The Stacks project

Lemma 33.18.4. Let $f : X \to S$ be a finite type morphism of affine schemes. Let $x \in X$ with image $s \in S$. Let

\[ r = \dim _{\kappa (x)} \Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x) = \dim _{\kappa (x)} \Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) = \dim _{\kappa (x)} T_{X/S, x} \]

Then there exists a factorization

\[ X \xrightarrow {\pi } \mathbf{A}^ r_ S \to S \]

of $f$ such that $\pi $ is unramified at $x$.

Proof. By Morphisms, Lemma 29.32.12 the first dimension is finite. The first equality follows as the restriction of $\Omega _{X/S}$ to the fibre is the module of differentials from Morphisms, Lemma 29.32.10. The last equality follows from Lemma 33.16.4. Thus we see that the statement makes sense.

To prove the lemma write $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$ and let $A \to B$ be the ring map corresponding to $f$. Let $\mathfrak q \subset B$ be the prime ideal corresponding to $x$. Choose a surjection of $A$-algebras $A[x_1, \ldots , x_ t] \to B$. Since $\Omega _{B/A}$ is generated by $\text{d}x_1, \ldots , \text{d}x_ t$ we see that their images in $\Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x)$ generate this as a $\kappa (x)$-vector space. After renumbering we may assume that $\text{d}x_1, \ldots , \text{d}x_ r$ map to a basis of $\Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x)$. We claim that $P = A[x_1, \ldots , x_ r] \to B$ is unramified at $\mathfrak q$. To see this it suffices to show that $\Omega _{B/P, \mathfrak q} = 0$ (Algebra, Lemma 10.151.3). Note that $\Omega _{B/P}$ is the quotient of $\Omega _{B/A}$ by the submodule generated by $\text{d}x_1, \ldots , \text{d}x_ r$. Hence $\Omega _{B/P, \mathfrak q} \otimes _{B_\mathfrak q} \kappa (\mathfrak q) = 0$ by our choice of $x_1, \ldots , x_ r$. By Nakayama's lemma, more precisely Algebra, Lemma 10.20.1 part (2) which applies as $\Omega _{B/P}$ is finite (see reference above), we conclude that $\Omega _{B/P, \mathfrak q} = 0$. $\square$

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Your email address will not be published.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