Proposition 16.5.3. Let $R \to \Lambda $ be a ring map. Let $I \subset R$ be an ideal. Assume that
$I$ is nilpotent,
$\Lambda /I\Lambda $ is a filtered colimit of smooth $R/I$-algebras, and
$R \to \Lambda $ is flat.
Ind-smoothness of an algebra is stable under infinitesimal deformations
Then $\Lambda $ is a filtered colimit of smooth $R$-algebras.
Proof. Since $I^ n = 0$ for some $n$, it follows by induction on $n$ that it suffices to consider the case where $I^2 = 0$. Let $\varphi : A \to \Lambda $ be an $R$-algebra map with $A$ of finite presentation over $R$. We have to find a factorization $A \to B \to \Lambda $ with $B$ smooth over $R$, see Algebra, Lemma 10.127.4. By Lemma 16.5.1 we may assume that $A = B/J$ with $B$ smooth over $R$ and $J \subset IB$ a finitely generated ideal. By Lemma 16.5.2 we can find a commutative diagram
\[ \xymatrix{ B \ar[rr]_\alpha \ar[rd]_\varphi & & B' \ar[ld]^\beta \\ & \Lambda } \]
of $R$-algebras with $B'$ smooth over $R$ such that $\alpha (J) = 0$. Thus $\alpha $ factors as $B \to A \to B'$ and the proof is complete. $\square$
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