The Stacks project

Lemma 37.62.2 (Variant over curves). Let $f : X \to S$ be a flat, finite type morphism of schemes. Assume $S$ is Nagata, integral with function field $K$, and regular of dimension $1$. Then there exists a finite extension $L/K$ such that in the diagram

\[ \xymatrix{ Y \ar[rd]_ g \ar[r]_-\nu & X \times _ S T \ar[d] \ar[r] & X \ar[d]_ f \\ & T \ar[r] & S } \]

the morphism $g$ is smooth at all generic points of fibres. Here $T$ is the normalization of $S$ in $\mathop{\mathrm{Spec}}(L)$ and $\nu : Y \to X \times _ S T$ is the normalization.

Proof. Choose a finite affine open covering $S = \bigcup \mathop{\mathrm{Spec}}(A_ i)$. Then $K$ is equal to the fraction field of $A_ i$ for all $i$. Let $X_ i = X \times _ S \mathop{\mathrm{Spec}}(A_ i)$. Choose $L_ i/K$ as in Theorem 37.62.1 for the morphism $X_ i \to \mathop{\mathrm{Spec}}(A_ i)$. Let $B_ i \subset L_ i$ be the integral closure of $A_ i$ and let $Y_ i$ be the normalized base change of $X$ to $B_ i$. Let $L/K$ be a finite extension dominating each $L_ i$. Let $T_ i \subset T$ be the inverse image of $\mathop{\mathrm{Spec}}(A_ i)$. For each $i$ we get a commutative diagram

\[ \xymatrix{ g^{-1}(T_ i) \ar[r] \ar[d] & Y_ i \ar[r] \ar[d] & X \times _ S \mathop{\mathrm{Spec}}(A_ i) \ar[d] \\ T_ i \ar[r] & \mathop{\mathrm{Spec}}(B_ i) \ar[r] & \mathop{\mathrm{Spec}}(A_ i) } \]

and in fact the left hand square is a normalized base change as discussed at the beginning of the section. In the proof of Theorem 37.62.1 we have seen that the smooth locus of $Y \to T$ contains the inverse image in $g^{-1}(T_ i)$ of the set of points where $Y_ i$ is smooth over $B_ i$. This proves the lemma. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 37.62: Reduced fibre theorem

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Your email address will not be published.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