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 \input{preamble} % OK, start here. % \begin{document} \title{Derived Categories of Stacks} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this chapter we write about derived categories associated to algebraic stacks. This means in particular derived categories of quasi-coherent sheaves, i.e., we prove analogues of the results on schemes (see Derived Categories of Schemes, Section \ref{perfect-section-introduction}) and algebraic spaces (see Derived Categories of Spaces, Section \ref{spaces-perfect-section-introduction}). The results in this chapter are different from those in \cite{LM-B} mainly because we consistently use the big sites''. Before reading this chapter please take a quick look at the chapters Sheaves on Algebraic Stacks'' and Cohomology of Algebraic Stacks'' where the terminology we use here is introduced. \section{Conventions, notation, and abuse of language} \label{section-conventions} \noindent We continue to use the conventions and the abuse of language introduced in Properties of Stacks, Section \ref{stacks-properties-section-conventions}. We use notation as explained in Cohomology of Stacks, Section \ref{stacks-cohomology-section-notation}. \section{The lisse-\'etale and the flat-fppf sites} \label{section-lisse-etale} \noindent The section is the analogue of Cohomology of Stacks, Section \ref{stacks-cohomology-section-lisse-etale} for derived categories. \begin{lemma} \label{lemma-shriek-derived} Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemmas \ref{stacks-cohomology-lemma-lisse-etale} and \ref{stacks-cohomology-lemma-lisse-etale-modules}. \begin{enumerate} \item The functor $g_! : \textit{Ab}(\mathcal{X}_{lisse,\etale}) \to \textit{Ab}(\mathcal{X}_\etale)$ has a left derived functor $$Lg_! : D(\mathcal{X}_{lisse,\etale}) \longrightarrow D(\mathcal{X}_\etale)$$ which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$. \item The functor $g_! : \textit{Mod}(\mathcal{X}_{lisse,\etale}, \mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_{\mathcal{X}})$ has a left derived functor $$Lg_! : D(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \longrightarrow D(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$$ which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$. \item The functor $g_! : \textit{Ab}(\mathcal{X}_{flat,fppf}) \to \textit{Ab}(\mathcal{X}_{fppf})$ has a left derived functor $$Lg_! : D(\mathcal{X}_{flat, fppf}) \longrightarrow D(\mathcal{X}_{fppf})$$ which is left adjoint to $g^{-1}$ and such that $g^{-1}Lg_! = \text{id}$. \item The functor $g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$ has a left derived functor $$Lg_! : D(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \longrightarrow D(\mathcal{O}_\mathcal{X})$$ which is left adjoint to $g^*$ and such that $g^*Lg_! = \text{id}$. \end{enumerate} Warning: It is not clear (a priori) that $Lg_!$ on modules agrees with $Lg_!$ on abelian sheaves, see Cohomology on Sites, Remark \ref{sites-cohomology-remark-when-derived-shriek-equal}. \end{lemma} \begin{proof} The existence of the functor $Lg_!$ and adjointness to $g^*$ is Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-existence-derived-lower-shriek}. (For the case of abelian sheaves use the constant sheaf $\mathbf{Z}$ as the structure sheaves.) Moreover, it is computed on a complex $\mathcal{H}^\bullet$ by taking a suitable left resolution $\mathcal{K}^\bullet \to \mathcal{H}^\bullet$ and applying the functor $g_!$ to $\mathcal{K}^\bullet$. Since $g^{-1}g_!\mathcal{K}^\bullet = \mathcal{K}^\bullet$ by Cohomology of Stacks, Lemmas \ref{stacks-cohomology-lemma-lisse-etale-modules} and \ref{stacks-cohomology-lemma-lisse-etale} we see that the final assertion holds in each case. \end{proof} \begin{lemma} \label{lemma-lisse-etale-functorial-derived} With assumptions and notation as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale-functorial}. We have $$g^{-1} \circ Rf_* = Rf'_* \circ (g')^{-1} \quad\text{and}\quad L(g')_! \circ (f')^{-1} = f^{-1} \circ Lg_!$$ on unbounded derived categories (both for the case of modules and for the case of abelian sheaves). \end{lemma} \begin{proof} Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{X}_\etale$ (resp.\ $\mathcal{X}_{fppf}$). We first show that the canonical (base change) map $$g^{-1} Rf_*\mathcal{F} \longrightarrow Rf'_* (g')^{-1}\mathcal{F}$$ is an isomorphism. To do this let $y$ be an object of $\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{flat,fppf}$). Say $y$ lies over the scheme $V$ such that $y : V \to \mathcal{Y}$ is smooth (resp.\ flat). Since $g^{-1}$ is the restriction we find that $$\left(g^{-1}R^pf_*\mathcal{F}\right)(y) = H^p_\tau(V \times_{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})$$ where $\tau = \etale$ (resp.\ $\tau = fppf$), see Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pushforward-restriction}. By Cohomology of Stacks, Equation (\ref{stacks-cohomology-equation-pushforward-lisse-etale}) for any sheaf $\mathcal{H}$ on $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$) $$f'_*\mathcal{H}(y) = \Gamma((V \times_{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H})$$ An object of $(V \times_{y, \mathcal{Y}} \mathcal{X})'$ can be seen as a pair $(x, \varphi)$ where $x$ is an object of $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$) and $\varphi : f(x) \to y$ is a morphism in $\mathcal{Y}$. We can also think of $\varphi$ as a section of $(f')^{-1}h_y$ over $x$. Thus $(V \times_\mathcal{Y} \mathcal{X})'$ is the localization of the site $\mathcal{X}_{lisse,\etale}$ (resp. $\mathcal{X}_{flat,fppf}$) at the sheaf of sets $(f')^{-1}h_y$, see Sites, Lemma \ref{sites-lemma-localize-topos-site}. The morphism $$\text{pr}' : (V \times_{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{lisse,\etale} \ (\text{resp. } \text{pr}' : (V \times_{y, \mathcal{Y}} \mathcal{X})' \to \mathcal{X}_{flat,fppf})$$ is the localization morphism. In particular, the pullback $(\text{pr}')^{-1}$ preserves injective abelian sheaves, see Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-cohomology-on-sheaf-sets}. At this point exactly the same argument as in Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-pushforward-restriction} shows that \begin{equation} \label{equation-higher-direct-image-lisse-etale} R^pf'_*\mathcal{H}(y) = H^p_\tau((V \times_{y, \mathcal{Y}} \mathcal{X})', \ (\text{pr}')^{-1}\mathcal{H}) \end{equation} where $\tau = \etale$ (resp.\ $\tau = fppf$). Since $(g')^{-1}$ is given by restriction we conclude that $$\left(R^pf'_*(g')^*\mathcal{F}\right)(y) = H^p_\tau((V \times_{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times_{y, \mathcal{Y}} \mathcal{X})'})$$ Finally, we can apply Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-cohomology-on-subcategory} to see that $$H^p_\tau((V \times_{y, \mathcal{Y}} \mathcal{X})', \ \text{pr}^{-1}\mathcal{F}|_{(V \times_{y, \mathcal{Y}} \mathcal{X})'}) = H^p_\tau(V \times_{y, \mathcal{Y}} \mathcal{X},\ \text{pr}^{-1}\mathcal{F})$$ are equal as desired; although we omit the verification of the assumptions of the lemma we note that the fact that $V \to \mathcal{Y}$ is smooth (resp.\ flat) is used to verify the second condition. \medskip\noindent The rest of the proof is formal. Since cohomology of abelian groups and sheaves of modules agree we also conclude that $g^{-1} Rf_*\mathcal{F} = Rf'_* (g')^{-1}\mathcal{F}$ when $\mathcal{F}$ is a sheaf of modules on $\mathcal{X}_\etale$ (resp.\ $\mathcal{X}_{fppf}$). \medskip\noindent Next we show that for $\mathcal{G}$ (either sheaf of modules or abelian groups) on $\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{flat,fppf}$) the canonical map $$L(g')_!(f')^{-1}\mathcal{G} \to f^{-1}Lg_!\mathcal{G}$$ is an isomorphism. To see this it is enough to prove for any injective sheaf $\mathcal{I}$ on $\mathcal{X}_\etale$ (resp.\ $\mathcal{X}_{fppf}$) that the induced map $$\Hom(L(g')_!(f')^{-1}\mathcal{G}, \mathcal{I}[n]) \leftarrow \Hom(f^{-1}Lg_!\mathcal{G}, \mathcal{I}[n])$$ is an isomorphism for all $n \in \mathbf{Z}$. (Hom's taken in suitable derived categories.) By the adjointness of $f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and their primed'' versions this follows from the isomorphism $g^{-1} Rf_*\mathcal{I} \to Rf'_* (g')^{-1}\mathcal{I}$ proved above. \medskip\noindent In the case of a bounded complex $\mathcal{G}^\bullet$ (of modules or abelian groups) on $\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{fppf}$) the canonical map \begin{equation} \label{equation-to-show} L(g')_!(f')^{-1}\mathcal{G}^\bullet \to f^{-1}Lg_!\mathcal{G}^\bullet \end{equation} is an isomorphism as follows from the case of a sheaf by the usual arguments involving truncations and the fact that the functors $L(g')_!(f')^{-1}$ and $f^{-1}Lg_!$ are exact functors of triangulated categories. \medskip\noindent Suppose that $\mathcal{G}^\bullet$ is a bounded above complex (of modules or abelian groups) on $\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{fppf}$). The canonical map (\ref{equation-to-show}) is an isomorphism because we can use the stupid truncations $\sigma_{\geq -n}$ (see Homology, Section \ref{homology-section-truncations}) to write $\mathcal{G}^\bullet$ as a colimit $\mathcal{G}^\bullet = \colim \mathcal{G}_n^\bullet$ of bounded complexes. This gives a distinguished triangle $$\bigoplus\nolimits_{n \geq 1} \mathcal{G}_n^\bullet \to \bigoplus\nolimits_{n \geq 1} \mathcal{G}_n^\bullet \to \mathcal{G}^\bullet \to \ldots$$ and each of the functors $L(g')_!$, $(f')^{-1}$, $f^{-1}$, $Lg_!$ commutes with direct sums (of complexes). \medskip\noindent If $\mathcal{G}^\bullet$ is an arbitrary complex (of modules or abelian groups) on $\mathcal{Y}_{lisse,\etale}$ (resp.\ $\mathcal{Y}_{fppf}$) then we use the canonical truncations $\tau_{\leq n}$ (see Homology, Section \ref{homology-section-truncations}) to write $\mathcal{G}^\bullet$ as a colimit of bounded above complexes and we repeat the argument of the paragraph above. \medskip\noindent Finally, by the adjointness of $f^{-1}$ and $Rf_*$, the adjointness of $Lg_!$ and $g^{-1}$, and their primed'' versions we conclude that the first identity of the lemma follows from the second in full generality. \end{proof} \begin{lemma} \label{lemma-higher-shriek-quasi-coherent} Let $\mathcal{X}$ be an algebraic stack. Notation as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale}. \begin{enumerate} \item Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{lisse,\etale}}$-module on the lisse-\'etale site of $\mathcal{X}$. For all $p \in \mathbf{Z}$ the sheaf $H^p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$. \item Let $\mathcal{H}$ be a quasi-coherent $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-module on the flat-fppf site of $\mathcal{X}$. For all $p \in \mathbf{Z}$ the sheaf $H^p(Lg_!\mathcal{H})$ is a locally quasi-coherent module with the flat base change property on $\mathcal{X}$. \end{enumerate} \end{lemma} \begin{proof} Pick a scheme $U$ and a surjective smooth morphism $x : U \to \mathcal{X}$. By Modules on Sites, Definition \ref{sites-modules-definition-site-local} there exists an \'etale (resp.\ fppf) covering $\{U_i \to U\}_{i \in I}$ such that each pullback $f_i^{-1}\mathcal{H}$ has a global presentation (see Modules on Sites, Definition \ref{sites-modules-definition-global}). Here $f_i : U_i \to \mathcal{X}$ is the composition $U_i \to U \to \mathcal{X}$ which is a morphism of algebraic stacks. (Recall that the pullback is'' the restriction to $\mathcal{X}/f_i$, see Sheaves on Stacks, Definition \ref{stacks-sheaves-definition-pullback} and the discussion following.) After refining the covering we may assume each $U_i$ is an affine scheme. Since each $f_i$ is smooth (resp.\ flat) by Lemma \ref{lemma-lisse-etale-functorial-derived} we see that $f_i^{-1}Lg_!\mathcal{H} = Lg_{i, !}(f'_i)^{-1}\mathcal{H}$. Using Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-check-lqc-fbc-on-covering} we reduce the statement of the lemma to the case where $\mathcal{H}$ has a global presentation and where $\mathcal{X} = (\Sch/X)_{fppf}$ for some affine scheme $X = \Spec(A)$. \medskip\noindent Say our presentation looks like $$\bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{H} \longrightarrow 0$$ where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}$ (resp.\ $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Note that the site $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$) has a final object, namely $X/X$ which is quasi-compact (see Cohomology on Sites, Section \ref{sites-cohomology-section-limits}). Hence we have $$\Gamma(\bigoplus\nolimits_{i \in I} \mathcal{O}) = \bigoplus\nolimits_{i \in I} A$$ by Sites, Lemma \ref{sites-lemma-directed-colimits-sections}. Hence the map in the presentation corresponds to a similar presentation $$\bigoplus\nolimits_{j \in J} A \longrightarrow \bigoplus\nolimits_{i \in I} A \longrightarrow M \longrightarrow 0$$ of an $A$-module $M$. Moreover, $\mathcal{H}$ is equal to the restriction to the lisse-\'etale (resp.\ flat-fppf) site of the quasi-coherent sheaf $M^a$ associated to $M$. Choose a resolution $$\ldots \to F_2 \to F_1 \to F_0 \to M \to 0$$ by free $A$-modules. The complex $$\ldots \mathcal{O} \otimes_A F_2 \to \mathcal{O} \otimes_A F_1 \to \mathcal{O} \otimes_A F_0 \to \mathcal{H} \to 0$$ is a resolution of $\mathcal{H}$ by free $\mathcal{O}$-modules because for each object $U/X$ of $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$) the structure morphism $U \to X$ is flat. Hence by construction the value of $Lg_!\mathcal{H}$ is $$\ldots \to \mathcal{O}_\mathcal{X} \otimes_A F_2 \to \mathcal{O}_\mathcal{X} \otimes_A F_1 \to \mathcal{O}_\mathcal{X} \otimes_A F_0 \to 0 \to \ldots$$ Since this is a complex of quasi-coherent modules on $\mathcal{X}_\etale$ (resp.\ $\mathcal{X}_{fppf}$) it follows from Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change} that $H^p(Lg_!\mathcal{H})$ is quasi-coherent. \end{proof} \section{Derived categories of quasi-coherent modules} \label{section-derived} \noindent Let $\mathcal{X}$ be an algebraic stack. As the inclusion functor $\QCoh(\mathcal{O}_\mathcal{X}) \to \textit{Mod}(\mathcal{O}_\mathcal{X})$ isn't exact, we cannot define $D_\QCoh(\mathcal{O}_\mathcal{X})$ as the full subcategory of $D(\mathcal{O}_\mathcal{X})$ consisting of complexes with quasi-coherent cohomology sheaves. In stead we define the category as follows. \begin{definition} \label{definition-derived} Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$ denote the category of locally quasi-coherent $\mathcal{O}_\mathcal{X}$-modules with the flat base change property. Let $\mathcal{P}_\mathcal{X} \subset \mathcal{M}_\mathcal{X}$ be the full subcategory consisting of parasitic objects. We define the {\it derived category of $\mathcal{O}_\mathcal{X}$-modules with quasi-coherent cohomology sheaves} as the Verdier quotient\footnote{This definition is different from the one in the literature, see \cite[6.3]{olsson_sheaves}, but it agrees with that definition by Lemma \ref{lemma-derived-quasi-coherent}.} $$D_\QCoh(\mathcal{O}_\mathcal{X}) = D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})/ D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$$ \end{definition} \noindent This definition makes sense: By Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change} we see that $\mathcal{M}_\mathcal{X}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ hence $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ is a strictly full, saturated triangulated subcategory of $D(\mathcal{O}_\mathcal{X})$, see Derived Categories, Lemma \ref{derived-lemma-cohomology-in-serre-subcategory}. Since parasitic modules form a Serre subcategory of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ (by Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-parasitic}) we see that $\mathcal{P}_\mathcal{X} = \text{Parasitic} \cap \mathcal{M}_\mathcal{X}$ is a weak Serre subcategory of $\textit{Mod}(\mathcal{O}_\mathcal{X})$ and hence $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ is a strictly full, saturated triangulated subcategory of $D(\mathcal{O}_\mathcal{X})$. Since clearly $$D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \subset D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$$ we conclude that the first is a strictly full, saturated triangulated subcategory of the second. Hence the Verdier quotient exists. A morphism $a : E \to E'$ of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ becomes an isomorphism in $D_\QCoh(\mathcal{O}_\mathcal{X})$ if and only if the cone $C(a)$ has parasitic cohomology sheaves, see Derived Categories, Section \ref{derived-section-quotients} and especially Lemma \ref{derived-lemma-operations}. \medskip\noindent Consider the functors $$D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \xrightarrow{H^i} \mathcal{M}_\mathcal{X} \xrightarrow{Q} \QCoh(\mathcal{O}_\mathcal{X})$$ Note that $Q$ annihilates the subcategory $\mathcal{P}_\mathcal{X}$, see Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-adjoint-kernel-parasitic}. By Derived Categories, Lemma \ref{derived-lemma-universal-property-quotient} we obtain a cohomological functor \begin{equation} \label{equation-Hi-quasi-coherent} H^i : D_\QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow \QCoh(\mathcal{O}_\mathcal{X}) \end{equation} Moreover, note that $E \in D_\QCoh(\mathcal{O}_\mathcal{X})$ is zero if and only if $H^i(E) = 0$ for all $i \in \mathbf{Z}$. \medskip\noindent Note that the categories $\mathcal{P}_\mathcal{X}$ and $\mathcal{M}_\mathcal{X}$ are also weak Serre subcategories of the abelian category $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ of modules in the \'etale topology, see Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change} and Lemma \ref{stacks-cohomology-lemma-parasitic}. Hence the statement of the following lemma makes sense. \begin{lemma} \label{lemma-compare-etale-fppf-QCoh} Let $\mathcal{X}$ be an algebraic stack. The comparison morphism $\epsilon : \mathcal{X}_{fppf} \to \mathcal{X}_\etale$ induces a commutative diagram $$\xymatrix{ D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] & D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] & D(\mathcal{O}_\mathcal{X}) \\ D_{\mathcal{P}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \ar[r] \ar[u]^{\epsilon^*} & D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \ar[r] \ar[u]^{\epsilon^*} & D(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \ar[u]^{\epsilon^*} }$$ Moreover, the left two vertical arrows are equivalences of triangulated categories, hence we also obtain an equivalence $$D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) / D_{\mathcal{P}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \longrightarrow D_\QCoh(\mathcal{O}_\mathcal{X})$$ \end{lemma} \begin{proof} Since $\epsilon^*$ is exact it is clear that we obtain a diagram as in the statement of the lemma. We will show the middle vertical arrow is an equivalence by applying Simplicial Spaces, Lemma \ref{spaces-simplicial-lemma-compare-topologies-derived-adequate-modules} to the following situation: $\mathcal{C} = \mathcal{X}$, $\tau = fppf$, $\tau' = \etale$, $\mathcal{O} = \mathcal{O}_\mathcal{X}$, $\mathcal{A} = \mathcal{M}_\mathcal{X}$, and $\mathcal{B}$ is the set of objects of $\mathcal{X}$ lying over affine schemes. To see the lemma applies we have to check conditions (1), (2), (3), (4). Conditions (1) and (2) are clear from the discussion above (explicitly this follows from Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change}). Condition (3) holds because every scheme has a Zariski open covering by affines. Condition (4) follows from Descent, Lemma \ref{descent-lemma-quasi-coherent-and-flat-base-change}. \medskip\noindent We omit the verification that the equivalence of categories $\epsilon^* : D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) \to D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ induces an equivalence of the subcategories of complexes with parasitic cohomology sheaves. \end{proof} \noindent It turns out that $D_\QCoh(\mathcal{O}_\mathcal{X})$ is the same as the derived category of complexes of modules with quasi-coherent cohomology sheaves on the lisse-\'etale or flat-fppf site. \begin{lemma} \label{lemma-derived-quasi-coherent} Let $\mathcal{X}$ be an algebraic stack. \begin{enumerate} \item Let $\mathcal{F}^\bullet$ be an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$. With $g$ as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale} for the lisse-\'etale site we have \begin{enumerate} \item $g^{-1}\mathcal{F}^\bullet$ is in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$, \item $g^{-1}\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet$ is in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$, \item $Lg_!\mathcal{H}^\bullet$ is in $D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ for $\mathcal{H}^\bullet$ in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$, and \item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors $$\xymatrix{ D_\QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \ar@<1ex>[l]^-{Lg_!} }$$ \end{enumerate} \item Let $\mathcal{F}^\bullet$ be an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. With $g$ as in Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-lisse-etale} for the flat-fppf site we have \begin{enumerate} \item $g^{-1}\mathcal{F}^\bullet$ is in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$, \item $g^{-1}\mathcal{F}^\bullet = 0$ if and only if $\mathcal{F}^\bullet$ is in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$, \item $Lg_!\mathcal{H}^\bullet$ is in $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ for $\mathcal{H}^\bullet$ in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$, and \item the functors $g^{-1}$ and $Lg_!$ define mutually inverse functors $$\xymatrix{ D_\QCoh(\mathcal{O}_\mathcal{X}) \ar@<1ex>[r]^-{g^{-1}} & D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}}) \ar@<1ex>[l]^-{Lg_!} }$$ \end{enumerate} \end{enumerate} \end{lemma} \begin{proof} The functor $g^{-1}$ is exact, hence (1)(a), (2)(a), (1)(b), and (2)(b) follow from Cohomology of Stacks, Lemmas \ref{stacks-cohomology-lemma-quasi-coherent} and \ref{stacks-cohomology-lemma-parasitic-in-terms-flat-fppf}. \medskip\noindent Proof of (1)(c) and (2)(c). The construction of $Lg_!$ in Lemma \ref{lemma-shriek-derived} (via Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-existence-derived-lower-shriek} which in turn uses Derived Categories, Proposition \ref{derived-proposition-left-derived-exists}) shows that $Lg_!$ on any object $\mathcal{H}^\bullet$ of $D(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ is computed as $$Lg_!\mathcal{H}^\bullet = \colim g_!\mathcal{K}_n^\bullet = g_! \colim \mathcal{K}_n^\bullet$$ (termwise colimits) where the quasi-isomorphism $\colim \mathcal{K}_n^\bullet \to \mathcal{H}^\bullet$ induces quasi-isomorphisms $\mathcal{K}_n^\bullet \to \tau_{\leq n} \mathcal{H}^\bullet$. Since $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ (resp.\ $\mathcal{M}_\mathcal{X} \subset \textit{Mod}(\mathcal{O}_\mathcal{X})$) is preserved under colimits we see that it suffices to prove (c) on bounded above complexes $\mathcal{H}^\bullet$ in $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$ (resp.\ $D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat,fppf}})$). In this case to show that $H^n(Lg_!\mathcal{H}^\bullet)$ is in $\mathcal{M}_\mathcal{X}$ we can argue by induction on the integer $m$ such that $\mathcal{H}^i = 0$ for $i > m$. If $m < n$, then $H^n(Lg_!\mathcal{H}^\bullet) = 0$ and the result holds. In general consider the distinguished triangle $$\tau_{\leq m - 1}\mathcal{H}^\bullet \to \mathcal{H}^\bullet \to H^m(\mathcal{H}^\bullet)[-m] \to \ldots$$ (Derived Categories, Remark \ref{derived-remark-truncation-distinguished-triangle}) and apply the functor $Lg_!$. Since $\mathcal{M}_\mathcal{X}$ is a weak Serre subcategory of the module category it suffices to prove (c) for two out of three. We have the result for $Lg_!\tau_{\leq m - 1}\mathcal{H}^\bullet$ by induction and we have the result for $Lg_!H^m(\mathcal{H}^\bullet)[-m]$ by Lemma \ref{lemma-higher-shriek-quasi-coherent}. Whence (c) holds. \medskip\noindent Let us prove (2)(d). By (2)(a) and (2)(b) the functor $g^{-1} = g^*$ induces a functor $$c : D_\QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$$ see Derived Categories, Lemma \ref{derived-lemma-universal-property-quotient}. Thus we have the following diagram of triangulated categories $$\xymatrix{ D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[rd]^{g^{-1}} \ar[rr]_q & & D_\QCoh(\mathcal{O}_\mathcal{X}) \ar[ld]^c \\ & D_\QCoh(\mathcal{O}_{\mathcal{X}_{flat, fppf}}) \ar@<1ex>[lu]^{Lg_!} }$$ where $q$ is the quotient functor, the inner triangle is commutative, and $g^{-1}Lg_! = \text{id}$. For any object of $E$ of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ the map $a : Lg_!g^{-1}E \to E$ maps to a quasi-isomorphism in $D(\mathcal{O}_{\mathcal{X}_{flat, fppf}})$. Hence the cone on $a$ maps to zero under $g^{-1}$ and by (2)(b) we see that $q(a)$ is an isomorphism. Thus $q \circ Lg_!$ is a quasi-inverse to $c$. \medskip\noindent In the case of the lisse-\'etale site exactly the same argument as above proves that $$D_{\mathcal{M}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X}) / D_{\mathcal{P}_\mathcal{X}}( \mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$$ is equivalent to $D_\QCoh(\mathcal{O}_{\mathcal{X}_{lisse,\etale}})$. Applying the last equivalence of Lemma \ref{lemma-compare-etale-fppf-QCoh} finishes the proof. \end{proof} \noindent The following lemma tells us that the quotient functor $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \to D_\QCoh(\mathcal{O}_\mathcal{X})$ is a Bousfield colocalization (insert future reference here). \begin{lemma} \label{lemma-bousfield-colocalization} Let $\mathcal{X}$ be an algebraic stack. Let $E$ be an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. There exists a canonical distinguished triangle $$E' \to E \to P \to E'[1]$$ in $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ such that $P$ is in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ and $$\Hom_{D(\mathcal{O}_\mathcal{X})}(E', P') = 0$$ for all $P'$ in $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. \end{lemma} \begin{proof} Consider the morphism of ringed topoi $g : \Sh(\mathcal{X}_{flat, fppf}) \longrightarrow \Sh(\mathcal{X}_{fppf})$. Set $E' = Lg_!g^{-1}E$ and let $P$ be the cone on the adjunction map $E' \to E$. Since $g^{-1}E' \to g^{-1}E$ is an isomorphism we see that $P$ is an object of $D_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ by Lemma \ref{lemma-derived-quasi-coherent} (2)(b). Finally, $\Hom(E', P') = \Hom(Lg_!g^{-1}E, P') = \Hom(g^{-1}E, g^{-1}P') = 0$ as $g^{-1}P' = 0$. \medskip\noindent Uniqueness. Suppose that $E'' \to E \to P'$ is a second distinguished triangle as in the statement of the lemma. Since $\Hom(E', P') = 0$ the morphism $E' \to E$ factors as $E' \to E'' \to E$, see Derived Categories, Lemma \ref{derived-lemma-representable-homological}. Similarly, the morphism $E'' \to E$ factors as $E'' \to E' \to E$. Consider the composition $\varphi : E' \to E'$ of the maps $E' \to E''$ and $E'' \to E'$. Note that $\varphi - 1 : E' \to E'$ fits into the commutative diagram $$\xymatrix{ E' \ar[d]^{\varphi - 1} \ar[r] & E \ar[d]^0 \\ E' \ar[r] & E }$$ hence factors through $P[-1] \to E$. Since $\Hom(E', P[-1]) = 0$ we see that $\varphi = 1$. Whence the maps $E' \to E''$ and $E'' \to E'$ are inverse to each other. \end{proof} \section{Derived pushforward of quasi-coherent modules} \label{section-derived-pushforward} \noindent As a first application of the material above we construct the derived pushforward. In Examples, Section \ref{examples-section-derived-push-quasi-coherent} the reader can find an example of a quasi-compact and quasi-separated morphism $f : \mathcal{X} \to \mathcal{Y}$ of algebraic stacks such that the direct image functor $Rf_*$ does not induce a functor $D_\QCoh(\mathcal{O}_\mathcal{X}) \to D_\QCoh(\mathcal{O}_\mathcal{Y})$. Thus restricting to bounded below complexes is necessary. \begin{proposition} \label{proposition-derived-direct-image-quasi-coherent} Let $f : \mathcal{X} \to \mathcal{Y}$ be a quasi-compact and quasi-separated morphism of algebraic stacks. The functor $Rf_*$ induces a commutative diagram $$\xymatrix{ D^{+}_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] \ar[d]^{Rf_*} & D^{+}_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] \ar[d]^{Rf_*} & D(\mathcal{O}_\mathcal{X}) \ar[d]^{Rf_*} \\ D^{+}_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] & D^{+}_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] & D(\mathcal{O}_\mathcal{Y}) }$$ and hence induces a functor $$Rf_{\QCoh, *} : D^{+}_\QCoh(\mathcal{O}_\mathcal{X}) \longrightarrow D^{+}_\QCoh(\mathcal{O}_\mathcal{Y})$$ on quotient categories. Moreover, the functor $R^if_\QCoh$ of Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-direct-image-quasi-coherent} are equal to $H^i \circ Rf_{\QCoh, *}$ with $H^i$ as in (\ref{equation-Hi-quasi-coherent}). \end{proposition} \begin{proof} We have to show that $Rf_*E$ is an object of $D^{+}_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$ for $E$ in $D^{+}_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. This follows from Cohomology of Stacks, Proposition \ref{stacks-cohomology-proposition-lcq-flat-base-change} and the spectral sequence $R^if_*H^j(E) \Rightarrow R^{i + j}f_*E$. The case of parasitic modules works the same way using Cohomology of Stacks, Lemma \ref{stacks-cohomology-lemma-pushforward-parasitic}. The final statement is clear from the definition of $H^i$ in (\ref{equation-Hi-quasi-coherent}). \end{proof} \section{Derived pullback of quasi-coherent modules} \label{section-derived-pullback} \noindent Derived pullback of complexes with quasi-coherent cohomology sheaves exists in general. \begin{proposition} \label{proposition-derived-pullback-quasi-coherent} Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. The exact functor $f^*$ induces a commutative diagram $$\xymatrix{ D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \ar[r] & D(\mathcal{O}_\mathcal{X}) \\ D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \ar[r] \ar[u]^{f^*} & D(\mathcal{O}_\mathcal{Y}) \ar[u]^{f^*} }$$ The composition $$D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \xrightarrow{f^*} D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X}) \xrightarrow{q_\mathcal{X}} D_\QCoh(\mathcal{O}_\mathcal{X})$$ is left derivable with respect to the localization $D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y}) \to D_\QCoh(\mathcal{O}_\mathcal{Y})$ and we may define $Lf^*_\QCoh$ as its left derived functor $$Lf_\QCoh^* : D_\QCoh(\mathcal{O}_\mathcal{Y}) \longrightarrow D_\QCoh(\mathcal{O}_\mathcal{X})$$ (see Derived Categories, Definitions \ref{derived-definition-right-derived-functor-defined} and \ref{derived-definition-everywhere-defined}). If $f$ is quasi-compact and quasi-separated, then $Lf^*_\QCoh$ and $Rf_{\QCoh, *}$ satisfy the following adjointness: $$\Hom_{D_\QCoh(\mathcal{O}_\mathcal{X})}(Lf^*_\QCoh A, B) = \Hom_{D_\QCoh(\mathcal{O}_\mathcal{Y})}(A, Rf_{\QCoh, *}B)$$ for $A \in D_\QCoh(\mathcal{O}_\mathcal{Y})$ and $B \in D^{+}_\QCoh(\mathcal{O}_\mathcal{X})$. \end{proposition} \begin{proof} To prove the first statement, we have to show that $f^*E$ is an object of $D_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ for $E$ in $D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. Since $f^* = f^{-1}$ is exact this follows immediately from the fact that $f^*$ maps $\mathcal{M}_\mathcal{Y}$ into $\mathcal{M}_\mathcal{X}$. \medskip\noindent Set $\mathcal{D} = D_{\mathcal{M}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. Let $S$ be the collection of morphisms in $\mathcal{D}$ whose cone is an object of $D_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. Set $\mathcal{D}' = D_\QCoh(\mathcal{O}_\mathcal{X})$. Set $F = q_\mathcal{X} \circ f^* : \mathcal{D} \to \mathcal{D}'$. Then $\mathcal{D}, S, \mathcal{D}', F$ are as in Derived Categories, Situation \ref{derived-situation-derived-functor} and Definition \ref{derived-definition-right-derived-functor-defined}. Let us prove that $LF(E)$ is defined for any object $E$ of $\mathcal{D}$. Namely, consider the triangle $$E' \to E \to P \to E'[1]$$ constructed in Lemma \ref{lemma-bousfield-colocalization}. Note that $s : E' \to E$ is an element of $S$. We claim that $E'$ computes $LF$. Namely, suppose that $s' : E'' \to E$ is another element of $S$, i.e., fits into a triangle $E'' \to E \to P' \to E''[1]$ with $P'$ in $D_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$. By Lemma \ref{lemma-bousfield-colocalization} (and its proof) we see that $E' \to E$ factors through $E'' \to E$. Thus we see that $E' \to E$ is cofinal in the system $S/E$. Hence it is clear that $E'$ computes $LF$. \medskip\noindent To see the final statement, write $B = q_\mathcal{X}(H)$ and $A = q_\mathcal{Y}(E)$. Choose $E' \to E$ as above. We will use on the one hand that $Rf_{\QCoh, *}(B) = q_\mathcal{Y}(Rf_*H)$ and on the other that $Lf^*_\QCoh(A) = q_\mathcal{X}(f^*E')$. \begin{align*} \Hom_{D_\QCoh(\mathcal{O}_\mathcal{X})}(Lf^*_\QCoh A, B) & = \Hom_{D_\QCoh(\mathcal{O}_\mathcal{X})}(q_\mathcal{X}(f^*E'), q_\mathcal{X}(H)) \\ & = \colim_{H \to H'} \Hom_{D(\mathcal{O}_\mathcal{X})}(f^*E', H') \\ & = \colim_{H \to H'} \Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*H') \\ & = \Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*H) \\ & = \Hom_{D_\QCoh(\mathcal{O}_\mathcal{Y})}(A, Rf_{\QCoh, *}B) \end{align*} Here the colimit is over morphisms $s : H \to H'$ in $D^+_{\mathcal{M}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$ whose cone $P(s)$ is an object of $D^+_{\mathcal{P}_\mathcal{X}}(\mathcal{O}_\mathcal{X})$. The first equality we've seen above. The second equality holds by construction of the Verdier quotient. The third equality holds by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}. Since $Rf_*P(s)$ is an object of $D^+_{\mathcal{P}_\mathcal{Y}}(\mathcal{O}_\mathcal{Y})$ by Proposition \ref{proposition-derived-direct-image-quasi-coherent} we see that $\Hom_{D(\mathcal{O}_\mathcal{Y})}(E', Rf_*P(s)) = 0$. Thus the fourth equality holds. The final equality holds by construction of $E'$. \end{proof} \input{chapters} \bibliography{my} \bibliographystyle{amsalpha} \end{document}