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 $i \geq 1$. $\square$

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