Lemma 76.13.5 (04AL)—The Stacks project

Processing math: 100%

Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.13: Formally smooth, étale, unramified transformations
Lemma 76.13.5 (cite)

numberstags

Lemma 76.13.5. Let $S$ be a scheme. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ . Let $a : F \to G$ be a representable transformation of functors.

If $a$ is smooth then $a$ is formally smooth.
If $a$ is étale, then $a$ is formally étale.
If $a$ is unramified, then $a$ is formally unramified.

Proof. Consider a solid commutative diagram

$\xymatrix{ F \ar[d]_ a & T \ar[d]^ i \ar[l] \\ G & T' \ar[l] \ar@{-->}[lu] }$

as in Definition 76.13.1. Then $F \times _ G T'$ is a scheme smooth (resp. étale, resp. unramified) over $T'$ . Hence by More on Morphisms, Lemma 37.11.7 (resp. Lemma 37.8.9, resp. Lemma 37.6.8) we can fill in (resp. uniquely fill in, resp. fill in at most one way) the dotted arrow in the diagram

$\xymatrix{ F \times _ G T' \ar[d] & T \ar[d]^ i \ar[l] \\ T' & T' \ar[l] \ar@{-->}[lu] }$

an hence we also obtain the corresponding assertion in the first diagram. $\square$

Comments (0)

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).

Comments (0)

Post a comment