Degree-p differential trace extends across normal surface valuations

For a monogenic algebra $B=A[z]\mathrm{/}(z^p-f)$ the differential presentation is $({\mathrm{\Omega }}_{A\mathrm{/}S}{\otimes }_{A}B\oplus Bdz)\mathrm{/}(Bdf)$, so modulo forms pulled back from $A$ the algebra is $({\wedge }^{q-1}({\mathrm{\Omega }}_{A\mathrm{/}S}\mathrm{/}Adf))\otimes {\mathrm{\Omega }}_{B\mathrm{/}A}$; define the trace by wedging a lift $\eta$ with $df$ and taking the coefficient of $z^{p-1}dz$. Changing the lift by a multiple of $df$ does not change the wedge, which proves well-definedness and the stated formula on $\eta \wedge z^{i}dz$.

Independence of the choice of $z$ follows from a universal coefficient calculation. Write $z={\sum }_{i=0}^{p-1}{\lambda }_i{w}^i$, with $w^p=g$; then $f=z^p={\sum }_i{\lambda }_i^p{g}^i$. Terms in $dz$ involving $d{\lambda }_i$ are pulled-back base forms and are killed by the trace. The coefficient of $w^{p-1}dw$ in the remaining part of $z^{j}dz$ is the coefficient of $w^{p-1}$ in $z^j{z}^{\mathrm{\prime }}(w)$ after reduction by $w^p=g$. If $j<p-1$, then $z^j{z}^{\mathrm{\prime }}(w)=(j+1{)}^{-1}(z^{j+1}{)}^{\mathrm{\prime }}(w)$; a derivative term can reduce to $w^{p-1}$ only from an original exponent divisible by $p$, whose derivative coefficient is zero in characteristic $p$. For $j=p-1$, first take universal coefficients in ${\mathbb{F}}_p[b,{\lambda }_0,\dots ,{\lambda }_{p-1}]$ and lift them to $\mathbb{Z}[b,{\lambda }_0,\dots ,{\lambda }_{p-1}]$. There $Z^{p-1}{Z}^{\mathrm{\prime }}=(1\mathrm{/}p)(Z^p{)}^{\mathrm{\prime }}$ for $Z={\sum }_i{\lambda }_i{w}^i$. In the expansion of $Z^p$, the coefficient of $w^{ip}$ is congruent modulo $p$ to ${\lambda }_i^p$; every other contribution to a reduced $w^{p-1}$ coefficient has coefficient divisible by $p$ and vanishes after division and reduction modulo $p$. Thus the coefficient of $w^{p-1}$ in $z^{p-1}{z}^{\mathrm{\prime }}$ after $w^p=g$ is ${\sum }_{i=1}^{p-1}i{\lambda }_i^p{g}^{i-1}$. Multiplying by $dg$ gives $df$, so the trace formula is unchanged under the generator change. This polynomial identity holds after every coefficient specialization and uses no division by $p$ in characteristic $p$.

To extend over $X$ test at height-one localizations $A={\mathcal{O}}_{X,x}$, which are discrete valuation rings by normality. The integral closure $B$ in the degree-$p$ field extension is a local discrete valuation ring because purely inseparable extensions have a unique prime, and it is a finite torsion-free $A$-module, hence free of rank $p$.

Writing $e$ for the ramification index and $f_{\mathrm{res}}$ for the residue degree, the length of $B\mathrm{/}{\pi }_{A}B$ is $p=e{f}_{\mathrm{res}}$; hence either $e=p,f_{\mathrm{res}}=1$, in which case a uniformizer $z$ with $z^p\in A$ generates $B$ over $A$ by Nakayama, or $e=1,f_{\mathrm{res}}=p$, in which case a lift $z$ of a residue-field primitive element has $z^p\in A$ by normality and Nakayama again gives $B=A[z]$.

In both cases of step 4.1 the monogenic formula of steps 1.1 and 2.1 applies at every height-one localization, taking regular relative forms to elements of ${\mathrm{\Omega }}_{A\mathrm{/}S}$; coherent relative differentials on $Y$ follow from the exact conormal sequence and finiteness.

The double-dual codimension-one lattice criterion for reflexive modules now extends the generic map ${\pi }_{\ast }{\wedge }^q{\mathrm{\Omega }}_{Y\mathrm{/}S}\to ({\wedge }^q{\mathrm{\Omega }}_{X\mathrm{/}S}{)}^{\ast \ast }$, and the extension is canonical because its target injects into the generic module; the argument supplies the discrete-valuation-ring basis and coordinate-invariance details required for the trace extension. The Axiom of Choice and the Axiom of Dependent Choice are inherited from the cited suppliers.