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83.14 Simplicial systems of the derived category: modules

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 83.13.

Definition 83.14.1. In Situation 83.3.3. 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

  1. for every $n$ an object $K_ n$ of $D(\mathcal{O}_ n)$,

  2. 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

\[ K_{\varphi \circ \psi } = K_\varphi \circ Lf_\varphi ^*K_\psi : Lf_{\varphi \circ \psi }^*K_ l = Lf_\varphi ^* Lf_\psi ^*K_ l \longrightarrow K_ n \]

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 83.14.2. In Situation 83.3.3 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 83.14.3. In Situation 83.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. Let $K$ be an object of $D(\mathcal{C}_{total})$. Set

\[ X_ n = (g_{n!}\mathcal{O}_ n) \otimes ^\mathbf {L}_\mathcal {O} K \quad \text{and}\quad Y_ n = (g_{n!}\mathcal{O}_ n \to \ldots \to g_{0!}\mathcal{O}_0)[-n] \otimes ^\mathbf {L}_\mathcal {O} K \]

as objects of $D(\mathcal{O})$ where the maps are as in Lemma 83.8.1. 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

  1. the distinguished triangles $Y_ n \to X_ n \to Y_{n - 1} \to Y_ n[1]$ define a Postnikov system (Derived Categories, Definition 13.40.1) for $\ldots \to X_2 \to X_1 \to X_0$,

  2. $K = \text{hocolim} Y_ n[n]$ in $D(\mathcal{O})$.

Proof. First, if $K = \mathcal{O}$, then this is the construction of Derived Categories, Example 13.40.2 applied to the complex

\[ \ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0 \]

in $\textit{Ab}(\mathcal{C}_{total})$ combined with the fact that this complex represents $K = \mathcal{O}$ in $D(\mathcal{C}_{total})$ by Lemma 83.10.1. 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 83.14.4. In Situation 83.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume

  1. $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,

  2. $K$ is cartesian,

  3. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i], K'_ i) = 0$ for $i > 0$, and

  4. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i + 1], K'_ i) = 0$ 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 83.13.4 except using Lemma 83.14.3 instead of Lemma 83.13.3. $\square$

Lemma 83.14.5. In Situation 83.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. If $K, K' \in D(\mathcal{O})$. Assume

  1. $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,

  2. $K$ is cartesian,

  3. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i - 1], K'_ i) = 0$ 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 83.13.5 except using Lemma 83.14.3 instead of Lemma 83.13.3. $\square$

Lemma 83.14.6. In Situation 83.3.3 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

  1. $f_\varphi ^{-1}\mathcal{O}_ n \to \mathcal{O}_ m$ is flat for $\varphi : [m] \to [n]$,

  2. $(K_ n, K_\varphi )$ is cartesian,

  3. $\mathop{\mathrm{Hom}}\nolimits (K_ i[t], K_ i) = 0$ 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 83.13.6 with the following changes

  1. use $g_ n^* = Lg_ n^*$ everywhere instead of $g_ n^{-1}$,

  2. use $f_\varphi ^* = Lf_\varphi ^*$ everywhere instead of $f_\varphi ^{-1}$,

  3. refer to Lemma 83.10.1 instead of Lemma 83.8.1,

  4. in the construction of $Y'_{m, n}$ use $\mathcal{O}_ m$ instead of $\mathbf{Z}$,

  5. compare with the proof of Lemma 83.14.3 rather than the proof of Lemma 83.13.3.

This ends the proof. $\square$


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