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 $A$.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #5946 by Zhouhang MAO on

Comment #6132 by Johan on

There are also: