Lemma 10.133.6. 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 10.133.2 and the fact that $A \to B$ is a presentation of $B$ over $A$. $\square$

There are also:

• 7 comment(s) on Section 10.133: The naive cotangent complex

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).