The Stacks project

Lemma 37.12.1. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Assume that $S$ 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

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

    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,

  3. same as in (2) but with $B' \to B$ ranging over small extensions (see Algebra, Definition 10.140.1), and

  4. same as in (2) but with $B' \to B$ ranging over small extensions such that $\alpha $ induces an isomorphism $\kappa (x) \to \kappa (\mathfrak m)$ where $\mathfrak m \subset B$ is the maximal ideal.

Proof. Choose an affine neighbourhood $V \subset S$ of $f(x)$ and choose an affine neighbourhood $U \subset X$ of $x$ such that $f(U) \subset V$. For any “test” diagram as in (2) the morphism $\alpha $ will map $\mathop{\mathrm{Spec}}(B)$ into $U$ and the morphism $\beta $ will map $\mathop{\mathrm{Spec}}(B')$ into $V$ (see Schemes, Section 26.13). Hence the lemma reduces to the morphism $f|_ U : U \to V$ of affines. (Indeed, $V$ is Noetherian and $f|_ U$ is of finite type, see Properties, Lemma 28.5.2 and Morphisms, Lemma 29.15.2.) In this affine case the lemma is identical to Algebra, Lemma 10.140.2. $\square$


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Your email address will not be published.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