Lemma 10.134.6. Let $A \to B$ be a surjective ring map with kernel $I$. Then $\mathop{N\! L}\nolimits _{B/A}$ is homotopy equivalent to the chain complex $(I/I^2 \to 0)$ with $I/I^2$ in degree $1$. In particular $H_1(L_{B/A}) = I/I^2$.

**Proof.**
Follows from Lemma 10.134.2 and the fact that $A \to B$ is a presentation of $B$ over $A$.
$\square$

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## Comments (1)

Comment #5946 by Zhouhang MAO on

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