The Stacks project

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. $\square$

Comments (0)

There are also:

  • 7 comment(s) on Section 37.11: Formally smooth morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 02H6. Beware of the difference between the letter 'O' and the digit '0'.