The Stacks project

Remark 91.12.5. In the situation of Theorem 91.12.4 let $I = \mathop{\mathrm{Ker}}(A \to B)$. Then $H^{-1}(L_{B/A}) = H_1(\mathcal{C}_{B/A}, \Omega ) = I/I^2$, see Lemma 91.11.2. Hence $H_ k(\mathcal{C}_{B/A}, \text{Sym}^ k(\Omega )) = \wedge ^ k_ B(I/I^2)$ by Remark 91.12.2. Thus the $E_1$-page looks like

\[ \begin{matrix} B \\ 0 \\ 0 & I/I^2 \\ 0 & H^{-2}(L_{B/A}) \\ 0 & H^{-3}(L_{B/A}) & \wedge ^2(I/I^2) \\ 0 & H^{-4}(L_{B/A}) & H_3(\mathcal{C}_{B/A}, \text{Sym}^2(\Omega )) \\ 0 & H^{-5}(L_{B/A}) & H_4(\mathcal{C}_{B/A}, \text{Sym}^2(\Omega )) & \wedge ^3(I/I^2) \end{matrix} \]

with horizontal differential. Thus we obtain edge maps $\text{Tor}_ i^ A(B, B) \to H^{-i}(L_{B/A})$, $i > 0$ and $\wedge ^ i_ B(I/I^2) \to \text{Tor}_ i^ A(B, B)$. Finally, we have $\text{Tor}_1^ A(B, B) = I/I^2$ and there is a five term exact sequence

\[ \text{Tor}_3^ A(B, B) \to H^{-3}(L_{B/A}) \to \wedge ^2_ B(I/I^2) \to \text{Tor}_2^ A(B, B) \to H^{-2}(L_{B/A}) \to 0 \]

of low degree terms.


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