The Stacks project

Lemma 91.18.10. 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 91.11.4 which induces an isomorphism on $H^0$ and $H^{-1}$. Since $f^{-1}L_{\mathcal{B}'/\mathcal{A}'} = L_{\mathcal{B}/\mathcal{A}}$ (Lemma 91.18.3) 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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