}}_{\mathcal{M}}(\mathcal{Z}/\mathcal{C})}\in\operatorname{Hom}_{\mathcal{C}}(\mathcal{C}_{{\mathrm{Sec}}(\mathcal{Z})},\mathcal{Z})$
(we will omit the subscript on $f$ when possible).

Now suppose we have a tower of algebraic stacks

$\mathcal{Z}\rightarrow\mathcal{W}\rightarrow\mathcal{C}\xrightarrow{p}\mathcal{M}$

where $\mathcal{Z},\mathcal{C}$, and $\mathcal{M}$ are as before and
$\mathcal{W}\rightarrow\mathcal{M}$ is locally finitely presented, quasi-
separated, and has affine stabilizers. To simplify the notation, let
$\mathfrak{S}(\mathcal{Z}):={\mathrm{Sec}}_{\mathcal{M}}({\mathcal{Z}}/{\mathcal{C}})$
and
$\mathfrak{S}(\mathcal{W}):={\mathrm{Sec}}_{\mathcal{M}}({\mathcal{W}}/{\mathcal{C}})$.
We have an induced map
$\mathfrak{S}(\mathcal{Z})\rightarrow\mathfrak{S}(\mathcal{W})$, and over this
map we have a canonical relative obstruction theory defined as follows. We
have a morphism in
$\operatorname{D_{qc}}(\mathcal{C}_{\mathfrak{S}(\mathcal{Z})})$ consisting of
canonical morphisms of cotangent complexes:

(55)
$\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}}\rightarrow\mathbb{L}_{\mathcal{C}_{\mathfrak{S}(\mathcal{Z})}/\mathcal{C}_{\mathfrak{S}(\mathcal{W})}}\xleftarrow{\sim}p^{*}\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}.$

Using the pair
$(\omega^{\bullet}_{\mathfrak{S}(\mathcal{Z})},tr_{\mathfrak{S}(\mathcal{Z})})$
defined in Proposition 3.4.1, we may apply the adjunction-like morphism $a$
defined in Section 2.3 to (55), obtaining

(56)
$\phi_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}:\mathbb{E}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}\rightarrow\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})},\quad\mathbb{E}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}:=\mathsf{R}p_{*}(\mathsf{L}f^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}}\otimes\omega^{\bullet}_{\mathfrak{S}(\mathcal{Z})}).$

For example, when $\mathcal{W}=\mathcal{C}$ we have
$\mathfrak{S}(\mathcal{W})=\mathcal{M}$ and we obtain an obstruction theory on
$\mathfrak{S}(\mathcal{Z})$ relative to $\mathcal{M}$. We refer the reader to
[CJW21, Appendix A] for functoriality properties of
${\mathrm{Sec}}_{\mathcal{M}}(\mathcal{Z}/\mathcal{C})$ and the obstruction
theories (56).

The main theorem of this article is the following.

######  Theorem 4.4.1.

The morphism (56) is an obstruction theory.

####  4.4.1. Proof of Theorem 4.4.1

We prove condition (3) of Lemma 4.3.2. To begin, fix a solid commuting diagram

(57)
${T}$${\mathfrak{S}(\mathcal{Z})}$${T^{\prime}}$${\mathfrak{S}(\mathcal{W})}$$\scriptstyle{m}$

with $m$ a smooth morphism and $T\rightarrow T^{\prime}$ a square-zero
extension of affine schemes with ideal sheaf
$I\in\operatorname{QCoh}(T_{\operatorname{lis-et}})$. Let $\mathcal{C}_{T}$
(resp. $\mathcal{C}_{T^{\prime}}$) denote the pullback of the universal curve
to $T$ (resp. $T^{\prime}$). We first observe that from the definition of the
moduli stacks we have a commuting diagram of algebraic stacks

(58)
${\mathcal{C}_{T}}$${\mathcal{Z}}$${\mathcal{W}}$${\mathcal{C}_{T^{\prime}}}$${\mathcal{C}_{T}}$${\mathcal{C}_{\mathfrak{S}(\mathcal{Z})}}$${\mathcal{C}_{\mathfrak{S}(\mathcal{W})}}$${\mathcal{C}_{T^{\prime}}}$${T}$${\mathfrak{S}(\mathcal{Z})}$${\mathfrak{S}(\mathcal{W})}$${T^{\prime}}$$\scriptstyle{f_{T}}$$\scriptstyle{p}$$\scriptstyle{m^{\prime}}$$\scriptstyle{p}$$\scriptstyle{f}$$\scriptstyle{f_{\mathfrak{S}(\mathcal{W})}}$$\scriptstyle{p}$$\scriptstyle{m}$

######  Claim 4.4.2 (Step 1).

The diagram (58) leads to a commuting diagram of Picard categories

(59)
${\underline{\mathrm{Def}}(f_{T})}$${\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{C}_{T},p^{*}I)}$${\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{C}_{T},p^{*}I)}$${\underline{\mathrm{Def}}(m)}$${\underline{\mathrm{Exal}}_{\mathfrak{S}(\mathcal{Z})}(T,I)}$${\underline{\mathrm{Exal}}_{\mathfrak{S}(\mathcal{W})}(T,I)}$$\scriptstyle{B}$$\scriptstyle{\Psi}$$\scriptstyle{B\circ
E}$$\scriptstyle{B}$$\scriptstyle{B\circ E}$

where the arrows $B$ and $E$ are as in Lemmas 4.2.2 and 4.2.3, the terms in
the leftmost column are fibers of the top and bottom horizontal maps and
$\Psi$ is an isomorphism.

###### Proof.

The right square in the diagram follows from the bottom two (fibered) squares
of (58) and the definitions of $B$ and $E$. Moreover, the element of
$\underline{\mathrm{Exal}}_{\mathfrak{S}(\mathcal{W})}(T,I)$ defined by $m$
and its horizontal square in (58) maps under $B\circ E$ to the element of
$\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{C}_{T},p^{*}I)$ defined by
$f_{T}$ and its horizontal square. By the definition (41) of
$\underline{\mathrm{Def}}$ we get the left square of (59).

To prove that $\Psi$ is an isomorphism suffices to check étale-locally on $T$;
i.e., it suffices to show that $\Psi$ induces an equivalence of categories
$\underline{\mathrm{Def}}(m)\rightarrow\underline{\mathrm{Def}}(f_{T})$. For
this we construct an inverse functor. Let $(k,\epsilon,\delta)$ be an element
of $\underline{\mathrm{Def}}(f_{T})$. We get an arrow
$k_{\delta}:T^{\prime}\rightarrow\mathfrak{S}(\mathcal{Z})$ determined by $k$
and $\delta$, making the resulting triangle over $\mathfrak{S}(\mathcal{W})$
strictly commutative. The 2-morphism $\epsilon$ determines a 2-morphism (also
denoted $\epsilon$) from $m$ to the composition $T\rightarrow
T^{\prime}\xrightarrow{k_{\delta}}\mathfrak{S}(\mathcal{Z})$. Hence our
functor sends the object $(k,\epsilon,\delta)$ to the object
$(k_{\delta},\epsilon,id)$. We leave it to the reader to check that this is
inverse to $\Psi$. ∎

######  Claim 4.4.3 (Step 2).

The diagram (58) leads to a morphism of distinguished triangles

(60)
${\mathsf{R}p_{*}\mathsf{R}\mathcal{H}om({\mathsf{L}f_{T}^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}}},{p^{*}I})}$${\mathsf{R}p_{*}\mathsf{R}\mathcal{H}om({\mathbb{L}_{\mathcal{C}_{T}/\mathcal{Z}}},{p^{*}I[1]})}$${\mathsf{R}p_{*}\mathsf{R}\mathcal{H}om({\mathbb{L}_{\mathcal{C}_{T}/\mathcal{W}}},{p^{*}I[1])}}$${\mathsf{R}\mathcal{H}om({\mathsf{L}m^{*}\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}},{I})}$${\mathsf{R}\mathcal{H}om({\mathbb{L}_{T/\mathfrak{S}(\mathcal{Z})}},{I[1]})}$${\mathsf{R}\mathcal{H}om({\mathbb{L}_{T/\mathfrak{S}(\mathcal{W})}},{I[1]})}$

where the leftmost vertical arrow has the property that there exists a
composition

(61)
$\mathsf{R}\mathcal{H}om({\mathsf{L}m^{*}\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}},{I})\rightarrow\mathsf{R}p_{*}\mathsf{R}\mathcal{H}om({\mathsf{L}f_{T}^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}}},{p^{*}I})\xrightarrow[\sim]{\exists}\mathsf{R}\mathcal{H}om(\mathsf{L}m^{*}(\mathsf{R}p_{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}}\otimes\omega^{\bullet}_{\mathfrak{S}(\mathcal{Z})}),I)$

equal to the map induced by
$\phi_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}:\mathsf{R}p_{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}}\otimes\omega^{\bullet}_{\mathfrak{S}(\mathcal{Z})}\rightarrow\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}$.
Applying the functor $ch\circ\tau_{\leq 0}\circ\mathsf{R}\Gamma$ to (60)
yields a commuting diagram of Picard categories

(62)
${\mathrm{Ext}^{0/-1}(\mathsf{L}f_{T}^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}},p^{*}I)}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{C}_{T}/\mathcal{Z}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{C}_{T}/\mathcal{W}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}(\mathsf{L}m^{*}\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})},I)}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{T/\mathfrak{S}(\mathcal{Z})},I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{T/\mathfrak{S}(\mathcal{W})},I[1])}$$\scriptstyle{A}$$\scriptstyle{\Phi}$$\scriptstyle{A\circ
D^{-1}\circ C}$$\scriptstyle{A}$$\scriptstyle{A\circ D^{-1}\circ C}$

where the arrows $A,$ $D$, and $C$ are defined as in Lemmas 4.2.2 and 4.2.3,
the terms in the leftmost column are the kernels of the top and bottom
horizontal maps, and if $\Phi$ is an isomorphism then $\Phi_{0}$ and
$\Phi_{-1}$ (defined in (53)) are isomorphisms.

###### Proof.

There is a morphism of distinguished triangles (see [Web20, Lem 2.2.12])

${\mathbb{L}_{\mathcal{C}_{T}/\mathcal{W}}}$${\mathbb{L}_{\mathcal{C}_{T}/\mathcal{Z}}}$${\mathsf{L}f_{T}^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}}[1]}$${p^{*}\mathbb{L}_{T/\mathfrak{S}(\mathcal{W})}}$${p^{*}\mathbb{L}_{T/\mathfrak{S}(\mathcal{Z})}}$${p^{*}\mathsf{L}m^{*}\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}[1]}$

(note that the vertical arrows are only defined in the derived category).
Applying
$\mathsf{R}p_{*}\mathsf{R}\mathcal{H}om^{\mathrm{qc}}(-,p^{*}I[1])$ to this
diagram and composing with the morphism (8) yields (60), but with
$\mathsf{R}\mathcal{H}om^{\mathrm{qc}}$ in place of $\mathsf{R}\mathcal{H}om$.
Now Lemma 2.4.3 (applied in the context of Example 3.0.1) produces the
composition (61) that is isomorphic to the map induced by
$\phi_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})}$, but still with
$\mathsf{R}\mathcal{H}om^{\mathrm{qc}}$ in place of $\mathsf{R}\mathcal{H}om$.
To replace $\mathsf{R}\mathcal{H}om^{\mathrm{qc}}$ with
$\mathsf{R}\mathcal{H}om$, we observe that all stacks in (58) are locally
Noetherian and all morphisms are locally of finite type, so by [Stacks, Tag
08PZ] all cotangent complexes are pseudo-coherent (in fact, in the derived
category $\operatorname{D}^{-}_{Coh}$ of the appropriate topos), and we may
make the replacement by [Stacks, Tag 0A6H] (recall that we are working on an
affine scheme $T$). Now (62) is produced by applying $ch\circ\tau_{\leq
0}\circ\mathsf{R}\Gamma$ and using [Stacks, Tag 08J6], and arguing as at the
end of the proof of Corollary 4.2.4. The map $\Phi$ being an isomorphism
implies $\Phi_{0}$ (resp. $\Phi_{-1}$) is an isomorphism by restricting $\Phi$
to isomorphism classes of objects (resp. automorphisms of the identity). ∎

######  Claim 4.4.4 (Step 3).

Condition (3) in Lemma 4.3.2 holds.

###### Proof.

We study the commuting cube formed by mapping the right square of (62) (on the
top floor) to the right square of (59) (on the ground) via (42) (vertical
maps):

(63)
${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{C}_{T}/\mathcal{Z}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{\mathcal{C}_{T}/\mathcal{W}},p^{*}I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{T/\mathfrak{S}(\mathcal{Z})},I[1])}$${\mathrm{Ext}^{0/-1}(\mathbb{L}_{T/\mathfrak{S}(\mathcal{W})},I[1])}$${\underline{\mathrm{Exal}}_{\mathcal{Z}}(\mathcal{C}_{T},p^{*}I)}$${\underline{\mathrm{Exal}}_{\mathcal{W}}(\mathcal{C}_{T},p^{*}I)}$${\underline{\mathrm{Exal}}_{\mathfrak{S}(\mathcal{Z})}(T,I)}$${\underline{\mathrm{Exal}}_{\mathfrak{S}(\mathcal{W})}(T,I)}$

This cube commutes by Lemmas 4.2.2 and 4.2.3. We note that Theorem 4.2.1
applies because the maps $\mathcal{C}_{T}\rightarrow\mathcal{Z}$ and
$\mathcal{C}_{T}\rightarrow\mathcal{W}$ are representable: for example,
representability of $\mathcal{C}_{T}\rightarrow\mathcal{Z}$ follows from the
fact that $m^{\prime}$ is representable and [Stacks, Tag 04Y5].

To prove (3a), restrict the diagram (63) to isomorphism classes of objects. As
in (48) we extend this diagram by the obstruction maps, obtaining a
commutative diagram

${\mathrm{Exal}_{\mathcal{W}}(\mathcal{C}_{T},p^{*}I)}$${\operatorname{Ext}^{1}(\mathbb{L}_{\mathcal{C}_{T}/\mathcal{W}},p^{*}I)}$${\operatorname{Ext}^{1}(\mathsf{L}f_{T}^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}},p^{*}I)}$${\mathrm{Exal}_{\mathfrak{S}(\mathcal{W})}(T,I)}$${\operatorname{Ext}^{1}(\mathbb{L}_{T/\mathfrak{S}(\mathcal{W})},I)}$${\operatorname{Ext}^{1}(\mathsf{L}m^{*}\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})},I)}$(42)$\scriptstyle{\sim}$$\scriptstyle{ob}$$\scriptstyle{B\circ
E}$$\scriptstyle{\eqref{eq:ft}}$$\scriptstyle{\sim}$$\scriptstyle{ob}$$\scriptstyle{\Phi_{1}^{\prime}}$

where the left square is a side of our cube and the right square is obtained
by applying the derived global sections functor $\mathsf{R}\Gamma$ to (60) and
then taking cohomology. By defintion of $B\circ E$ and commutativity of the
diagram, the map labeled $\Phi_{1}^{\prime}$ sends $o(m)$ to $o(f_{T})$. By
(61) the map $\Phi_{1}^{\prime}$ is quasi-isomorphic to $\Phi_{1}$, where
$\Phi_{1}$ is defined as in Lemma 4.3.2. By Corollary 4.2.4, the element
$o(f_{T})$ (resp $o(m)$) vanishes if and only if $\mathrm{Def}(f_{T})$ (resp
$\mathrm{Def}(m)$) is nonempty. Since the map
$\Psi:\mathrm{Def}(m)\rightarrow\mathrm{Def}(f_{T})$ from (59) is an
isomorphism we see that (3a) holds.

To prove (3b), we may assume the diagram (63) was formed from the trivial
example of (57) (see Example 4.1.3). In this case the terms in the left column
of (59) are kernels (not just fibers) of the horizontal maps, so (63) induces
the following commuting square of kernels:

${\mathrm{Ext}^{0/-1}(\mathsf{L}f_{T}^{*}\mathbb{L}_{\mathcal{Z}/\mathcal{W}},p^{*}I)}$${\underline{\mathrm{Def}}(f_{T})}$${\mathrm{Ext}^{0/-1}(\mathsf{L}m^{*}\mathbb{L}_{\mathfrak{S}(\mathcal{Z})/\mathfrak{S}(\mathcal{W})},I)}$${\underline{\mathrm{Def}}(m)}$$\scriptstyle{\sim}$$\scriptstyle{\Phi}$$\scriptstyle{\sim}$$\scriptstyle{\Psi}$

The horizontal maps are induced by the instances of (42) in the diagram (63)
and they are isomorphisms because (42) is an isomorphism. Since $\Psi$ is an
isomorphism, $\Phi$ is an isomorphism as well. ∎

##  Appendix A Descent theorems for lisse-étale sheaves on algebraic stacks

In this section, we recall the unbounded cohomological descent theorem in
[LO08, Ex 2.2.5] for quasi-coherent sheaves in the lisse-étale site of an
algebraic stack (Proposition A.2.3), and then we use it to prove a new descent
theorem (Propositions A.3.12) that is needed in this paper. In this section,
if $\mathcal{X}$ is an algebraic stack we use $\operatorname{Le}(\mathcal{X})$
to denote the lisse-étale site, and if $U$ is an algebraic space we use
$\operatorname{Et}(U)$ to denote its small étale site ([Stacks, Tag 03ED]).

###  A.1. Morphisms from étale to lisse-étale sites

If $U$ is an algebraic space and $m:U\to\mathcal{X}$ is a smooth morphism,
there is an induced functor of sites
$\operatorname{Et}(U)\to\operatorname{Le}(\mathcal{X})$ (also denoted $m$)
that sends a scheme $V$ with an étale map $V\to U$ to the composition $V\to
U\to\mathcal{X}$.

######  Remark A.1.1.

We make the following observations about the functor $m$:

  1. (1)

The functor $m:\operatorname{Et}(U)\to\operatorname{Le}(\mathcal{X})$ is
cocontinuous and hence induces a morphism of topoi
$m:U_{\mathrm{et}}\to\mathcal{X}_{\operatorname{lis-et}}$ by [Stacks, Tag
00XI]. The functor $m^{-1}:\mathcal{X}_{\operatorname{lis-et}}\to
U_{\mathrm{et}}$ is just restriction.

  2. (2)

Since $m^{-1}$ is restriction, we have
$m^{-1}\mathscr{O}_{\mathcal{X}}=\mathscr{O}_{U}$ and
$m^{-1}\mathcal{F}=m^{*}\mathcal{F}$ when $\mathcal{F}$ is a sheaf of
$\mathscr{O}_{\mathcal{X}}$-modules.

  3. (3)

The functor $m:\operatorname{Et}(U)\to\operatorname{Le}(\mathcal{X})$ is also
continuous, and hence $m^{-1}$ has a left adjoint by [Stacks, Tag 04BG]. Since
$m$ commutes with fiber products and equalizers, the left adjoint is exact by
[Stacks, Tag 04BH]. In particular $m^{-1}$ preserves injectives.

Suppose we have the following commuting diagram of algebraic stacks where $V$
and $U$ are algebraic spaces and $m$ is smooth.

${V}$${U}$${X}$${\mathcal{X}}$$\scriptstyle{m^{\prime}}$$\scriptstyle{f^{\prime}}$$\scriptstyle{m}$$\scriptstyle{f}$

  1. (4)

If $f$ is representable and $V=U\times_{\mathcal{X}}X$, then for
$\mathcal{F}\in X_{\operatorname{lis-et}}$ we have a canonical identification
$f^{\prime}_{*}m^{\prime-1}\mathcal{F}=m^{-1}f_{*}\mathcal{F}$, where
$f^{\prime}_{*}:V_{\mathrm{et}}\to U_{\mathrm{et}}$ (resp.
$f_{*}:X_{\operatorname{lis-et}}\to\mathcal{X}_{\operatorname{lis-et}}$) is
the usual pushforward of étale (resp. lisse-étale) sheaves induced by a
continuous functor of sites. (Note that $f_{*}$ may not have an exact left
adjoint.) Indeed, if $W$ is a scheme and $W\to U$ is étale, then we have

$(f^{\prime}_{*}m^{\prime-1}\mathcal{F})(W)=\mathcal{F}(W\times_{U}V)\quad\quad\quad\quad(m^{-1}f_{*}\mathcal{F})(W)=\mathcal{F}(W\times_{\mathcal{X}}X)$

but there is a natural identification of algebraic spaces $W\times_{U}V\simeq
W\times_{\mathcal{X}}X$.

  2. (5)

If $f$ is smooth, then $f_{*}$ has an exact left adjoint and we let
$f^{*}:\mathrm{Mod}(\mathscr{O}_{\mathcal{X}_{\operatorname{lis-
et}}})\to\mathrm{Mod}(\mathscr{O}_{X_{\operatorname{lis-et}}})$ be the induced
pullback of $\mathscr{O}$-modules. In this case,
$f^{\prime*}m^{\prime*}\mathcal{F}=m^{*}f^{*}\mathcal{F}$ for
$\mathcal{F}\in\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-et}}).$
Indeed, by part (2) above (since $f$ is representable) the functors $m^{*}$
and $m^{\prime*}f^{*}$ are just restriction, but $f^{\prime*}$ is the pullback
functor from $\operatorname{QCoh}(U_{\mathrm{et}})$ to
$\operatorname{QCoh}(U_{X,\mathrm{et}})$. Hence the desired equality holds by
the Cartesian property of $\mathcal{F}$.

###  A.2. The first descent theorem

In this section, we recall Lazslow-Olsson’s theorem for unbounded
cohomological descent for lisse-étale sheaves on an algebraic stack
(Proposition A.2.3). To begin we recall the following general construction
(which will be used multiple times in this appendix).

######  Construction A.2.1.

Let $I$ be a category and let $\mathcal{C}$ be a functor from $I$ to the
2-category of categories (see [Stacks, Tag 003N]); that is, for each $i\in I$
we have a category $\mathcal{C}_{i}$, and for each morphism $\phi:i\to j$ in
$I$ we have a functor
$\phi^{*}_{\mathcal{C}}:\mathcal{C}_{i}\to\mathcal{C}_{j}$ and these are
compatible with compositions. We define a _category of systems_
$\mathcal{C}_{total}$ whose objects are tuples
$\mathcal{F}:=(\mathcal{F}_{i},\mathcal{F}(\phi))$ with
$\mathcal{F}_{i}\in\mathcal{C}_{i}$ and
$\mathcal{F}(\phi):\phi^{*}_{\mathcal{C}}\mathcal{F}_{i}\to\mathcal{F}_{j}$,
such that the following diagrams commute.

${\phi^{*}_{\mathcal{C}}\psi^{*}_{\mathcal{C}}\mathcal{F}_{\ell}}$${\mathcal{F}_{n}}$${\phi^{*}_{\mathcal{C}}\mathcal{F}_{m}}$$\scriptstyle{\mathcal{F}(\psi\circ\phi)}$$\scriptstyle{\phi^{*}_{\mathcal{C}}\mathcal{F}(\psi)}$$\scriptstyle{\mathcal{F}(\phi)}$

A morpism from $(\mathcal{F}_{i},\mathcal{F}(\phi))$ to
$(\mathcal{G}_{i},\mathcal{G}(\phi))$ in $\mathcal{C}_{total}$ is a collection
of morphisms $\alpha_{i}:\mathcal{F}_{i}\to\mathcal{G}_{i}$ compatible with
the $\mathcal{F}(\phi)$ and $\mathcal{G}(\phi)$. The _category of Cartesian
systems_ $\mathcal{C}^{cart}_{total}$ is the full subcategory of
$\mathcal{C}_{total}$ whose objects have the property that every
$\mathcal{F}(\phi)$ is an isomorphism.

######  Remark A.2.2.

Suppose we are given two functors $\mathcal{C},\mathcal{D}$ from $I$ to the
2-category of categories, and suppose we have functors
$\Lambda_{i}:\mathcal{C}_{i}\to\mathcal{D}_{i}$ such that the squares

${\mathcal{C}_{i}}$${\mathcal{D}_{i}}$${\mathcal{C}_{j}}$${\mathcal{D}_{j}}$$\scriptstyle{\Lambda_{i}}$$\scriptstyle{\phi^{*}_{C}}$$\scriptstyle{\phi^{*}_{D}}$$\scriptstyle{\Lambda_{j}}$

2-commute, and the 2-morphisms respect (vertical) compositions of squares.
Then we have a functor $\Lambda:\mathcal{C}_{total}\to\mathcal{D}_{total}$
given by the rule
$\Lambda(\mathcal{F}_{i},\mathcal{F}(\phi))=(\Lambda_{i}(\mathcal{F}_{i}),\Lambda_{j}(\mathcal{F}(\phi)))$.

Let $\mathcal{X}$ be an algebraic stack and let $U\to\mathcal{X}$ be a smooth
cover by an algebraic space. Let $U_{\bullet}$ be the simplicial algebraic
space that is equal to the 0-coskeleton of $U\to\mathcal{X}$. We apply
Construction A.2.1 to the category $I:=\Delta^{+}$, were $\Delta^{+}$ is the
subcategory of the simplicial category $\Delta$ with the same objects but only
the injective morphisms. For $i\in\Delta^{+}$ we set
$\mathcal{C}_{i}:=U_{i,\mathrm{et}}$ and for $\phi:i\to j$ we let
$\phi^{*}:U_{i,\mathrm{et}}\to U_{j,\mathrm{et}}$ be the usual inverse image
functor for this morphism of topoi. The resulting category of systems is
called the strictly simplicial topos in [Ols07, Sec 2.1] and [LO08, Ex 2.1.5],
and we notate it $U^{+}_{\bullet,\mathrm{et}}$. The structure sheaves
$\mathscr{O}_{U_{i}}$ define a distinguished ring object
$\mathscr{O}_{U^{+}_{\bullet}}$ in $U^{+}_{\bullet,\mathrm{et}}$. A _quasi-
coherent sheaf_ in $U^{+}_{\bullet,\mathrm{et}}$ is an
$\mathscr{O}_{U^{+}_{\bullet}}$-module $(\mathcal{F}_{i},\mathcal{F}(\phi))$
such that each $\mathcal{F}_{i}$ is in
$\operatorname{QCoh}(U_{i,\mathrm{et}})$ and the morphism
$\phi^{*}\mathcal{F}_{i}\otimes_{\phi^{*}\mathscr{O}_{U_{i}}}\mathscr{O}_{U_{j}}\to\mathcal{F}_{j}$
induced by $\mathcal{F}(\phi^{*})$ is an isomorphism. Observe that the
category of quasi-coherent sheaves
$\operatorname{QCoh}(U_{\bullet,\mathrm{et}}^{+})$ is equal to the category of
Cartesian systems with
$\mathcal{C}_{i}=\operatorname{QCoh}(U_{i,\mathrm{et}})$ and $\phi^{*}$ equal
to the usual pullback of quasi-coherent sheaves.

There is a functor
$\varpi^{*}:\mathrm{Mod}(\mathscr{O}_{\mathcal{X}_{\operatorname{lis-
et}}})\to\mathrm{Mod}(\mathscr{O}_{U^{+}_{\bullet,\mathrm{et}}})$ given as
follows: for $\mathcal{F}\in\mathcal{X}_{\operatorname{lis-et}}$ set
$(\varpi^{*}\mathcal{F})_{i}=m_{i}^{-1}\mathcal{F}\otimes_{m_{i}^{-1}\mathscr{O}_{\mathcal{X}}}\mathscr{O}_{U_{i}}$
where $m_{i}^{-1}:\mathcal{X}_{\operatorname{lis-et}}\to U_{i,\mathrm{et}}$ is
defined using the projection $U_{i}\to\mathcal{X}$ and Remark A.1.1.1, and let
$\mathcal{F}(\phi)$ be the identity for each $\phi$. Note that $\varpi^{*}$ is
exact and sends quasi-coherent sheaves to quasi-coherent sheaves. The
following proposition is due to Laszo-Olsson.

######  Proposition A.2.3 (Laszlo-Olsson).

The morphism $\varpi^{*}:\operatorname{QCoh}(\mathcal{X}_{\operatorname{lis-
et}})\to\operatorname{QCoh}(U^{+}_{\bullet,\mathrm{et}})$ is an exact
equivalence of categories. We use $\varpi_{*}$ to denote the quasi-inverse.
Moreover, $\varpi^{*}:\operatorname{D_{qc}}(\mathcal{X}_{\operatorname{lis-
et}})\to\operatorname{D_{qc}}(U^{+}_{\bullet,\mathrm{et}})$ is an equivalence,
and we use $\mathsf{R}\varpi_{*}$ to denote the quasi-inverse.

######  Remark A.2.4.

The equivalence of categories of quasi-coherent sheaves is proved in [Ols16,
Prop 9.2.13]. The equivalence of unbounded derived categories is proved in
[LO08, Ex 2.2.5] (using [Ols07, Thm 6.14]) under the assumption that
$\mathcal{X}$ is quasi-separated (a standing assumption for both [Ols07] and
[LO08]). This assumption is not needed for Proposition A.2.3. Indeed, [LO08,
Thm 2.2.3] appears as [Stacks, Tag 0D7V] without the quasi-separated
hypothesis, and one may check directly that the necessary portions of [Ols07]
(namely Proposition 4.4, Lemma 4.5, and Lemma 4.8) do not use this hypothesis.

######  Remark A.2.5.

The equivalences $(\varpi^{*},\varpi_{*})$ are functorial as follows. Let
$X\to\mathcal{X}$ be a smooth morphism of algebraic stacks (inducing a
morphism of lisse-étale topoi), let $V\to X$ be a smooth surjective morphism
from a scheme $V$, and let $V\to U$ be a morphism commuting with maps to
$\mathcal{X}$. It follows from Remark A.1.1.5 that there is an identification
$f^{*}_{\bullet,\mathrm{et}}\varpi^{*}\simeq\varpi^{*}f^{*}$ (where
$f^{*}_{\bullet,\mathrm{et}}$ is given by $f^{*}_{i}$ at level $i$), and since
$\varpi^{*}$ is an equivalence we also have
$\varpi_{*}f^{*}_{\bullet,\mathrm{et}}\simeq f^{*}\varpi_{*}$.

###  A.3. The second descent theorem: hypercovers

In this section we prove an unbounded cohomological descent theorem in the
lisse-etale topology for very smooth hypercovers of algebraic stacks
(Proposition A.3.12).

####  A.3.1. Very smooth hypercovers

Recall that if $\mathcal{U}\to\mathcal{X}$ and $\mathcal{V}\to\mathcal{X}$ are
representable morphisms of algebraic stacks, then the category
$\operatorname{Hom}_{\mathcal{X}}(\mathcal{U},\mathcal{V})$ is isomorphic to a
set.

######  Definition A.3.1.

The _enlarged smooth site_ $\operatorname{Es}(\mathcal{X})$ of $\mathcal{X}$
is the category with objects given by morphisms $f:\mathcal{U}\to\mathcal{X}$
where $\mathcal{U}$ is an algebraic stack and $f$ is smooth and representable,
and with arrows from $\mathcal{U}\to\mathcal{X}$ to
$\mathcal{V}\to\mathcal{X}$ given by the set
$\operatorname{Hom}_{\mathcal{X}}(\mathcal{U},\mathcal{V})$. A covering is a
set of smooth maps $\\{\mathcal{U}_{i}\to\mathcal{U}\\}_{i\in I}$ that are
jointly surjective.

######  Remark A.3.2.

The site $\operatorname{Es}(\mathcal{X})$ contains
$id:\mathcal{X}\to\mathcal{X}$ as the final object.

######  Remark A.3.3.

The morphisms in $\operatorname{Es}(\mathcal{X})$ are all representable.

######  Definition A.3.4.

A _smooth hypercover_ of $\mathcal{X}$ is a simplicial object $X_{\bullet}$ in
$\operatorname{Es}(\mathcal{X})$ such that

  1. (1)

$X_{0}\to\mathcal{X}$ is surjective (note that it will also be smooth)

  2. (2)

$X_{n+1}\to(\mathrm{cosk}_{n}\mathrm{sk}_{n}X_{\bullet})_{n+1}$ is smooth and
surjective for $n\geq 0$.

######  Remark A.3.5.

A smooth hypercover of $\mathcal{X}$ is a hypercover of the final object in
$\operatorname{Es}(\mathcal{X})$ in the sense of [Stacks, Tag 01G5]. Moreover,
if $\mathcal{X}$ is an algebraic space, then a smooth hypercover of
$\mathcal{X}$ is also an fppf hypercover in the sense of [Stacks, Tag 0DH4].

######  Definition A.3.6.

A _very smooth hypercover_ of $\mathcal{X}$ is a smooth hypercover
$X_{\bullet}$ such that every degeneracy and face map $X_{i}\to X_{j}$ is
smooth.

If $X_{\bullet}$ is a smooth hypercover of $\mathcal{X}$ and
$f:\mathcal{Y}\to\mathcal{X}$ is a morphism of algebraic stacks, we can
pullback $X_{\bullet}$ to a simplicial object $Y_{\bullet}$ in
$\operatorname{Es}(\mathcal{Y})$: define
$Y_{i}=X_{i}\times_{\mathcal{X}}\mathcal{Y}$.

######  Remark A.3.7.

If $\mathcal{Y}\to\mathcal{X}$ is a morphism of algebraic stacks and
$X_{\bullet}$ is a (very) smooth hypercover of $\mathcal{X}$, then
$Y_{\bullet}$ is a (very) smooth hypercover of $\mathcal{Y}$. This follows
from [Stacks, Tag 0DAZ].

######  Remark A.3.8.

From [Stacks, Tag 0DEQ] and the proof of [Stacks, Tag 0DAV] it follows that if
$\mathcal{X}$ is an algebraic stack then a very smooth hypercover of
$\mathcal{X}$ exists. In fact, we may take $X_{i}$ to be a disjoint union of
affine schemes.

######  Remark A.3.9.

Let $\mathcal{X}\to\mathcal{Y}$ be a morphism of algebraic stacks. Then we can
find very smooth hypercovers $X_{\bullet}\to\mathcal{X}$ and
$Y_{\bullet}\to\mathcal{Y}$ with $X_{i}$ and $Y_{i}$ disjoint unions of affine
schemes, with a morphism $X_{\bullet}\to Y_{\bullet}$ commuting with the
augmentations and the given morphism $\mathcal{X}\to\mathcal{Y}$. This follows
from analyzing the construction of $X_{\bullet}$ and $Y_{\bullet}$ in [Stacks,
0DAV], using the fact that the functors $\operatorname{cosk}_{n}$ are finite
limits and hence commute with pullback (see the proof of [Stacks, Tag 0DAZ]).

####  A.3.2. The lisse-etale topos of a very smooth hypercover

Recall that if $\mathcal{X}\to\mathcal{Y}$ is a smooth morphism of algebraic
stacks then there is a morphism of sites
$\operatorname{Le}{\mathcal{X}}\to\operatorname{Le}{\mathcal{Y}}$ (see e.g.
[Stacks, Tag 00X1] and [Ols07, Sec 3.3]), and in fact a morphism of ringed
topoi $(\mathcal{X}_{\operatorname{lis-
et}},\mathscr{O}_{\mathcal{X}})\to(\mathcal{Y}_{\operatorname{lis-
et}},\mathscr{O}_{\mathcal{Y}})$. We follow [Stacks, Tag 09WB] by defining the
category of sites to be the category whose objects are sites and whose
morphisms are morphsisms of sites. If $\mathcal{C}_{\bullet}$ is a simplicial
object in this category, then for each morphism $\varphi:[i]\to[j]$ of the
simplicial category $\Delta$ we have a morphism of sites
$f_{\varphi}:\mathcal{C}_{i}\to\mathcal{C}_{j}$.

######  Definition A.3.10.

Let $X_{\bullet}$ be a very smooth hypercover of $\mathcal{X}$. We construct
an associated site $\operatorname{Le}(X_{\bullet})$ as follows: let
$\mathcal{C}_{\bullet}$ be the simplicial object in the category of sites with
$\mathcal{C}_{i}:=\operatorname{Le}(X_{i})$ and $f_{\varphi}$ equal to the
given morphism of sites (it is important that all the face and degeneracy maps
are smooth). Define $\operatorname{Le}(X_{\bullet})$ to be the site
$\mathcal{C}_{total}$ in [Stacks, Tag 09WC], and use
$X_{\bullet,\operatorname{lis-et}}$ to denote the corresponding topos.

######  Remark A.3.11.

By [Stacks, Tag 09WF], a sheaf on $\operatorname{Le}(X_{\bullet})$ is given by
a system $(\mathcal{F}_{i},\mathcal{F}(\varphi))$ where $\mathcal{F}_{i}$ is a
sheaf on $\operatorname{Le}(X_{i})$ and
$\mathcal{F}(\varphi):f_{\varphi}^{-1}\mathcal{F}_{i}\to\mathcal{F}_{j}$ are
compatible morphisms.

Using Remark A.3.11, define a sheaf
$\mathscr{O}_{X_{\bullet,\operatorname{lis-et}}}$ on
$X_{\bullet,\operatorname{lis-et}}$ to be the sheaf equal to
$\mathscr{O}_{X_{i}}$ on $X_{i}$ with transition maps induced by the morphisms
of ringed topoi already given. This makes $X_{\bullet,\operatorname{lis-et}}$
a ringed site. An $\mathscr{O}_{X_{\bullet,\operatorname{lis-et}}}$-module