Lemma 10.138.11 (06A7)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.138: Formally smooth maps
Lemma 10.138.11 (cite)

Lemma 10.138.11. Let $A \to B \to C$ be ring maps. Assume $A \to C$ is surjective (so also $B \to C$ is) and $A \to B$ formally smooth. Denote $I = \mathop{\mathrm{Ker}}(A \to C)$ and $J = \mathop{\mathrm{Ker}}(B \to C)$ . Then the sequence

$0 \to I/I^2 \to J/J^2 \to \Omega _{B/A} \otimes _ B B/J \to 0$

of Lemma 10.134.7 is split exact.

Proof. Since $A \to B$ is formally smooth there exists a ring map $\sigma : B \to A/I^2$ whose composition with $A \to B$ equals the quotient map $A \to A/I^2$ . Then $\sigma$ induces a map $J/J^2 \to I/I^2$ which is inverse to the map $I/I^2 \to J/J^2$ . $\square$

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