The Stacks project

Lemma 90.4.3. Let $A \to B$ be a ring map. Let $\pi $ and $i$ be as in (90.4.0.1). There is a canonical isomorphism

\[ L_{B/A} = L\pi _!(Li^*\Omega _{\mathcal{O}/A}) = L\pi _!(i^*\Omega _{\mathcal{O}/A}) = L\pi _!(\Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B}) \]

in $D(B)$.

Proof. For an object $\alpha : P \to B$ of the category $\mathcal{C}$ the module $\Omega _{P/A}$ is a free $P$-module. Thus $\Omega _{\mathcal{O}/A}$ is a flat $\mathcal{O}$-module. Hence $Li^*\Omega _{\mathcal{O}/A} = i^*\Omega _{\mathcal{O}/A}$ is the sheaf of $\underline{B}$-modules which associates to $\alpha : P \to A$ the $B$-module $\Omega _{P/A} \otimes _{P, \alpha } B$. By Lemma 90.4.2 we see that the right hand side is computed by the value of this sheaf on the standard resolution which is our definition of the left hand side (Definition 90.3.2). $\square$


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