The Stacks project

Lemma 54.2.4. Let $S$ be a scheme over $\mathbf{F}_ p$. Let $f : Y \to X$ be a finite morphism of Noetherian normal integral schemes over $S$. Assume

  1. the extension of function fields is purely inseparable of degree $p$, and

  2. $\Omega _{X/S}$ is a coherent $\mathcal{O}_ X$-module (for example if $X$ is of finite type over $S$).

For $i \geq 1$ there is a canonical map

\[ \text{Tr} : f_*\Omega ^ i_{Y/S} \longrightarrow (\Omega _{X/S}^ i)^{**} \]

whose stalk in the generic point of $X$ recovers the trace map of Lemma 54.2.2.

Proof. The exact sequence $f^*\Omega _{X/S} \to \Omega _{Y/S} \to \Omega _{Y/X} \to 0$ shows that $\Omega _{Y/S}$ and hence $f_*\Omega _{Y/S}$ are coherent modules as well. Thus it suffices to prove the trace map in the generic point extends to stalks at $x \in X$ with $\dim (\mathcal{O}_{X, x}) = 1$, see Divisors, Lemma 31.12.14. Thus we reduce to the case discussed in the next paragraph.

Assume $X = \mathop{\mathrm{Spec}}(A)$ and $Y = \mathop{\mathrm{Spec}}(B)$ with $A$ a discrete valuation ring and $B$ finite over $A$. Since the induced extension $L/K$ of fraction fields is purely inseparable, we see that $B$ is local too. Hence $B$ is a discrete valuation ring too. Then either

  1. $B/A$ has ramification index $p$ and hence $B = A[x]/(x^ p - a)$ where $a \in A$ is a uniformizer, or

  2. $\mathfrak m_ B = \mathfrak m_ A B$ and the residue field $B/\mathfrak m_ A B$ is purely inseparable of degree $p$ over $\kappa _ A = A/\mathfrak m_ A$. Choose any $x \in B$ whose residue class is not in $\kappa _ A$ and then we'll have $B = A[x]/(x^ p - a)$ where $a \in A$ is a unit.

Let $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$ be an affine open such that $X$ maps into $\mathop{\mathrm{Spec}}(\Lambda )$. Then we can apply Lemma 54.2.2 to see that the trace map extends to $\Omega ^ i_{B/\Lambda } \to \Omega ^ i_{A/\Lambda }$ for all $i \geq 1$. $\square$

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Your email address will not be published.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 0AX6. Beware of the difference between the letter 'O' and the digit '0'.