The Stacks project

Lemma 85.13.3. In Situation 85.3.3 suppose given $K_0 \in D(\mathcal{C}_0)$ and an isomorphism

\[ \alpha : f_{\delta _1^1}^{-1}K_0 \longrightarrow f_{\delta _0^1}^{-1}K_0 \]

satisfying the cocycle condition. Set $\tau ^ n_ i : [0] \to [n]$, $0 \mapsto i$ and set $K_ n = f_{\tau ^ n_ n}^{-1}K_0$. Then the $K_ n$ form a cartesian simplicial system of the derived category.

Proof. Please compare with Lemma 85.12.4 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 $K_\varphi : f_\varphi ^{-1}K_ n \to K_ m$ using

\[ \xymatrix{ f_\varphi ^{-1}K_ n \ar@{=}[r] & f_\varphi ^{-1} f_{\tau ^ n_ n}^{-1}K_0 \ar@{=}[r] & f_{\tau ^ m_{\varphi (n)}}^{-1}K_0 \ar@{=}[r] & f_{\tau ^ m_{\varphi (n)m}}^{-1}f_{\delta ^1_1}^{-1}K_0 \ar[d]_{f_{\tau ^ m_{\varphi (n)m}}^{-1}\alpha } \\ & K_ m \ar@{=}[r] & f_{\tau ^ m_ m}^{-1}K_0 \ar@{=}[r] & f_{\tau ^ m_{\varphi (n)m}}^{-1}f_{\delta ^1_0}^{-1}K_0 } \]

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 system in the derived category. $\square$

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