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
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
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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