The Stacks project

Remark 15.84.2. The following two statements follow from Lemma 15.84.1, Algebra, Definition 10.137.1, and Algebra, Proposition 10.138.8.

1. A ring map $A \to B$ is smooth if and only if $A \to B$ is of finite presentation and $\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N) = 0$ for every $B$-module $N$.

2. A ring map $A \to B$ is formally smooth if and only if $\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}, N) = 0$ for every $B$-module $N$.