The Stacks project

Lemma 10.131.10. In diagram (, suppose that $S \to S'$ is surjective with kernel $I \subset S$, and assume that $R' = R$. Moreover, assume that there exists an $R$-algebra map $S' \to S$ which is a right inverse to $S \to S'$. Then the exact sequence of $S'$-modules of Lemma 10.131.9 turns into a short exact sequence

\[ 0 \longrightarrow I/I^2 \longrightarrow \Omega _{S/R} \otimes _ S S' \longrightarrow \Omega _{S'/R} \longrightarrow 0 \]

which is even a split short exact sequence.

Proof. Let $\beta : S' \to S$ be the right inverse to the surjection $\alpha : S \to S'$, so $S = I \oplus \beta (S')$. Clearly we can use $\beta : \Omega _{S'/R} \to \Omega _{S/R}$, to get a right inverse to the map $\Omega _{S/R} \otimes _ S S' \to \Omega _{S'/R}$. On the other hand, consider the map

\[ D : S \longrightarrow I/I^2, \quad x \longmapsto x - \beta (\alpha (x)) \]

It is easy to show that $D$ is an $R$-derivation (omitted). Moreover $x D(s) = 0$ if $x \in I, s \in S$. Hence, by the universal property $D$ induces a map $\tau : \Omega _{S/R} \otimes _ S S' \to I/I^2$. We omit the verification that it is a left inverse to $\text{d} : I/I^2 \to \Omega _{S/R} \otimes _ S S'$. Hence we win. $\square$

Comments (3)

Comment #1086 by Nuno Cardoso on

Typo: It should be "assume that there exists an -algebra map which is a right inverse to " instead of "assume that there exists an -algebra map which is a right inverse to ".

Comment #8559 by on

The readers might find interesting to know the following: In Matsumura's Commutative Algebra (not to be confused with Commutative Ring Theory), Ch. 10, (26.I), in Theorem 58 it is stated a sufficient and equivalent condition for the map from Lemma 10.131.9 to have an -linear retraction (namely, that there is an -algebra map which is a right inverse to the canonical map ). It is immediate to verify that "there exists an -algebra map which is a right inverse to " implies the condition between parentheses.

There are also:

  • 14 comment(s) on Section 10.131: Differentials

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02HP. Beware of the difference between the letter 'O' and the digit '0'.