The Stacks project

Remark 91.11.5. Adopt notation as in Remark 91.11.1. The arguments given there show that the differential

\[ H_2(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) \longrightarrow H_0(\mathcal{C}_{B/A}, \text{Tor}_1^\mathcal {O}(\mathcal{J}, \underline{B})) \]

of the spectral sequence is an isomorphism. Let $\mathcal{C}'_{B/A}$ denote the full subcategory of $\mathcal{C}_{B/A}$ consisting of surjective maps $P \to B$. The agreement of the cotangent complex with the naive cotangent complex (Lemma 91.11.3) shows that we have an exact sequence of sheaves

\[ 0 \to \underline{H_1(L_{B/A})} \to \mathcal{J}/\mathcal{J}^2 \xrightarrow {\text{d}} \Omega \to \underline{H_2(L_{B/A})} \to 0 \]

on $\mathcal{C}'_{B/A}$. It follows that $\mathop{\mathrm{Ker}}(d)$ and $\mathop{\mathrm{Coker}}(d)$ on the whole category $\mathcal{C}_{B/A}$ have vanishing higher homology groups, since these are computed by the homology groups of constant simplicial abelian groups by Lemma 91.4.1. Hence we conclude that

\[ H_ n(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2) \to H_ n(L_{B/A}) \]

is an isomorphism for all $n \geq 2$. Combined with the remark above we obtain the formula $H_2(L_{B/A}) = H_0(\mathcal{C}_{B/A}, \text{Tor}_1^\mathcal {O}(\mathcal{J}, \underline{B}))$.


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