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

46.10.0.1
\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 (46.10.0.1) 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).


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 070T. Beware of the difference between the letter 'O' and the digit '0'.