The Stacks project

Remark 92.7.6. Let $A \to B \to C$ be ring maps with $B \to C$ injective. Recall the notation $P_\bullet $, $Q_\bullet $, $\overline{Q}_\bullet $ of Remark 92.7.5. Let $R_\bullet $ be the standard resolution of $C$ over $B$. In this remark we explain how to get the canonical identification of $\Omega _{\overline{Q}_\bullet /B} \otimes _{\overline{Q}_\bullet } C$ with $L_{C/B} = \Omega _{R_\bullet /B} \otimes _{R_\bullet } C$. Let $S_\bullet \to B$ be the standard resolution of $B$ over $B$. Note that the functoriality map $S_\bullet \to R_\bullet $ identifies $R_ n$ as a polynomial algebra over $S_ n$ because $B \to C$ is injective. For example in degree $0$ we have the map $B[B] \to B[C]$, in degree $1$ the map $B[B[B]] \to B[B[C]]$, and so on. Thus $\overline{R}_\bullet = R_\bullet \otimes _{S_\bullet } B$ is a simplicial polynomial algebra over $B$ as well and it follows (as in Remark 92.7.5) from Cohomology on Sites, Lemma 21.39.12 that $\overline{R}_\bullet \to C$ is a resolution. Since we have a commutative diagram

\[ \xymatrix{ Q_\bullet \ar[r] & R_\bullet \\ P_\bullet \ar[u] \ar[r] & S_\bullet \ar[u] \ar[r] & B } \]

we obtain a canonical map $\overline{Q}_\bullet = Q_\bullet \otimes _{P_\bullet } B \to \overline{R}_\bullet $. Thus the maps

\[ L_{C/B} = \Omega _{R_\bullet /B} \otimes _{R_\bullet } C \longrightarrow \Omega _{\overline{R}_\bullet /B} \otimes _{\overline{R}_\bullet } C \longleftarrow \Omega _{\overline{Q}_\bullet /B} \otimes _{\overline{Q}_\bullet } C \]

are quasi-isomorphisms (Remark 92.5.5) and composing one with the inverse of the other gives the desired identification.

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