The Stacks project

Lemma 76.20.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Assume that $Y$ is locally Noetherian and $f$ locally of finite type. The following are equivalent:

  1. $f$ is smooth at $x$,

  2. for every solid commutative diagram

    \[ \xymatrix{ X \ar[d]_ f & \mathop{\mathrm{Spec}}(B) \ar[d]^ i \ar[l]^-\alpha \\ Y & \mathop{\mathrm{Spec}}(B') \ar[l]_-{\beta } \ar@{-->}[lu] } \]

    where $B' \to B$ is a surjection of local rings with $\mathop{\mathrm{Ker}}(B' \to B)$ of square zero, and $\alpha $ mapping the closed point of $\mathop{\mathrm{Spec}}(B)$ to $x$ there exists a dotted arrow making the diagram commute, and

  3. same as in (2) but with $B' \to B$ ranging over small extensions (see Algebra, Definition 10.141.1).

Proof. Condition (1) means there is an open subspace $X' \subset X$ such that $X' \to Y$ is smooth. Hence (1) implies conditions (2) and (3) by Lemma 76.19.6. Condition (2) implies condition (3) trivially. Assume (3). Choose a commutative diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \\ Y & V \ar[l] } \]

with $U$ and $V$ affine, horizontal arrows étale and such that there is a point $u \in U$ mapping to $x$. Next, consider a diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] & \mathop{\mathrm{Spec}}(B) \ar[d]^ i \ar[l]^-\alpha \\ Y & V \ar[l] & \mathop{\mathrm{Spec}}(B') \ar[l]_-{\beta } } \]

as in (3) but for $u \in U \to V$. Let $\gamma : \mathop{\mathrm{Spec}}(B') \to X$ be the arrow we get from our assumption that (3) holds for $X$. Because $U \to X$ is étale and hence formally étale (Lemma 76.16.8) the morphism $\gamma $ has a unique lift to $U$ compatible with $\alpha $. Then because $V \to Y$ is étale hence formally étale this lift is compatible with $\beta $. Hence (3) holds for $u \in U \to V$ and we conclude that $U \to V$ is smooth at $u$ by More on Morphisms, Lemma 37.12.1. This proves that $X \to Y$ is smooth at $x$, thereby finishing the proof. $\square$


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Your email address will not be published.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