## 83.13 Simplicial systems of the derived category

In this section we are going to prove a special case of [Proposition 3.2.9, BBD] in the setting of derived categories of abelian sheaves. The case of modules is discussed in Section 83.14.

Definition 83.13.1. In Situation 83.3.3. A simplicial system of the derived category consists of the following data

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

2. for every $\varphi : [m] \to [n]$ a map $K_\varphi : f_\varphi ^{-1}K_ m \to K_ n$ in $D(\mathcal{C}_ n)$

subject to the condition that

$K_{\varphi \circ \psi } = K_\varphi \circ f_\varphi ^{-1}K_\psi : f_{\varphi \circ \psi }^{-1}K_ l = f_\varphi ^{-1} f_\psi ^{-1}K_ l \longrightarrow K_ n$

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.

We have given this notion a ridiculously long name intentionally. The goal is to show that a simplicial system of the derived category comes from an object of $D(\mathcal{C}_{total})$ under certain hypotheses.

Lemma 83.13.2. In Situation 83.3.3. If $K \in D(\mathcal{C}_{total})$ is an object, then $(K_ n, K(\varphi ))$ is a simplicial system of the derived category. If $K$ is cartesian, so is the system.

Proof. This is obvious. $\square$

Lemma 83.13.3. In Situation 83.3.3. Let $K$ be an object of $D(\mathcal{C}_{total})$. Set

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

as objects of $D(\mathcal{C}_{total})$ where the maps are as in Lemma 83.8.1. With the evident canonical maps $Y_ n \to X_ n$ and $Y_0 \to Y_1 \to Y_2 \to \ldots$ we have

1. the distinguished triangles $Y_ n \to X_ n \to Y_{n - 1} \to Y_ n$ 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 $D(\mathcal{C}_{total})$.

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

$\ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}$

in $\textit{Ab}(\mathcal{C}_{total})$ combined with the fact that this complex represents $K = \mathbf{Z}$ in $D(\mathcal{C}_{total})$ by Lemma 83.8.1. The general case follows from this, the fact that the exact functor $- \otimes ^\mathbf {L}_\mathbf {Z} K$ sends Postnikov systems to Postnikov systems, and that $- \otimes ^\mathbf {L}_\mathbf {Z} K$ commutes with homotopy colimits. $\square$

Lemma 83.13.4. In Situation 83.3.3. If $K, K' \in D(\mathcal{C}_{total})$. Assume

1. $K$ is cartesian,

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

3. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i + 1], K'_ i) = 0$ for $i \geq 0$.

Then any map $K \to K'$ which induces the zero map $K_0 \to K'_0$ is zero.

Proof. Consider the objects $X_ n$ and the Postnikov system $Y_ n$ associated to $K$ in Lemma 83.13.3. As $K = \text{hocolim} Y_ n[n]$ the map $K \to K'$ induces a compatible family of morphisms $Y_ n[n] \to K'$. By (1) and Lemma 83.12.9 we have $X_ n = g_{n!}K_ n$. Since $Y_0 = X_0$ we find that $K_0 \to K'_0$ being zero implies $Y_0 \to K'$ is zero. Suppose we've shown that the map $Y_ n[n] \to K'$ is zero for some $n \geq 0$. From the distinguished triangle

$Y_ n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1] \to Y_ n[n + 1]$

we get an exact sequence

$\mathop{\mathrm{Hom}}\nolimits (X_{n + 1}[n + 1], K') \to \mathop{\mathrm{Hom}}\nolimits (Y_{n + 1}[n + 1], K') \to \mathop{\mathrm{Hom}}\nolimits (Y_ n[n], K')$

As $X_{n + 1}[n + 1] = g_{n + 1!}K_{n + 1}[n + 1]$ the first group is equal to

$\mathop{\mathrm{Hom}}\nolimits (K_{n + 1}[n + 1], K'_{n + 1})$

which is zero by assumption (2). By induction we conclude all the maps $Y_ n[n] \to K'$ are zero. Consider the defining distinguished triangle

$\bigoplus Y_ n[n] \to \bigoplus Y_ n[n] \to K \to (\bigoplus Y_ n[n])$

for the homotopy colimit. Arguing as above, we find that it suffices to show that

$\mathop{\mathrm{Hom}}\nolimits ((\bigoplus Y_ n[n]), K') = \prod \mathop{\mathrm{Hom}}\nolimits (Y_ n[n + 1], K')$

is zero for all $n \geq 0$. To see this, arguing as above, it suffices to show that

$\mathop{\mathrm{Hom}}\nolimits (K_ n[n + 1], K'_ n) = 0$

for all $n \geq 0$ which follows from condition (3). $\square$

Lemma 83.13.5. In Situation 83.3.3. If $K, K' \in D(\mathcal{C}_{total})$. Assume

1. $K$ is cartesian,

2. $\mathop{\mathrm{Hom}}\nolimits (K_ i[i - 1], K'_ i) = 0$ 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 $D(\mathcal{C}_{total})$.

Proof. Let $\{ K_ n \to K'_ n\} _{n \geq 0}$ be a morphism of simplicial systems of the derived category. Consider the objects $X_ n$ and Postnikov system $Y_ n$ associated to $K$ of Lemma 83.13.3. By (1) and Lemma 83.12.9 we have $X_ n = g_{n!}K_ n$. In particular, the map $K_0 \to K'_0$ induces a morphism $X_0 \to K'$. Since $\{ K_ n \to K'_ n\}$ is a morphism of systems, a computation (omitted) shows that the composition

$X_1 \to X_0 \to K'$

is zero. As $Y_0 = X_0$ and as $Y_1$ fits into a distinguished triangle

$Y_1 \to X_1 \to Y_0 \to Y_1$

we conclude that there exists a morphism $Y_1 \to K'$ whose composition with $X_0 = Y_0 \to Y_1$ is the morphism $X_0 \to K'$ given above. Suppose given a map $Y_ n[n] \to K'$ for $n \geq 1$. From the distinguished triangle

$X_{n + 1}[n] \to Y_ n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1]$

we get an exact sequence

$\mathop{\mathrm{Hom}}\nolimits (Y_{n + 1}[n + 1], K') \to \mathop{\mathrm{Hom}}\nolimits (Y_ n[n], K') \to \mathop{\mathrm{Hom}}\nolimits (X_{n + 1}[n], K')$

As $X_{n + 1}[n] = g_{n + 1!}K_{n + 1}[n]$ the last group is equal to

$\mathop{\mathrm{Hom}}\nolimits (K_{n + 1}[n], K'_{n + 1})$

which is zero by assumption (2). By induction we get a system of maps $Y_ n[n] \to K'$ compatible with transition maps and reducing to the given map on $Y_0$. This produces a map

$\gamma : K = \text{hocolim} Y_ n[n] \longrightarrow K'$

This map in any case has the property that the diagram

$\xymatrix{ X_0 \ar[rd] \ar[r] & K \ar[d]^\gamma \\ & K' }$

is commutative. Restricting to $\mathcal{C}_0$ we deduce that the map $\gamma _0 : K_0 \to K'_0$ is the same as the first map $K_0 \to K'_0$ of the morphism of simplicial systems. Since $K$ is cartesian, this easily gives that $\{ \gamma _ n\}$ is the map of simplicial systems we started out with. $\square$

Lemma 83.13.6. In Situation 83.3.3. Let $(K_ n, K_\varphi )$ be a simplicial system of the derived category. Assume

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

2. $\mathop{\mathrm{Hom}}\nolimits (K_ i[t], K_ i) = 0$ for $i \geq 0$ and $t > 0$.

Then there exists a cartesian object $K$ of $D(\mathcal{C}_{total})$ whose associated simplicial system is isomorphic to $(K_ n, K_\varphi )$.

Proof. Set $X_ n = g_{n!}K_ n$ in $D(\mathcal{C}_{total})$. For each $n \geq 1$ we have

$\mathop{\mathrm{Hom}}\nolimits (X_ n, X_{n - 1}) = \mathop{\mathrm{Hom}}\nolimits (K_ n, g_ n^{-1}g_{n - 1!}K_{n - 1}) = \bigoplus \nolimits _{\varphi : [n - 1] \to [n]} \mathop{\mathrm{Hom}}\nolimits (K_ n, f_\varphi ^{-1}K_{n - 1})$

Thus we get a map $X_ n \to X_{n - 1}$ corresponding to the alternating sum of the maps $K_\varphi ^{-1} : K_ n \to f_\varphi ^{-1}K_{n - 1}$ where $\varphi$ runs over $\delta ^ n_0, \ldots , \delta ^ n_ n$. We can do this because $K_\varphi$ is invertible by assumption (1). Please observe the similarity with the definition of the maps in the proof of Lemma 83.8.1. We obtain a complex

$\ldots \to X_2 \to X_1 \to X_0$

in $D(\mathcal{C}_{total})$. We omit the computation which shows that the compositions are zero. By Derived Categories, Lemma 13.40.6 if we have

$\mathop{\mathrm{Hom}}\nolimits (X_ i[i - j - 2], X_ j) = 0\text{ for }i > j + 2$

then we can extend this complex to a Postnikov system. The group is equal to

$\mathop{\mathrm{Hom}}\nolimits (K_ i[i - j - 2], g_ i^{-1}g_{j!}K_ j)$

Again using that $(K_ n, K_\varphi )$ is cartesian we see that $g_ i^{-1}g_{j!}K_ j$ is isomorphic to a finite direct sum of copies of $K_ i$. Hence the group vanishes by assumption (2). Let the Postnikov system be given by $Y_0 = X_0$ and distinguished sequences $Y_ n \to X_ n \to Y_{n - 1} \to Y_ n$ for $n \geq 1$. We set

$K = \text{hocolim} Y_ n[n]$

To finish the proof we have to show that $g_ m^{-1}K$ is isomorphic to $K_ m$ for all $m$ compatible with the maps $K_\varphi$. Observe that

$g_ m^{-1} K = \text{hocolim} g_ m^{-1}Y_ n[n]$

and that $g_ m^{-1}Y_ n[n]$ is a Postnikov system for $g_ m^{-1}X_ n$. Consider the isomorphisms

$g_ m^{-1}X_ n = \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}K_ n \xrightarrow {\bigoplus K_\varphi } \bigoplus \nolimits _{\varphi : [n] \to [m]} K_ m$

These maps define an isomorphism of complexes

$\xymatrix{ \ldots \ar[r] & g_ m^{-1}X_2 \ar[r] \ar[d] & g_ m^{-1}X_1 \ar[r] \ar[d] & g_ m^{-1}X_0 \ar[d] \\ \ldots \ar[r] & \bigoplus \nolimits _{\varphi :  \to [m]} K_ m \ar[r] & \bigoplus \nolimits _{\varphi :  \to [m]} K_ m \ar[r] & \bigoplus \nolimits _{\varphi :  \to [m]} K_ m }$

in $D(\mathcal{C}_ m)$ where the arrows in the bottom row are as in the proof of Lemma 83.8.1. The squares commute by our choice of the arrows of the complex $\ldots \to X_2 \to X_1 \to X_0$; we omit the computation. The bottom row complex has a postnikov tower given by

$Y'_{m, n} = \left(\bigoplus \nolimits _{\varphi : [n] \to [m]} \mathbf{Z} \to \ldots \to \bigoplus \nolimits _{\varphi :  \to [m]} \mathbf{Z}\right)[-n] \otimes ^\mathbf {L}_\mathbf {Z} K_ m$

and $\text{hocolim} Y'_{m, n} = K_ m$ (please compare with the proof of Lemma 83.13.3 and Derived Categories, Example 13.40.2). Applying the second part of Derived Categories, Lemma 13.40.6 the vertical maps in the big diagram extend to an isomorphism of Postnikov systems provided we have

$\mathop{\mathrm{Hom}}\nolimits (g_ m^{-1}X_ i[i - j - 1], \bigoplus \nolimits _{\varphi : [j] \to [m]} K_ m) = 0\text{ for }i > j + 1$

The is true if $\mathop{\mathrm{Hom}}\nolimits (K_ m[i - j - 1], K_ m) = 0$ for $i > j + 1$ which holds by assumption (2). Choose an isomorphism given by $\gamma _{m, n} : g_ m^{-1}Y_ n \to Y'_{m, n}$ of Postnikov systems in $D(\mathcal{C}_ m)$. By uniqueness of homotopy colimits, we can find an isomorphism

$g_ m^{-1} K = \text{hocolim} g_ m^{-1}Y_ n[n] \xrightarrow {\gamma _ m} \text{hocolim} Y'_{m, n} = K_ m$

compatible with $\gamma _{m, n}$.

We still have to prove that the maps $\gamma _ m$ fit into commutative diagrams

$\xymatrix{ f_\varphi ^{-1}g_ m^{-1}K \ar[d]_{f_\varphi ^{-1}\gamma _ m} \ar[r]_{K(\varphi )} & g_ n^{-1}K \ar[d]^{\gamma _ n} \\ f_\varphi ^{-1}K_ m \ar[r]^{K_\varphi } & K_ n }$

for every $\varphi : [m] \to [n]$. Consider the diagram

$\xymatrix{ f_\varphi ^{-1}(\bigoplus _{\psi :  \to [m]} f_\psi ^{-1}K_0) \ar@{=}[r] \ar[d]_{f_\varphi ^{-1}(\bigoplus K_\psi )} & f_\varphi ^{-1}g_ m^{-1}X_0 \ar[d] \ar[r]_{X_0(\varphi )} & g_ n^{-1}X_0 \ar[d] & \bigoplus _{\chi :  \to [n]} f_\chi ^{-1}K_0 \ar@{=}[l] \ar[d]^{\bigoplus K_\chi } \\ f_\varphi ^{-1}(\bigoplus _{\psi :  \to [m]} K_ m) \ar@{=}[d] & f_\varphi ^{-1}g_ m^{-1}K \ar[d]_{f_\varphi ^{-1}\gamma _ m} \ar[r]_{K(\varphi )} & g_ n^{-1}K \ar[d]^{\gamma _ n} & \bigoplus _{\chi :  \to [n]} K_ n \ar@{=}[d] \\ f_\varphi ^{-1}Y'_{0, m} \ar[r] & f_\varphi ^{-1}K_ m \ar[r]^{K_\varphi } & K_ n & Y'_{0, n} \ar[l] }$

The top middle square is commutative as $X_0 \to K$ is a morphism of simplicial objects. The left, resp. the right rectangles are commutative as $\gamma _ m$, resp. $\gamma _ n$ is compatible with $\gamma _{0, m}$, resp. $\gamma _{0, n}$ which are the arrows $\bigoplus K_\psi$ and $\bigoplus K_\chi$ in the diagram. Going around the outer rectangle of the diagram is commutative as $(K_ n, K_\varphi )$ is a simplical system and the map $X_0(\varphi )$ is given by the obvious identifications $f_\varphi ^{-1}f_\psi ^{-1}K_0 = f_{\varphi \circ \psi }^{-1}K_0$. Note that the arrow $\bigoplus _\psi K_ m \to Y'_{0, m} \to K_ m$ induces an isomorphism on any of the direct summands (because of our explicit construction of the Postnikov systems $Y'_{i, j}$ above). Hence, if we take a direct summand of the upper left and corner, then this maps isomorphically to $f_\varphi ^{-1}g_ m^{-1}K$ as $\gamma _ m$ is an isomorphism. Working out what the above says, but looking only at this direct summand we conclude the lower middle square commutes as we well. This concludes the proof. $\square$

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