Lemma 91.16.2. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}_0$ be a surjection of sheaves of rings whose kernel is an ideal sheaf $\mathcal{I}$ of square zero. For every object $K_0$ in $D^-(\mathcal{O}_0)$ there is a canonical map
\[ \omega (K_0) : K_0 \longrightarrow K_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I}[2] \]
in $D(\mathcal{O}_0)$ such that for any map $K_0 \to L_0$ in $D^-(\mathcal{O}_0)$ the diagram
\[ \xymatrix{ K_0 \ar[d] \ar[rr]_-{\omega (K_0)} & & (K_0 \otimes ^\mathbf {L}_{\mathcal{O}_0} \mathcal{I})[2] \ar[d] \\ L_0 \ar[rr]^-{\omega (L_0)} & & (L_0 \otimes ^\mathbf {L}_{\mathcal{O}_0} \mathcal{I})[2] } \]
commutes.
Proof.
Represent $K_0$ by any complex $\mathcal{K}_0^\bullet $ of $\mathcal{O}_0$-modules. Apply Lemma 91.16.1 with $\mathcal{G}^ n = 0$ for all $n$. Denote $d : \mathcal{F}^ n \to \mathcal{F}^{n + 1}$ the maps produced by the lemma. Then we see that $d \circ d : \mathcal{F}^ n \to \mathcal{F}^{n + 2}$ is zero modulo $\mathcal{I}$. Since $\mathcal{F}^ n$ is flat, we see that $\mathcal{I}\mathcal{F}^ n = \mathcal{F}^ n \otimes _{\mathcal{O}} \mathcal{I} = \mathcal{F}^ n_0 \otimes _{\mathcal{O}_0} \mathcal{I}$. Hence we obtain a canonical map of complexes
\[ d \circ d : \mathcal{F}_0^\bullet \longrightarrow (\mathcal{F}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I})[2] \]
Since $\mathcal{F}_0^\bullet $ is a bounded above complex of flat $\mathcal{O}_0$-modules, it is K-flat and may be used to compute derived tensor product. Moreover, the map of complexes $\mathcal{F}_0^\bullet \to \mathcal{K}_0^\bullet $ is a quasi-isomorphism by construction. Therefore the source and target of the map just constructed represent $K_0$ and $K_0 \otimes _{\mathcal{O}_0}^\mathbf {L} \mathcal{I}[2]$ and we obtain our map $\omega (K_0)$.
Let us show that this procedure is compatible with maps of complexes. Namely, let $\mathcal{L}_0^\bullet $ represent another object of $D^-(\mathcal{O}_0)$ and suppose that
\[ \mathcal{K}_0^\bullet \longrightarrow \mathcal{L}_0^\bullet \]
is a map of complexes. Apply Lemma 91.16.1 for the complex $\mathcal{L}_0^\bullet $, the flat modules $\mathcal{F}^ n$, the maps $\mathcal{F}^ n \to \mathcal{F}^{n + 1}$, and the compositions $\mathcal{F}^ n \to \mathcal{K}_0^ n \to \mathcal{L}_0^ n$ (we apologize for the reversal of letters used). We obtain flat modules $\mathcal{G}^ n$, maps $\mathcal{F}^ n \to \mathcal{G}^ n$, maps $\mathcal{G}^ n \to \mathcal{G}^{n + 1}$, and maps $\mathcal{G}^ n \to \mathcal{L}_0^ n$ with all properties as in the lemma. Then it is clear that
\[ \xymatrix{ \mathcal{F}_0^\bullet \ar[d] \ar[r] & (\mathcal{F}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I})[2] \ar[d] \\ \mathcal{G}_0^\bullet \ar[r] & (\mathcal{G}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I})[2] } \]
is a commutative diagram of complexes.
To see that $\omega (K_0)$ is well defined, suppose that we have two complexes $\mathcal{K}_0^\bullet $ and $(\mathcal{K}'_0)^\bullet $ of $\mathcal{O}_0$-modules representing $K_0$ and two systems $(\mathcal{F}^ n, d : \mathcal{F}^ n \to \mathcal{F}^{n + 1}, \mathcal{F}^ n \to \mathcal{K}_0^ n)$ and $((\mathcal{F}')^ n, d : (\mathcal{F}')^ n \to (\mathcal{F}')^{n + 1}, (\mathcal{F}')^ n \to \mathcal{K}_0^ n)$ as above. Then we can choose a complex $(\mathcal{K}''_0)^\bullet $ and quasi-isomorphisms $\mathcal{K}_0^\bullet \to (\mathcal{K}''_0)^\bullet $ and $(\mathcal{K}'_0)^\bullet \to (\mathcal{K}''_0)^\bullet $ realizing the fact that both complexes represent $K_0$ in the derived category. Next, we apply the result of the previous paragraph to
\[ (\mathcal{K}_0)^\bullet \oplus (\mathcal{K}'_0)^\bullet \longrightarrow (\mathcal{K}''_0)^\bullet \]
This produces a commutative diagram
\[ \xymatrix{ \mathcal{F}_0^\bullet \oplus (\mathcal{F}'_0)^\bullet \ar[d] \ar[r] & (\mathcal{F}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I})[2] \oplus ((\mathcal{F}'_0)^\bullet \otimes _{\mathcal{O}_0} \mathcal{I})[2] \ar[d] \\ \mathcal{G}_0^\bullet \ar[r] & (\mathcal{G}_0^\bullet \otimes _{\mathcal{O}_0} \mathcal{I})[2] } \]
Since the vertical arrows give quasi-isomorphisms on the summands we conclude the desired commutativity in $D(\mathcal{O}_0)$.
Having established well-definedness, the statement on compatibility with maps is a consequence of the result in the second paragraph.
$\square$
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