The Stacks project

Lemma 37.6.7. Let $f : X \to S$ be a morphism of schemes. Then $f$ is formally unramified if and only if $\Omega _{X/S} = 0$.

Proof. We recall some of the arguments of the proof of Morphisms, Lemma 29.32.5. Let $W \subset X \times _ S X$ be an open such that $\Delta : X \to X \times _ S X$ induces a closed immersion into $W$. Let $\mathcal{J} \subset \mathcal{O}_ W$ be the ideal sheaf of this closed immersion. Let $X' \subset W$ be the closed subscheme defined by the quasi-coherent sheaf of ideals $\mathcal{J}^2$. Consider the two morphisms $p_1, p_2 : X' \to X$ induced by the two projections $X \times _ S X \to X$. Note that $p_1$ and $p_2$ agree when composed with $\Delta : X \to X'$ and that $X \to X'$ is a closed immersion defined by a an ideal whose square is zero. Moreover there is a short exact sequence

\[ 0 \to \mathcal{J}/\mathcal{J}^2 \to \mathcal{O}_{X'} \to \mathcal{O}_ X \to 0 \]

and $\Omega _{X/S} = \mathcal{J}/\mathcal{J}^2$. Moreover, $\mathcal{J}/\mathcal{J}^2$ is generated by the local sections $p_1^\sharp (f) - p_2^\sharp (f)$ for $f$ a local section of $\mathcal{O}_ X$.

Suppose that $f : X \to S$ is formally unramified. By assumption this means that $p_1 = p_2$ when restricted to any affine open $T' \subset X'$. Hence $p_1 = p_2$. By what was said above we conclude that $\Omega _{X/S} = \mathcal{J}/\mathcal{J}^2 = 0$.

Conversely, suppose that $\Omega _{X/S} = 0$. Then $X' = X$. Take any pair of morphisms $f'_1, f'_2 : T' \to X$ fitting as dotted arrows in the diagram of Definition 37.6.1. This gives a morphism $(f'_1, f'_2) : T' \to X \times _ S X$. Since $f'_1|_ T = f'_2|_ T$ and $|T| =|T'|$ we see that the image of $T'$ under $(f'_1, f'_2)$ is contained in the open $W$ chosen above. Since $(f'_1, f'_2)(T) \subset \Delta (X)$ and since $T$ is defined by an ideal of square zero in $T'$ we see that $(f'_1, f'_2)$ factors through $X'$. As $X' = X$ we conclude $f_1' = f'_2$ as desired. $\square$

Comments (0)

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Your email address will not be published.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 02H9. Beware of the difference between the letter 'O' and the digit '0'.