Situation 2.1. 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.Footnote ¹ This means we have natural isomorphisms

(2.1)$$ \begin{align} f^*(X) \otimes f^*(Y) \rightarrow f^*(X\otimes Y) \end{align} $$

(2.2)$$ \begin{align} \mathscr{O}_{\mathscr{C}} \rightarrow f^*\mathscr{O}_{\mathscr{D}}. \end{align} $$

Situation 2.3. We have a diagram of categories and functors

(2.6)

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

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

(2.9)

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 diagram (2.9) is compatibility of the $(m^{\prime *}f^*, f_*m^{\prime }_*)$ and $(m_*g_*, g^*m^*)$ adjunctions—see [Reference LipmanLip09, (3.6.2)].

Situation 2.5. 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

  (2.10)$$ \begin{align} \pi: Y \otimes f_*(X) \rightarrow f_*(f^*Y\otimes X) \end{align} $$
  defined in [Reference HallHal, Appendix A] is an isomorphism,

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

• • The canonical morphism

  (2.11)$$ \begin{align} \varphi: f^{*}Y \otimes f^{!}\mathscr{O}_{\mathscr{D}} \rightarrow f^{!}Y \end{align} $$
  defined in [Reference Fausk, Hu and MayFHM03, (5.5)] is an isomorphism.

Proof. Since C is invertible, it follows from [Reference MayMay01, Lem 2.9] that C is dualisable and that the coevaluation map defined there is an isomorphism. It follows from the definition of the coevaluation map that the unit in equation (2.12) is an isomorphism when $X=\mathscr {O}_{\mathscr {C}}$. For general X, there is a commuting square

where the map labelled $\nu $ (defined in [Reference Lewis and McClureLew86, p. 120]) is an isomorphism since C is dualisable (see [Reference Lewis and McClureLew86, 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 $.

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

$$\begin{align*}f^C_!f^*(Z) \xrightarrow{{(2.13)}} f^C_!f^!_C(Z) \xrightarrow{\epsilon^{f^C_!}} Z \end{align*}$$

is equal to

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

To see this, expand the $(f_!^C, f^C_!)$-counit in terms of the $(\otimes , \operatorname {\mathrm {Hom}})$-counit and the $(f_!, f^!)$-counit; commute the morphism $\varphi $ in the definition of equation (2.13) with the $(\otimes , \operatorname {\mathrm {Hom}})$-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 $(\otimes , \operatorname {\mathrm {Hom}})$ adjunction. Now equation (2.14) is equivalent to the diagram

whose commutativity is proved in [Reference LipmanLip09, Lem 3.4.7(iv)].

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 [Reference Fausk, Hu and MayFHM03, (4.1)] and [Stacks, Tag 0A9Q]). For future reference, we summarise 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 diagram (2.15).

To show the commutativity of diagram (2.15), we use the Yoneda embedding: for an arbitrary $T \in \mathscr {D}$, it suffices to show the commutativity of

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

where $\tilde \gamma $ is defined to equal the isomorphisms in diagram (2.14). This second diagram commutes by Lemma 2.8.

There are isomorphisms ${\color {gray}1}={\color {gray}4}$ and ${\color {gray}3}={\color {gray}6}$ given by $(\otimes , \operatorname {\mathrm {Hom}})$ adjunction and an isomorphism ${\color {gray}2}={\color {gray}5}$ given by

(2.16)$$ \begin{align} \begin{aligned} {}^{\color{gray}2}\mathscr{D}(T, &f_*\operatorname{\mathrm{Hom}}(f^*X, f^*Y)) = \mathscr{C}(f^*T, \operatorname{\mathrm{Hom}}(f^*X, f^*Y)) = \mathscr{C}(f^*T \otimes f^*X, f^*Y)\\ &= {}^{{\color{gray}7}}\mathscr{C}(f^*(T\otimes X), f^*Y) = \mathscr{C}(f^!_C(T\otimes X), f^!_CY)= {}^{\color{gray}5}\mathscr{D}(f^C_!(f^!_C(T\otimes X)), Y), \end{aligned} \end{align} $$

where the equalities are $(f^*, f_*)$-adjunction, $(\otimes , \operatorname {\mathrm {Hom}})$-adjunction, the isomorphism in equation (2.1), the isomorphism in equation (2.13) and $(f^C_!, f^!_C)$-adjunction. The square with corners ${\color {gray}1}, {\color {gray}3}, {\color {gray}4},$ and ${\color {gray}6}$ commutes by functoriality of the $(\otimes , \operatorname {\mathrm {Hom}})$ adjunction. We take the commutativity of the square with corners ${\color {gray}2}, {\color {gray}3}, {\color {gray}5},$ and ${\color {gray}6}$ as the definition of $\gamma $. The final square commutes as follows. By the definition of the vertical map in diagram (2.15) and adjunction, the composition ${\color {gray}1} \rightarrow {\color {gray}2} \rightarrow {\color {gray}7}$ is equal to

By functoriality of equation (2.1), this composition is equivalent to

$$\begin{align*}{}^{\color{gray}1}\mathscr{D}(T, \operatorname{\mathrm{Hom}}(X,Y)) \xrightarrow{\otimes X}\mathscr{D}(T\otimes X, \operatorname{\mathrm{Hom}}(X,Y)\otimes X) \xrightarrow{\epsilon^\otimes} {}^{\color{gray}4}\mathscr{D}(T\otimes X, Y) \xrightarrow{f^*}{}^{\color{gray}7}\mathscr{C}(f^*(T\otimes X), f^*Y). \end{align*}$$

The first two arrows are precisely the $(\otimes , \operatorname {\mathrm {Hom}})$ adjunction ${\color {gray}1} = {\color {gray}4}.$ Finally, the composition

where the arrow comes from the previous formula and the two equalities come from equation (2.16), is induced by the counit $\epsilon ^{f^C_!}$ as desired.

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

$$ \begin{align*} \mathscr{D}(T, &f_*\operatorname{\mathrm{Hom}}(f^*X, f^*Y)) = \mathscr{C}(f^*T, \operatorname{\mathrm{Hom}}(f^*X, f^*Y)) = \mathscr{C}(f^*T \otimes f^*X, f^*Y) \\ &= \mathscr{C}(f^*T \otimes f^*X, f^!_CY) = \mathscr{D}(f_!^C(f^*T \otimes f^*X), Y) \xrightarrow{\sim} \mathscr{D}(T \otimes f_!^Cf^*X, Y)\\ &= \mathscr{D}(T ,\operatorname{\mathrm{Hom}}( f_!^Cf^*X, Y)) = \mathscr{D}(T ,\operatorname{\mathrm{Hom}}( f_!^Cf^!_CX, Y)), \end{align*} $$

where the equalities are $(f^*, f_*)$ adjunction, $(\otimes , \operatorname {\mathrm {Hom}})$ adjunction, the isomorphism in equation (2.13), $(f^C_!, f^!_C)$ adjunction, the projection formula, $(\otimes , \operatorname {\mathrm {Hom}})$ adjunction and finally the isomorphism in equation (2.13) again. In particular, if we follow $\gamma $ with the inverse of the last equality, we have defined a natural isomorphism

(2.17)$$ \begin{align} f_*\operatorname{\mathrm{Hom}}(Z, f^*Y) \xrightarrow{\sim} \operatorname{\mathrm{Hom}}( f_!^CZ, Y) \end{align} $$

that is functorial in both arguments. (The content of this statement is that to define equation (2.17), it is not necessary for Z to be of the form $f^*X$.)

Situation 2.10. 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.

Situation 2.12. We have a diagram of closed symmetric monoidal categories as in diagram (2.6) 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 diagram (2.6) are in Situation 2.10, and these data are compatible as follows:

• • The basechange map in equation (2.7) is an isomorphism.

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

  (2.19)

  

Proof. The proof of [Reference Chen, Janda and WebbCJW21, Lem A.2.1] works in this more general situation.

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

(2.22)

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

(2.23)

The commutativity of the top triangle is Lemma 2.8 with $X=m^*Y$ and $Y=\mathscr {O}_{\mathscr {T}}$, and the commutativity of the bottom triangle is Lemma 2.9. We note that the definition in equation (2.17) is equal to $\gamma $ followed by the equality induced by equation (2.13). Finally, by Lemma 2.13, the top row of diagram (2.22) followed by the top row of diagram (2.23) agrees with the top row of diagram (2.21) (after inserting a copy of equation (2.7)).

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 [Reference Abramovich, Olsson and VistoliAOV11, Def 2.1(v)], there is a fibre square

(3.2)

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 basechange [Reference Abramovich, Olsson and VistoliAOV08, Cor 3.3], and we have

$$\begin{align*}\mathscr{O}_{C, \bar c} \simeq (\mathscr{O}_{T, q(\bar c)}[x,y]/(xy-t))^{\mu_n} \simeq \mathscr{O}_{T, q(\bar c)}[x^n, y^n]/(x^ny^n-t^n).\end{align*}$$

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 diagram (3.2) as the stack

(3.3)$$ \begin{align} [(\operatorname{\mathrm{Spec}}(\mathscr{O}_{C, \bar c}[x,y]/(x^n-u, y^n-v, xy-t))/\mu_n]. \end{align} $$

Now write $\mathscr {O}_{C, \bar c}$ as the inverse limit of affine schemes $\operatorname {\mathrm {Spec}}(R_i)$ with étale maps to C. Since $\operatorname {\mathrm {Spec}}(\mathscr {O}_{C, \bar c}[x,y]/(x^n-u, y^n-v, xy-t)) \to \operatorname {\mathrm {Spec}}(\mathscr {O}_{C, \bar c})$ is finitely presented, there is an index $i_0$ and elements $u, v, t \in R_{i_0}$ such that $\operatorname {\mathrm {Spec}}(R_{i_0}[x,y]/(x^n-u, y^n-v, xy-t))$ pulls back to the affine scheme in equation (3.3). Define $\mu _n$ to act on $\operatorname {\mathrm {Spec}}(R_{i_0}[x,y]/(x^n-u, y^n-v, xy-t))$ 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 $\operatorname {\mathrm {Spec}}(R_i)$. Observe that for $i \geq i_0$, we have two stacks $[(\operatorname {\mathrm {Spec}}(R_{i}[x,y]/(x^n-u, y^n-v, xy-t))/\mu _n]$ and $\mathcal {C}_{R_i}$ defined over $\operatorname {\mathrm {Spec}}(R_i)$ and an isomorphism between their pullbacks to $\operatorname {\mathrm {Spec}}(\mathscr {O}_{C, \bar c})$. By [Reference Laumon and Moret-BaillyLM00, Prop 4.18(i)], there is an index $ j\geq i_0$ and an isomorphism $[(\operatorname {\mathrm {Spec}}(R_{j}[x,y]/(x^n-u, y^n-v, xy-t))/\mu _n] \simeq \mathcal {C}_{R_j}$. We may set $R:= R_j$.

Proof. For part (1), by flat basechange [Reference Hall and RydhHR17, Lem 1.2(4)], we may assume $\mathcal {M}$ is an affine scheme, but this is [Reference Abramovich, Olsson and VistoliAOV11, Prop 2.6]. Now (1) implies part (2) by definition, part (3) by [Reference Hall and RydhHR17, Cor 4.12] and part (4) by [Reference Hall and RydhHR17, Cor 4.13]. For part (5), we recall that perfection is a flat-local property of complexes in the sense of [Reference Hall and RydhHR17, Lem 4.1], so we may use basechange [Reference Hall and RydhHR17, Cor 4.13] to reduce to the case when $\mathcal {M}$ is a Noetherian affine scheme. Now the result follows from Lemma 3.4 below.

Proof of Lemma 3.4.

We will repeatedly use the fact that if X is a scheme, there are equivalences of categories $\operatorname {\mathrm {QCoh}}(X_{\operatorname {lis-et}}) \simeq \operatorname {\mathrm {QCoh}}(X_{\mathrm {zar}}) $ and $\operatorname {\mathrm {D_{qc}}}(X_{\operatorname {lis-et}}) \simeq \operatorname {\mathrm {D_{qc}}}(X_{\mathrm {zar}})$, 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 $\mathcal {F} \in \operatorname {\mathrm {D_{qc}}}(\mathcal {C}_{\operatorname {lis-et}})$ 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 [Reference OlssonOls07, Rmk 6.10, Prop 6.12] that the sheaves $H^i(\mathcal {F})$ are coherent and vanish for $i\gg 0$. By [Reference AlperAlp13, 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 $\mathcal {F} \in \operatorname {\mathrm {D_{qc}}}( \mathcal {C}_{\operatorname {lis-et}})$ 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 $\mathcal {G} \in \operatorname {\mathrm {QCoh}}(T)$, 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 [Reference Hall and RydhHR17, p. 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

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

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

Proof. Existence of $f^!$ is [Reference Hall and RydhHR17, Thm 4.14(1)], and that equation (2.11) is an isomorphism follows from [Reference Fausk, Hu and MayFHM03, Prop 5.4]. The isomorphism in equation (3.4) is [Reference Fausk, Hu and MayFHM03, Prop 4.3] (see also [Stacks, Tag 0A9Q]).

Proof of Lemma 3.7.

We explain why the proof of [Reference LipmanLip09, 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 [Reference LipmanLip09, Prop 4.6.8], the morphism in equation (3.6) satisfies a cocycle condition for squares stacked horizontally. This implies that it is enough to prove the Lemma when $\mathcal {Y}'$ is an affine scheme and m is smooth or when $\mathcal {Y}'$ and $\mathcal {Y}$ are both affine (see [Reference LipmanLip09, pp. 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 $\mathcal {F} \in \operatorname {\mathrm {D_{qc}}}(\mathcal {Y}_{\operatorname {lis-et}})$, and let $\mathcal {P}$ be a perfect, compact generator for $\operatorname {\mathrm {D_{qc}}}(\mathcal {X}_{\operatorname {lis-et}})$ (see Remark 3.10). Since $\mathcal {Y}'$ and $\mathcal {Y}$ are concentrated by assumption, the morphisms m and $m'$ are also concentrated [Reference Hall and RydhHR17, Lem 2.5], and we have functors $\mathsf {R} m_*$ and $\mathsf {R} m^{\prime }_*$. To show that equation (3.6) is an isomorphism, we claim that it suffices to show the induced map

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

is an isomorphism. First, $\mathsf {L} {m'}^*\mathcal {P}$ is perfect,Footnote ³ so we may replace the functors $\mathsf {R}\mathcal {H}om_{\mathscr {O}_{\mathcal {X}'}}^{\mathrm {qc}}$ with $\mathsf {R}\mathcal {H}om_{\mathscr {O}_{\mathcal {X}'}}$ (see Example 3.1). Next, if equation (3.7) 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 $\operatorname {\mathrm {D_{qc}}}(\mathcal {X}{\prime }_{\operatorname {lis-et}})$ by [Reference Hall and RydhHR17, Cor 2.8]—this is where we use that m (hence $m'$) is quasi-affine. We conclude that equation (3.6) is an isomorphism (see Remark 3.10).

To show that equation (3.7) is an isomorphism, we cite the bottom two cells of the commuting diagram on [Reference LipmanLip09, p. 182] to reduce to proving a certain morphism

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

is an isomorphism.Footnote ⁴ (In the cited diagram, the map notated $u^*\delta $ is, in our notation, equal to $\mathsf {L} m^*$ applied to the isomorphism in equation (3.4).) We will not bother to write the definition of $\mathsf {R} m_*(4.4.1)^*_{\mathrm {pc}}$ because [Reference LipmanLip09, Lem 4.6.4] gives a commuting diagram

(3.8)

The arrows labelled (2.7) are isomorphisms by [Reference Hall and RydhHR17, Cor 4.13]. The arrows labelled $\rho $ are defined in [Reference Fausk, Hu and MayFHM03, (3.3)]; and by [Reference Fausk, Hu and MayFHM03, 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 [Reference LipmanLip09, (3.5.4.5)] by [Reference LipmanLip09, Exercise 3.5.6(a)]. Finally, in diagram (3.8), we know that equation (3.4) is an isomorphism; this concludes the proof.

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 dualisable, so we have

$$\begin{align*}p^!\mathscr{O}_T = r^!q^!\mathscr{O}_T = r^*q^!\mathscr{O}_T \otimes r^!\mathscr{O}_C,\end{align*}$$

where the second equality uses [Reference Fausk, Hu and MayFHM03, Thm 8.4] and the fact that $q^!\mathscr {O}_T$ is dualisable. 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 3.2, we have a local description of $\mathcal {C} \to C$ near $\bar c$ given by the diagram (3.1). Note that a right adjoint to pushforward exists for every horizontal map in diagram (3.1). It follows from [Reference NeemanNee17, Lem 0.1] that the pullback of $r^!\mathscr {O}_C$ to $[V/\mu _n]$ is equal to $\tau ^!\mathscr {O}_U$ (note that [Reference NeemanNee17, Lem 0.1] applies since $U \to C$ is étale and $r_*$ preserves pseudo-coherent objects by Lemma 3.4). 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.4), $r^!\mathscr {O}_C$ is invertible if and only if $\tau ^!\mathscr {O}_U$ is.

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

$$\begin{align*}\rho^!\mathscr{O}_U = \sigma^*\tau^!\mathscr{O}_U \otimes \sigma^!\mathscr{O}_{[V/\mu_n]}, \end{align*}$$

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 3.13 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 argument identical to the one used in the previous paragraph.

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

(3.9)$$ \begin{align} \mathsf{R}\mathcal{H}om_A(B, A). \end{align} $$

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 equation (3.9), 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 3.2 lists two possibilities for A. In case (1), the computation of equation (3.9) 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

$$\begin{align*}\ldots \xrightarrow{d_3} R^{\oplus 2r-2} \xrightarrow{d_2} R^{\oplus 2r-2} \xrightarrow{d_1} R^{\oplus 2r-1} \xrightarrow{d_0} A \rightarrow 0, \end{align*}$$

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

$$\begin{align*}\begin{array}{llllll} d_0:&e \mapsto 1 & \quad d_i,\;i\;\text{odd}:& f_i \mapsto vf_i-t^ig_{r-i} & \quad d_i,\;i>0\;\text{even:}&f_i \mapsto uf_i+t^ig_{r-i}\\ &f_i \mapsto x^i&&g_{r-i} \mapsto ug_{r-i}-t^{r-i}f_i&&g_{r-i}\mapsto t^{r-i}f_i+vg_{r-i}\\ &g_i \mapsto y^i &&&& \end{array} \end{align*}$$

For details, see [Reference WebbWeb20, pp. 25–28].

Proof of Proposition 3.14.

The idea is 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 3.11). When both $\mathcal {N}$ and $\mathcal {M}$ are quasi-separated Noetherian algebraic spaces, we define equation (3.11) to be the basechange map in equation (3.6) (it is an isomorphism by Lemma 3.7). The commuting diagram (3.12) follows from the definition of equation (3.6); see [Reference LipmanLip09, Rmk 4.4(d)]. The cocycle condition on equation (3.11) is [Reference LipmanLip09, 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.12), and let $\mathcal {C}_{M,\bullet }$ be its pullback to $\mathcal {C}_{\mathcal {M}}$ (see Remark A.13). We have associated categories of quasi-coherent sheaves $\operatorname {\mathrm {QCoh}}(M_{\bullet , \operatorname {lis-et}})$ and $\operatorname {\mathrm {QCoh}}(\mathcal {C}_{M, \bullet , \operatorname {lis-et}})$ as in Section A.3.2. By Remark A.14, 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 in equation (3.11) defines an object $\omega _{M, \bullet }$ of $\operatorname {\mathrm {QCoh}}(\mathcal {C}_{M, \bullet , \operatorname {lis-et}})$. By Proposition A.18, the sheaf $\omega _{M, \bullet }$ corresponds to a unique quasi-coherent sheaf $\omega _{\mathcal {M}}$ in $\operatorname {\mathrm {QCoh}}(\mathcal {C}_{\mathcal {M}, \operatorname {lis-et}})$ 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.20, the complex $\mathsf {R} p_*\omega ^{\bullet }_{\mathcal {M}}$ is represented by the element of $\operatorname {\mathrm {D_{qc}}}(M_{\bullet , \operatorname {lis-et}})$ 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 diagram (3.12). Now, from Proposition A.18 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 3.3).

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 fibre square (3.10) 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.15). 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 $m^{\prime *}_n\omega _{M_n} \xrightarrow {\text {(3.11)}}\omega _{N_n}$. Under the identifications $(a^*m^{\prime *}\omega _{\mathcal {M}})|_{M_n} \simeq m^{\prime *}_n\omega _{M_n}$ of Remark A.20, these isomorphisms are compatible with the transition maps for the sheaves $a^*{m'}^*\omega _{\mathcal {M}}$ and $a^*\omega _{\mathcal {N}}$ in $\operatorname {\mathrm {QCoh}}(\mathcal {C}_{N, \bullet , \operatorname {lis-et}})$ because equation (3.11) 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 diagram (3.12) commute, apply the equivalences $a^*$ and use Remark A.20 to get a collection of commuting diagrams indexed by $n \in \mathbb {Z}_{\geq 0}$.

Proof. Applying Lemma 4.5 to the maps $ \mathcal {X} \rightarrow \mathcal {Y} \xrightarrow {r} \mathcal {Z}$, we get a commuting diagram

(4.9)

where ${\underline {R}}$ is the same as the map B in the lemma. When we restrict diagram (4.9) to isomorphism classes of objects, we get the commuting square in the following diagram:

(4.10)

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

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

The set $\mathrm {Def}(f)$ is nonempty if and only if, in diagram (4.10), the fibre of R over the element $[g]\in \mathrm {Exal}_{\mathcal {Z}}(\mathcal {X}, I)$ defined by diagram (4.2) is nonempty. From the long exact sequence for $\operatorname {\mathrm {Ext}}^i(-, I)$ applied to equation (4.11), we see that this happens if and only if the image of $[g]$ in $\operatorname {\mathrm {Ext}}^1(\mathsf {L} f^*\mathbb {L}_{\mathcal {Y}/\mathcal {Z}}, I)$ (under the maps given in diagram (4.10)) is 0. We define

(4.12)$$ \begin{align} o(f) = ob(\alpha^{-1}([g])). \end{align} $$

If $\mathrm {Def}(f)$ is not empty, then by Lemma 4.9 below, $\underline {\mathrm {Def}}(f)$ is isomorphic to the kernel of the morphism of Picard categories

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

It follows from diagram (4.9) and [Reference OlssonOls06, Lem 2.29] applied to the distinguished triangle

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

induced from equation (4.11) 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

$$\begin{align*}(\tau_{\leq -1}Cone(\tau_{\leq 0}\beta)))[-1] = (\tau_{\leq -1}Cone(\beta))[-1]=\tau_{\leq 0}Cone(\beta[-1]),\end{align*}$$

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

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

(4.13)$$ \begin{align} \mathcal{K} \xrightarrow{(1_{\mathcal{K}}, \sigma)} \mathcal{K} \times_{\mathcal{X}} \mathcal{F} \rightarrow \mathcal{F}. \end{align} $$

On the other hand, we have the composition

(4.14)$$ \begin{align} \mathcal{F} \xrightarrow{(pr_1, -\sigma)} \mathcal{P} \times_{\mathcal{X}} \mathcal{P} \xrightarrow{\mu} \mathcal{P} \xrightarrow{f} \mathcal{Q}, \end{align} $$

where $pr_1: \mathcal {F} \rightarrow \mathcal {P}$ is the canonical morphism and $-\sigma $ is $\sigma $ followed by the inverse morphism. The composition of equation (4.14) 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 equation (4.13).

Proof of Lemma 4.11.

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

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

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

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

implies that $\Phi _1$ is injective and $\Phi _0$ and $\Phi _{-1}$ are isomorphisms. Combined with Corollary 4.7, this proves (2) (with $\mathcal {X}$ an arbitrary scheme). Now (2) implies (3) using Example 4.3.

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 $I \in \operatorname {\mathrm {QCoh}}(\mathcal {X}_{\operatorname {lis-et}})$ be arbitrary. We will show that if $i \geq -1$, then $\operatorname {\mathrm {Ext}}^0(H^i( \mathsf {L} f^*C), I)=0$, which implies $f^* H^i(C) = H^i(\mathsf {L} f^*C) =0$ (the first equality is [Reference Hall and RydhHR17, (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 4.7 and assumption (3a) that if $\Phi _1(o(f))=0$, then $o(f)=0$, so it suffices to show that every element of $\operatorname {\mathrm {Ext}}^1(\mathsf {L} f^*\mathbb {L}_{\mathcal {Y}/\mathcal {Z}}, I)$ is equal to $o(f)$ for some diagram (4.2), or equivalently that the map $ob$ in equation (4.12) is surjective. This follows from the long exact sequence

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

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 equation (4.16) shows that $\operatorname {\mathrm {Ext}}^i(\mathsf {L} f^*C, I)=0$ for every $i\leq 1$. By [Stacks, Tag 07AA], there is a spectral sequence whose second page is

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

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