## 90.11 Comparison with the naive cotangent complex

The naive cotangent complex was introduced in Algebra, Section 10.134.

The first case is where we have a surjection of rings.

slogan
Lemma 90.11.2. 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 (90.11.1.2) 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 90.11.1 we obtain what we want using Lemma 90.4.4.

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).
$\square$

Lemma 90.11.3. Let $A \to B$ be a ring map. Then $\tau _{\geq -1}L_{B/A}$ is canonically quasi-isomorphic to the naive cotangent complex.

**Proof.**
Consider $P = A[B] \to B$ with kernel $I$. The naive cotangent complex $\mathop{N\! L}\nolimits _{B/A}$ of $B$ over $A$ is the complex $I/I^2 \to \Omega _{P/A} \otimes _ P B$, see Algebra, Definition 10.134.1. Observe that in (90.11.1.2) we have already constructed a canonical map

\[ c : \mathop{N\! L}\nolimits _{B/A} \longrightarrow \tau _{\geq -1}L_{B/A} \]

Consider the distinguished triangle (90.7.0.1)

\[ L_{P/A} \otimes _ P^\mathbf {L} B \to L_{B/A} \to L_{B/P} \to (L_{P/A} \otimes _ P^\mathbf {L} B)[1] \]

associated to the ring maps $A \to A[B] \to B$. We know that $L_{P/A} = \Omega _{P/A}[0] = \mathop{N\! L}\nolimits _{P/A}$ in $D(P)$ (Lemma 90.4.7 and Algebra, Lemma 10.134.3) and that $\tau _{\geq -1}L_{B/P} = I/I^2[1] = \mathop{N\! L}\nolimits _{B/P}$ in $D(B)$ (Lemma 90.11.2 and Algebra, Lemma 10.134.6). To show $c$ is a quasi-isomorphism it suffices by Algebra, Lemma 10.134.4 and the long exact cohomology sequence associated to the distinguished triangle to show that the maps $L_{P/A} \to L_{B/A} \to L_{B/P}$ are compatible on cohomology groups with the corresponding maps $\mathop{N\! L}\nolimits _{P/A} \to \mathop{N\! L}\nolimits _{B/A} \to \mathop{N\! L}\nolimits _{B/P}$ of the naive cotangent complex. We omit the verification.
$\square$

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