Lemma 37.11.7 (Infinitesimal lifting criterion). Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

The morphism $f$ is smooth, and

the morphism $f$ is locally of finite presentation and formally smooth.

Lemma 37.11.7 (Infinitesimal lifting criterion). Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

The morphism $f$ is smooth, and

the morphism $f$ is locally of finite presentation and formally smooth.

**Proof.**
Assume $f : X \to S$ is locally of finite presentation and formally smooth. Consider a pair of affine opens $\mathop{\mathrm{Spec}}(A) = U \subset X$ and $\mathop{\mathrm{Spec}}(R) = V \subset S$ such that $f(U) \subset V$. By Lemma 37.11.5 we see that $U \to V$ is formally smooth. By Lemma 37.11.6 we see that $R \to A$ is formally smooth. By Morphisms, Lemma 29.21.2 we see that $R \to A$ is of finite presentation. By Algebra, Proposition 10.138.13 we see that $R \to A$ is smooth. Hence by the definition of a smooth morphism we see that $X \to S$ is smooth.

Conversely, assume that $f : X \to S$ is smooth. Consider a solid commutative diagram

\[ \xymatrix{ X \ar[d]_ f & T \ar[d]^ i \ar[l]^ a \\ S & T' \ar[l] \ar@{-->}[lu] } \]

as in Definition 37.11.1. We will show the dotted arrow exists thereby proving that $f$ is formally smooth.

Let $\mathcal{F}$ be the sheaf of sets on $T'$ of Lemma 37.9.4 in the special case discussed in Remark 37.9.6. Let

\[ \mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ T}(a^*\Omega _{X/S}, \mathcal{C}_{T/T'}) \]

be the sheaf of $\mathcal{O}_ T$-modules with action $\mathcal{H} \times \mathcal{F} \to \mathcal{F}$ as in Lemma 37.9.5. Our goal is simply to show that $\mathcal{F}(T) \not= \emptyset $. In other words we are trying to show that $\mathcal{F}$ is a trivial $\mathcal{H}$-torsor on $T$ (see Cohomology, Section 20.4). There are two steps: (I) To show that $\mathcal{F}$ is a torsor we have to show that $\mathcal{F}_ t \not= \emptyset $ for all $t \in T$ (see Cohomology, Definition 20.4.1). (II) To show that $\mathcal{F}$ is the trivial torsor it suffices to show that $H^1(T, \mathcal{H}) = 0$ (see Cohomology, Lemma 20.4.3 – we may use either cohomology of $\mathcal{H}$ as an abelian sheaf or as an $\mathcal{O}_ T$-module, see Cohomology, Lemma 20.13.3).

First we prove (I). To see this, for every $t \in T$ we can choose an affine open $U \subset T$ neighbourhood of $t$ such that $a(U)$ is contained in an affine open $\mathop{\mathrm{Spec}}(A) = W \subset X$ which maps to an affine open $\mathop{\mathrm{Spec}}(R) = V \subset S$. By Morphisms, Lemma 29.34.2 the ring map $R \to A$ is smooth. Hence by Algebra, Proposition 10.138.13 the ring map $R \to A$ is formally smooth. Lemma 37.11.6 in turn implies that $W \to V$ is formally smooth. Hence we can lift $a|_ U : U \to W$ to a $V$-morphism $a' : U' \to W \subset X$ showing that $\mathcal{F}(U) \not= \emptyset $.

Finally we prove (II). By Morphisms, Lemma 29.32.13 we see that $\Omega _{X/S}$ is of finite presentation (it is even finite locally free by Morphisms, Lemma 29.34.12). Hence $a^*\Omega _{X/S}$ is of finite presentation (see Modules, Lemma 17.11.4). Hence the sheaf $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ T}(a^*\Omega _{X/S}, \mathcal{C}_{T/T'})$ is quasi-coherent by the discussion in Schemes, Section 26.24. Thus by Cohomology of Schemes, Lemma 30.2.2 we have $H^1(T, \mathcal{H}) = 0$ as desired. $\square$

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