Lemma 92.18.10. Let \mathcal{C} be a site. Let \mathcal{A} \to \mathcal{B} be a homomorphism of sheaves of rings on \mathcal{C}. There is a canonical map L_{\mathcal{B}/\mathcal{A}} \to \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} which identifies the naive cotangent complex with the truncation \tau _{\geq -1}L_{\mathcal{B}/\mathcal{A}}.
Proof. Let \mathcal{P}_\bullet be the standard resolution of \mathcal{B} over \mathcal{A}. Let \mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{A}[\mathcal{B}] \to \mathcal{B}). Recall that \mathcal{P}_0 = \mathcal{A}[\mathcal{B}]. The map of the lemma is given by the commutative diagram
\xymatrix{ L_{\mathcal{B}/\mathcal{A}} \ar[d] & \ldots \ar[r] & \Omega _{\mathcal{P}_2/\mathcal{A}} \otimes _{\mathcal{P}_2} \mathcal{B} \ar[r] \ar[d] & \Omega _{\mathcal{P}_1/\mathcal{A}} \otimes _{\mathcal{P}_1} \mathcal{B} \ar[r] \ar[d] & \Omega _{\mathcal{P}_0/\mathcal{A}} \otimes _{\mathcal{P}_0} \mathcal{B} \ar[d] \\ \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} & \ldots \ar[r] & 0 \ar[r] & \mathcal{I}/\mathcal{I}^2 \ar[r] & \Omega _{\mathcal{P}_0/\mathcal{A}} \otimes _{\mathcal{P}_0} \mathcal{B} }
We construct the downward arrow with target \mathcal{I}/\mathcal{I}^2 by sending a local section \text{d}f \otimes b to the class of (d_0(f) - d_1(f))b in \mathcal{I}/\mathcal{I}^2. Here d_ i : \mathcal{P}_1 \to \mathcal{P}_0, i = 0, 1 are the two face maps of the simplicial structure. This makes sense as d_0 - d_1 maps \mathcal{P}_1 into \mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{P}_0 \to \mathcal{B}). We omit the verification that this rule is well defined. Our map is compatible with the differential \Omega _{\mathcal{P}_1/\mathcal{A}} \otimes _{\mathcal{P}_1} \mathcal{B} \to \Omega _{\mathcal{P}_0/\mathcal{A}} \otimes _{\mathcal{P}_0} \mathcal{B} as this differential maps a local section \text{d}f \otimes b to \text{d}(d_0(f) - d_1(f)) \otimes b. Moreover, the differential \Omega _{\mathcal{P}_2/\mathcal{A}} \otimes _{\mathcal{P}_2} \mathcal{B} \to \Omega _{\mathcal{P}_1/\mathcal{A}} \otimes _{\mathcal{P}_1} \mathcal{B} maps a local section \text{d}f \otimes b to \text{d}(d_0(f) - d_1(f) + d_2(f)) \otimes b which are annihilated by our downward arrow. Hence a map of complexes.
To see that our map induces an isomorphism on the cohomology sheaves H^0 and H^{-1} we argue as follows. Let \mathcal{C}' be the site with the same underlying category as \mathcal{C} but endowed with the chaotic topology. Let f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}') be the morphism of topoi whose pullback functor is sheafification. Let \mathcal{A}' \to \mathcal{B}' be the given map, but thought of as a map of sheaves of rings on \mathcal{C}'. The construction above gives a map L_{\mathcal{B}'/\mathcal{A}'} \to \mathop{N\! L}\nolimits _{\mathcal{B}'/\mathcal{A}'} on \mathcal{C}' whose value over any object U of \mathcal{C}' is just the map
L_{\mathcal{B}(U)/\mathcal{A}(U)} \to \mathop{N\! L}\nolimits _{\mathcal{B}(U)/\mathcal{A}(U)}
of Remark 92.11.4 which induces an isomorphism on H^0 and H^{-1}. Since f^{-1}L_{\mathcal{B}'/\mathcal{A}'} = L_{\mathcal{B}/\mathcal{A}} (Lemma 92.18.3) and f^{-1}\mathop{N\! L}\nolimits _{\mathcal{B}'/\mathcal{A}'} = \mathop{N\! L}\nolimits _{\mathcal{B}/\mathcal{A}} (Modules on Sites, Lemma 18.35.3) the lemma is proved. \square
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