In this section we are going to prove a special case of [Proposition 3.2.9, BBD] in the setting of derived categories of $\mathcal{O}$-modules. The (slightly) easier case of abelian sheaves is discussed in Section 84.13.
Definition 84.14.1. In Situation 84.3.3. Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. A simplicial system of the derived category of modules consists of the following data
for every $n$ an object $K_ n$ of $D(\mathcal{O}_ n)$,
for every $\varphi : [m] \to [n]$ a map $K_\varphi : Lf_\varphi ^*K_ m \to K_ n$ in $D(\mathcal{O}_ n)$
subject to the condition that
for any morphisms $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$ of $\Delta $. We say the simplicial system is cartesian if the maps $K_\varphi $ are isomorphisms for all $\varphi $. Given two simplicial systems of the derived category there is an obvious notion of a morphism of simplicial systems of the derived category of modules.
We have given this notion a ridiculously long name intentionally. The goal is to show that a simplicial system of the derived category of modules comes from an object of $D(\mathcal{O})$ under certain hypotheses.
Lemma 84.14.2. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K \in D(\mathcal{O})$ is an object, then $(K_ n, K(\varphi ))$ is a simplicial system of the derived category of modules. If $K$ is cartesian, so is the system.
Proof. This is immediate from the definitions. $\square$
Lemma 84.14.3. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Suppose given $K_0 \in D(\mathcal{O}_0)$ and an isomorphism satisfying the cocycle condition. Set $\tau ^ n_ i : [0] \to [n]$, $0 \mapsto i$ and set $K_ n = Lf_{\tau ^ n_ n}^*K_0$. The objects $K_ n$ form the members of a cartesian simplicial system of the derived category of modules.
\[ \alpha : L(f_{\delta _1^1})^*K_0 \longrightarrow L(f_{\delta _0^1})^*K_0 \]
Proof. Please compare with Lemmas 84.13.3 and 84.12.4 and its proof (also to see the cocycle condition spelled out). The construction is analogous to the construction discussed in Descent, Section 35.3 from which we borrow the notation $\tau ^ n_ i : [0] \to [n]$, $0 \mapsto i$ and $\tau ^ n_{ij} : [1] \to [n]$, $0 \mapsto i$, $1 \mapsto j$. Given $\varphi : [n] \to [m]$ we define $K_\varphi : L(f_\varphi )^*K_ n \to K_ m$ using
We omit the verification that the cocycle condition implies the maps compose correctly (in their respective derived categories) and hence give rise to a simplicial systems of the derived category of modules. $\square$
Lemma 84.14.4. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $K$ be an object of $D(\mathcal{C}_{total})$. Set as objects of $D(\mathcal{O})$ where the maps are as in Lemma 84.8.1. With the evident canonical maps $Y_ n \to X_ n$ and $Y_0 \to Y_1[1] \to Y_2[2] \to \ldots $ we have
the distinguished triangles $Y_ n \to X_ n \to Y_{n - 1} \to Y_ n[1]$ define a Postnikov system (Derived Categories, Definition 13.41.1) for $\ldots \to X_2 \to X_1 \to X_0$,
$K = \text{hocolim} Y_ n[n]$ in $D(\mathcal{O})$.
\[ X_ n = (g_{n!}\mathcal{O}_ n) \otimes ^\mathbf {L}_\mathcal {O} K \quad \text{and}\quad Y_ n = (g_{n!}\mathcal{O}_ n \to \ldots \to g_{0!}\mathcal{O}_0)[-n] \otimes ^\mathbf {L}_\mathcal {O} K \]
Proof. First, if $K = \mathcal{O}$, then this is the construction of Derived Categories, Example 13.41.2 applied to the complex
in $\textit{Ab}(\mathcal{C}_{total})$ combined with the fact that this complex represents $K = \mathcal{O}$ in $D(\mathcal{C}_{total})$ by Lemma 84.10.1. The general case follows from this, the fact that the exact functor $- \otimes ^\mathbf {L}_\mathcal {O} K$ sends Postnikov systems to Postnikov systems, and that $- \otimes ^\mathbf {L}_\mathcal {O} K$ commutes with homotopy colimits. $\square$
Lemma 84.14.5. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume
$f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,
$K$ is cartesian,
$\mathop{\mathrm{Hom}}\nolimits (K_ i[i], K'_ i) = 0$ for $i > 0$, and
$\mathop{\mathrm{Hom}}\nolimits (K_ i[i + 1], K'_ i) = 0$ for $i \geq 0$.
Then any map $K \to K'$ which induces the zero map $K_0 \to K'_0$ is zero.
Proof. The proof is exactly the same as the proof of Lemma 84.13.5 except using Lemma 84.14.4 instead of Lemma 84.13.4. $\square$
Lemma 84.14.6. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume
$f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,
$K$ is cartesian,
$\mathop{\mathrm{Hom}}\nolimits (K_ i[i - 1], K'_ i) = 0$ for $i > 1$.
Then any map $\{ K_ n \to K'_ n\} $ between the associated simplicial systems of $K$ and $K'$ comes from a map $K \to K'$ in $D(\mathcal{O})$.
Proof. The proof is exactly the same as the proof of Lemma 84.13.6 except using Lemma 84.14.4 instead of Lemma 84.13.4. $\square$
Lemma 84.14.7. In Situation 84.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $(K_ n, K_\varphi )$ be a simplicial system of the derived category of modules. Assume
$f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,
$(K_ n, K_\varphi )$ is cartesian,
$\mathop{\mathrm{Hom}}\nolimits (K_ i[t], K_ i) = 0$ for $i \geq 0$ and $t > 0$.
Then there exists a cartesian object $K$ of $D(\mathcal{O})$ whose associated simplicial system is isomorphic to $(K_ n, K_\varphi )$.
Proof. The proof is exactly the same as the proof of Lemma 84.13.7 with the following changes
use $g_ n^* = Lg_ n^*$ everywhere instead of $g_ n^{-1}$,
use $f_\varphi ^* = Lf_\varphi ^*$ everywhere instead of $f_\varphi ^{-1}$,
in the construction of $Y'_{m, n}$ use $\mathcal{O}_ m$ instead of $\mathbf{Z}$,
compare with the proof of Lemma 84.14.4 rather than the proof of Lemma 84.13.4.
This ends the proof. $\square$
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