Lemma 85.13.4 (0D9I)—The Stacks project

Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.13: Simplicial systems of the derived category
Lemma 85.13.4 (cite)

Lemma 85.13.4. In Situation 85.3.3. 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 85.8.1. 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 $D(\mathcal{C}_{total})$.

Proof. First, if $K = \mathbf{Z}$, then this is the construction of Derived Categories, Example 13.41.2 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 85.8.1. 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. $\square$

The Stacks project

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