The Stacks project

Lemma 37.12.2. Let $f : X \to S$ be a morphism of schemes. Assume that $S$ is locally Noetherian and $f$ locally of finite type. The following are equivalent:

  1. $f$ is smooth,

  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 small extension of Artinian local rings and $\beta $ of finite type (!) there exists a dotted arrow making the diagram commute.

Proof. If $f$ is smooth, then the infinitesimal lifting criterion (Lemma 37.11.7) says $f$ is formally smooth and (2) holds.

Assume (2). The set of points $x \in X$ where $f$ is not smooth forms a closed subset $T$ of $X$. By the discussion in Morphisms, Section 29.16, if $T \not= \emptyset $ there exists a point $x \in T \subset X$ such that the morphism

\[ \mathop{\mathrm{Spec}}(\kappa (x)) \to X \to S \]

is of finite type (namely, pick any point $x$ of $T$ which is closed in an affine open of $X$). By Morphisms, Lemma 29.16.2 given any local Artinian ring $B'$ with residue field $\kappa (x)$ then any morphism $\beta : \mathop{\mathrm{Spec}}(B') \to S$ is of finite type. Thus we see that all the diagrams used in Lemma 37.12.1 (4) correspond to diagrams as in the current lemma (2). Whence $X \to S$ is smooth a $x$ a contradiction. $\square$

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Your email address will not be published.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