The cohomology of the cotangent complex of a surjective ring map is trivial in degree zero; it is the kernel modulo its square in degree $-1$.

Lemma 91.11.2. Let $A \to B$ be a surjective ring map with kernel $I$. Then $H^0(L_{B/A}) = 0$ and $H^{-1}(L_{B/A}) = I/I^2$. This isomorphism comes from the map (91.11.1.2) for the object $(A \to B)$ of $\mathcal{C}_{B/A}$.

Proof. We will show below (using the surjectivity of $A \to B$) that there exists a short exact sequence

$0 \to \pi ^{-1}(I/I^2) \to \mathcal{J}/\mathcal{J}^2 \to \Omega \to 0$

of sheaves on $\mathcal{C}_{B/A}$. Taking $L\pi _!$ and the associated long exact sequence of homology, and using the vanishing of $H_1(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2)$ and $H_0(\mathcal{C}_{B/A}, \mathcal{J}/\mathcal{J}^2)$ shown in Remark 91.11.1 we obtain what we want using Lemma 91.4.4.

What is left is to verify the local statement mentioned above. For every object $U = (P \to B)$ of $\mathcal{C}_{B/A}$ we can choose an isomorphism $P = A[E]$ such that the map $P \to B$ maps each $e \in E$ to zero. Then $J = \mathcal{J}(U) \subset P = \mathcal{O}(U)$ is equal to $J = IP + (e; e \in E)$. The value on $U$ of the short sequence of sheaves above is the sequence

$0 \to I/I^2 \to J/J^2 \to \Omega _{P/A} \otimes _ P B \to 0$

Verification omitted (hint: the only tricky point is that $IP \cap J^2 = IJ$; which follows for example from More on Algebra, Lemma 15.30.9). $\square$

Comment #1278 by on

Suggested slogan: The cohomology of the cotangent complex of a ring map is trivial in degree zero; it is the kernel modulo its square in degree -1.

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