Formally smooth morphism

Let $f\text{\hspace{0.17em}⁣}:X\to S$ be a morphism of schemes (Schemes and morphisms over a base) and let $i\text{\hspace{0.17em}⁣}:T_0\hookrightarrow T$ be a square-zero thickening, namely a closed immersion (Closed immersions of schemes) whose ideal sheaf $\mathcal{I}=\ker ({\mathcal{O}}_T\to i_{\ast }{\mathcal{O}}_{T_0})$ (Ideal sheaves) satisfies ${\mathcal{I}}^2=0$.

Formally smooth. The morphism $f$ is formally smooth if for every commutative $S$-diagram

$\begin{array}{ccc}{T}_0& \stackrel{a}{\to }& X\\ \downarrow i& & \downarrow f\\ T& \stackrel{b}{\to }& S\end{array}$

every point $t\in T$ has an open neighbourhood $U\subseteq T$ over which a lift exists: there is an $S$-morphism $U\to X$ extending $a{\mathrm{\mid }}_{T_0\cap U}$. In other words, lifts exist Zariski locally on the test scheme $T$, and there is no uniqueness requirement.

Equivalent formulation. Since the lifting problem is local on $T$, it is equivalent to require that the sheaf-theoretic lifting problem ${\mathrm{Hom}}_S(T,X)\to {\mathrm{Hom}}_S(T_0,X)$ be surjective locally on $T$; equivalently, by the universal property of the fibre product, the projection $X{\times }_{S}T\to T$ admits a section locally on $T$ over the given morphism $T_0\to X{\times }_{S}T$. No finite-type, finite-presentation or flatness hypothesis is imposed, and no uniqueness of lifts is asserted; in particular a formally smooth morphism need not be an open immersion or a submersion in any topological sense, and "formally smooth" is not by itself the same condition as "the relative differentials are locally free" nor as "smooth of finite presentation", which is treated elsewhere.