The Stacks project

46.10 Derived categories of adequate modules, II

Let $S$ be a scheme. Denote $\mathcal{O}_ S$ the structure sheaf of $S$ and $\mathcal{O}$ the structure sheaf of the big site $(\mathit{Sch}/S)_\tau $. In Descent, Remark 35.8.4 we constructed a morphism of ringed sites
\begin{equation} \label{adequate-equation-compare-big-small} f : ((\mathit{Sch}/S)_\tau , \mathcal{O}) \longrightarrow (S_{Zar}, \mathcal{O}_ S). \end{equation}

In the previous sections have seen that the functor $f_* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ S)$ transforms adequate sheaves into quasi-coherent sheaves, and induces an exact functor $v : \textit{Adeq}(\mathcal{O}) \to \mathit{QCoh}(\mathcal{O}_ S)$, and in fact that $f_* = v$ induces an equivalence $\textit{Adeq}(\mathcal{O})/\mathcal{C} \to \mathit{QCoh}(\mathcal{O}_ S)$ where $\mathcal{C}$ is the subcategory of parasitic adequate modules. Moreover, the functor $f^*$ transforms quasi-coherent modules into adequate modules, and induces a functor $u : \mathit{QCoh}(\mathcal{O}_ S) \to \textit{Adeq}(\mathcal{O})$ which is a left adjoint to $v$.

There is a very similar relationship between $D_{\textit{Adeq}}(\mathcal{O})$ and $D_\mathit{QCoh}(S)$. First we explain why the category $D_{\textit{Adeq}}(\mathcal{O})$ is independent of the chosen topology.

Remark 46.10.1. Let $S$ be a scheme. Let $\tau , \tau ' \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $. Denote $\mathcal{O}_\tau $, resp. $\mathcal{O}_{\tau '}$ the structure sheaf $\mathcal{O}$ viewed as a sheaf on $(\mathit{Sch}/S)_\tau $, resp. $(\mathit{Sch}/S)_{\tau '}$. Then $D_{\textit{Adeq}}(\mathcal{O}_\tau )$ and $D_{\textit{Adeq}}(\mathcal{O}_{\tau '})$ are canonically isomorphic. This follows from Cohomology on Sites, Lemma 21.29.1. Namely, assume $\tau $ is stronger than the topology $\tau '$, let $\mathcal{C} = (\mathit{Sch}/S)_{fppf}$, and let $\mathcal{B}$ the collection of affine schemes over $S$. Assumptions (1) and (2) we've seen above. Assumption (3) is clear and assumption (4) follows from Lemma 46.5.8.

Remark 46.10.2. Let $S$ be a scheme. The morphism $f$ see ( induces adjoint functors $Rf_* : D_{\textit{Adeq}}(\mathcal{O}) \to D_\mathit{QCoh}(S)$ and $Lf^* : D_\mathit{QCoh}(S) \to D_{\textit{Adeq}}(\mathcal{O})$. Moreover $Rf_* Lf^* \cong \text{id}_{D_\mathit{QCoh}(S)}$.

We sketch the proof. By Remark 46.10.1 we may assume the topology $\tau $ is the Zariski topology. We will use the existence of the unbounded total derived functors $Lf^*$ and $Rf_*$ on $\mathcal{O}$-modules and their adjointness, see Cohomology on Sites, Lemma 21.19.1. In this case $f_*$ is just the restriction to the subcategory $S_{Zar}$ of $(\mathit{Sch}/S)_{Zar}$. Hence it is clear that $Rf_* = f_*$ induces $Rf_* : D_{\textit{Adeq}}(\mathcal{O}) \to D_\mathit{QCoh}(S)$. Suppose that $\mathcal{G}^\bullet $ is an object of $D_\mathit{QCoh}(S)$. We may choose a system $\mathcal{K}_1^\bullet \to \mathcal{K}_2^\bullet \to \ldots $ of bounded above complexes of flat $\mathcal{O}_ S$-modules whose transition maps are termwise split injectives and a diagram

\[ \xymatrix{ \mathcal{K}_1^\bullet \ar[d] \ar[r] & \mathcal{K}_2^\bullet \ar[d] \ar[r] & \ldots \\ \tau _{\leq 1}\mathcal{G}^\bullet \ar[r] & \tau _{\leq 2}\mathcal{G}^\bullet \ar[r] & \ldots } \]

with the properties (1), (2), (3) listed in Derived Categories, Lemma 13.29.1 where $\mathcal{P}$ is the collection of flat $\mathcal{O}_ S$-modules. Then $Lf^*\mathcal{G}^\bullet $ is computed by $\mathop{\mathrm{colim}}\nolimits f^*\mathcal{K}_ n^\bullet $, see Cohomology on Sites, Lemmas 21.18.1 and 21.18.2 (note that our sites have enough points by √Čtale Cohomology, Lemma 59.30.1). We have to see that $H^ i(Lf^*\mathcal{G}^\bullet ) = \mathop{\mathrm{colim}}\nolimits H^ i(f^*\mathcal{K}_ n^\bullet )$ is adequate for each $i$. By Lemma 46.5.11 we conclude that it suffices to show that each $H^ i(f^*\mathcal{K}_ n^\bullet )$ is adequate.

The adequacy of $H^ i(f^*\mathcal{K}_ n^\bullet )$ is local on $S$, hence we may assume that $S = \mathop{\mathrm{Spec}}(A)$ is affine. Because $S$ is affine $D_\mathit{QCoh}(S) = D(\mathit{QCoh}(\mathcal{O}_ S))$, see the discussion in Derived Categories of Schemes, Section 36.3. Hence there exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{K}_ n^\bullet $ where $\mathcal{F}^\bullet $ is a bounded above complex of flat quasi-coherent modules. Then $f^*\mathcal{F}^\bullet \to f^*\mathcal{K}_ n^\bullet $ is a quasi-isomorphism, and the cohomology sheaves of $f^*\mathcal{F}^\bullet $ are adequate.

The final assertion $Rf_* Lf^* \cong \text{id}_{D_\mathit{QCoh}(S)}$ follows from the explicit description of the functors above. (In plain English: if $\mathcal{F}$ is quasi-coherent and $p > 0$, then $L_ pf^*\mathcal{F}$ is a parasitic adequate module.)

Remark 46.10.3. Remark 46.10.2 above implies we have an equivalence of derived categories

\[ D_{\textit{Adeq}}(\mathcal{O})/D_\mathcal {C}(\mathcal{O}) \longrightarrow D_\mathit{QCoh}(S) \]

where $\mathcal{C}$ is the category of parasitic adequate modules. Namely, it is clear that $D_\mathcal {C}(\mathcal{O})$ is the kernel of $Rf_*$, hence a functor as indicated. For any object $X$ of $D_{\textit{Adeq}}(\mathcal{O})$ the map $Lf^*Rf_*X \to X$ maps to a quasi-isomorphism in $D_\mathit{QCoh}(S)$, hence $Lf^*Rf_*X \to X$ is an isomorphism in $D_{\textit{Adeq}}(\mathcal{O})/D_\mathcal {C}(\mathcal{O})$. Finally, for $X, Y$ objects of $D_{\textit{Adeq}}(\mathcal{O})$ the map

\[ Rf_* : \mathop{\mathrm{Hom}}\nolimits _{D_{\textit{Adeq}}(\mathcal{O})/D_\mathcal {C}(\mathcal{O})}(X, Y) \to \mathop{\mathrm{Hom}}\nolimits _{D_\mathit{QCoh}(S)}(Rf_*X, Rf_*Y) \]

is bijective as $Lf^*$ gives an inverse (by the remarks above).

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