Proposition 10.138.8. Let $R \to S$ be a ring map. Consider a formally smooth $R$-algebra $P$ and a surjection $P \to S$ with kernel $J$. The following are equivalent

$S$ is formally smooth over $R$,

for some $P \to S$ as above there exists a section to $P/J^2 \to S$,

for all $P \to S$ as above there exists a section to $P/J^2 \to S$,

for some $P \to S$ as above the sequence $0 \to J/J^2 \to \Omega _{P/R} \otimes S \to \Omega _{S/R} \to 0$ is split exact,

for all $P \to S$ as above the sequence $0 \to J/J^2 \to \Omega _{P/R} \otimes S \to \Omega _{S/R} \to 0$ is split exact, and

the naive cotangent complex $\mathop{N\! L}\nolimits _{S/R}$ is quasi-isomorphic to a projective $S$-module placed in degree $0$.

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