The Stacks project

Remark 91.11.1. Let $A \to B$ be a ring map. Working on $\mathcal{C}_{B/A}$ as in Section 91.4 let $\mathcal{J} \subset \mathcal{O}$ be the kernel of $\mathcal{O} \to \underline{B}$. Note that $L\pi _!(\mathcal{J}) = 0$ by Lemma 91.5.7. Set $\Omega = \Omega _{\mathcal{O}/A} \otimes _\mathcal {O} \underline{B}$ so that $L_{B/A} = L\pi _!(\Omega )$ by Lemma 91.4.3. It follows that $L\pi _!(\mathcal{J} \to \Omega ) = L\pi _!(\Omega ) = L_{B/A}$. Thus, for any object $U = (P \to B)$ of $\mathcal{C}_{B/A}$ we obtain a map

91.11.1.1
\begin{equation} \label{cotangent-equation-comparison-map-A} (J \to \Omega _{P/A} \otimes _ P B) \longrightarrow L_{B/A} \end{equation}

where $J = \mathop{\mathrm{Ker}}(P \to B)$ in $D(A)$, see Cohomology on Sites, Remark 21.39.4. Continuing in this manner, note that $L\pi _!(\mathcal{J} \otimes _\mathcal {O}^\mathbf {L} \underline{B}) = L\pi _!(\mathcal{J}) = 0$ by Lemma 91.5.6. Since $\text{Tor}_0^\mathcal {O}(\mathcal{J}, \underline{B}) = \mathcal{J}/\mathcal{J}^2$ the spectral sequence

\[ H_ p(\mathcal{C}_{B/A}, \text{Tor}_ q^\mathcal {O}(\mathcal{J}, \underline{B})) \Rightarrow H_{p + q}(\mathcal{C}_{B/A}, \mathcal{J} \otimes _\mathcal {O}^\mathbf {L} \underline{B}) = 0 \]

(dual of Derived Categories, Lemma 13.21.3) implies that $H_0(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) = 0$ and $H_1(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) = 0$. It follows that the complex of $\underline{B}$-modules $\mathcal{J}/\mathcal{J}^2 \to \Omega $ satisfies $\tau _{\geq -1}L\pi _!(\mathcal{J}/\mathcal{J}^2 \to \Omega ) = \tau _{\geq -1}L_{B/A}$. Thus, for any object $U = (P \to B)$ of $\mathcal{C}_{B/A}$ we obtain a map

91.11.1.2
\begin{equation} \label{cotangent-equation-comparison-map} (J/J^2 \to \Omega _{P/A} \otimes _ P B) \longrightarrow \tau _{\geq -1}L_{B/A} \end{equation}

in $D(B)$, see Cohomology on Sites, Remark 21.39.4.


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