The Stacks project

Lemma 59.101.3. Let $S = \mathop{\mathrm{Spec}}(A)$ be an affine scheme. Let $M^\bullet $ be a complex of $A$-modules. Consider the complex $\mathcal{F}^\bullet $ of presheaves of $\mathcal{O}$-modules on $(\textit{Aff}/S)_{fppf}$ given by the rule

\[ (U/S) = (\mathop{\mathrm{Spec}}(B)/\mathop{\mathrm{Spec}}(A)) \longmapsto M^\bullet \otimes _ A B \]

Then this is a complex of modules and the canonical map

\[ M^\bullet \longrightarrow R\Gamma ((\textit{Aff}/S)_{fppf}, \mathcal{F}^\bullet ) \]

is a quasi-isomorphism.

Proof. Each $\mathcal{F}^ n$ is a sheaf of modules as it agrees with the restriction of the module $\mathcal{G}^ n = (\widetilde{M}^ n)^ a$ of Lemma 59.101.1 to $(\textit{Aff}/S)_{fppf} \subset (\mathit{Sch}/S)_{fppf}$. Since this inclusion defines an equivalence of ringed topoi (Topologies, Lemma 34.7.11), we have

\[ R\Gamma ((\textit{Aff}/S)_{fppf}, \mathcal{F}^\bullet ) = R\Gamma ((\mathit{Sch}/S)_{fppf}, \mathcal{G}^\bullet ) \]

We observe that $M^\bullet = R\Gamma (S, \widetilde{M}^\bullet )$ for example by Derived Categories of Schemes, Lemma 36.3.5. Hence we are trying to show the comparison map

\[ R\Gamma (S, \widetilde{M}^\bullet ) \longrightarrow R\Gamma ((\mathit{Sch}/S)_{fppf}, (\widetilde{M}^\bullet )^ a) \]

is an isomorphism. If $M^\bullet $ is bounded below, then this holds by Descent, Proposition 35.9.3 and the first spectral sequence of Derived Categories, Lemma 13.21.3. For the general case, let us write $M^\bullet = \mathop{\mathrm{lim}}\nolimits M_ n^\bullet $ with $M_ n^\bullet = \tau _{\geq -n}M^\bullet $. Whence the system $M_ n^ p$ is eventually constant with value $M^ p$. We claim that

\[ (\widetilde{M}^\bullet )^ a = R\mathop{\mathrm{lim}}\nolimits (\widetilde{M}_ n^\bullet )^ a \]

Namely, it suffices to show that the natural map from left to right induces an isomorphism on cohomology over any affine object $U = \mathop{\mathrm{Spec}}(B)$ of $(\mathit{Sch}/S)_{fppf}$. For $i \in \mathbf{Z}$ and $n > |i|$ we have

\[ H^ i(U, (\widetilde{M}_ n^\bullet )^ a) = H^ i(\tau _{\geq -n}M^\bullet \otimes _ A B) = H^ i(M^\bullet \otimes _ A B) \]

The first equality holds by the bounded below case treated above. Thus we see from Cohomology on Sites, Lemma 21.23.2 that the claim holds. Then we finally get

\begin{align*} R\Gamma ((\mathit{Sch}/S)_{fppf}, (\widetilde{M}^\bullet )^ a) & = R\Gamma ((\mathit{Sch}/S)_{fppf}, R\mathop{\mathrm{lim}}\nolimits (\widetilde{M}_ n^\bullet )^ a) \\ & = R\mathop{\mathrm{lim}}\nolimits R\Gamma ((\mathit{Sch}/S)_{fppf}, (\widetilde{M}_ n^\bullet )^ a) \\ & = R\mathop{\mathrm{lim}}\nolimits M_ n^\bullet \\ & = M^\bullet \end{align*}

as desired. The second equality holds because $R\mathop{\mathrm{lim}}\nolimits $ commutes with $R\Gamma $, see Cohomology on Sites, Lemma 21.23.2. $\square$

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