Lemma 10.134.16. Let $R \to S$ be a ring map of finite type. Let $g \in S$. For any presentations $\alpha : R[x_1, \ldots , x_ n] \to S$, and $\beta : R[y_1, \ldots , y_ m] \to S_ g$ we have

$(I/I^2)_ g \oplus S^{\oplus m}_ g \cong J/J^2 \oplus S_ g^{\oplus n}$

as $S_ g$-modules where $I = \mathop{\mathrm{Ker}}(\alpha )$ and $J = \mathop{\mathrm{Ker}}(\beta )$.

Proof. By Lemma 10.134.15, we see that it suffices to prove this for a single choice of $\alpha$ and $\beta$. Thus we may take $\beta$ the presentation of Lemma 10.134.12 and the result is clear. $\square$

There are also:

• 9 comment(s) on Section 10.134: 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).