Surface p basis subfield separation

For a finite $J\subset I$ write $k_J=k^p(b_i:i\mathrm{\notin }J)$; the $p$-monomials in the variables $b_i$, $i\in J$, form a $k_J$-basis of $k$ by the defining property of a $p$-basis and the finite-support condition. Consequently the products of these monomials with the $X$- and $Y$-monomials of exponents less than $p$ form an explicit $A_J$-basis of $A=k[\text{\hspace{0.17em}⁣}[X]\text{\hspace{0.17em}⁣}][Y]$, because $A$ is a finite free module over $k[\text{\hspace{0.17em}⁣}[X^p]\text{\hspace{0.17em}⁣}][Y^p]$ with the monomials of exponents less than $p$ as a basis; so $A$ is finite free over $A_J=k_J[\text{\hspace{0.17em}⁣}[X^p]\text{\hspace{0.17em}⁣}][Y^p]$.

The family $(K_J)$ over finite $J$ is downward directed by inclusion of the finite sets, with union the whole index set corresponding to increasing $J$; only the downward directed structure and its cofinal refinements are used below.

The intersection of the fields $K_J$ is $K^p$. The inclusion $K^p\subseteq {\bigcap }_J{K}_J$ is immediate. For the reverse inclusion, fix $x\in {\bigcap }_J{K}_J$ and write $x=f\mathrm{/}{g}^p$ with $f,g\in A$, $g\mathrm{\ne }0$ (replace a denominator $g_0$ by $g_0^p$). For each finite $J$, separately write $x=a_J\mathrm{/}{c}_J$ with $a_J,c_J\in A_J$, $c_J\mathrm{\ne }0$. Then $x=(a_J{c}_J^{p-1})\mathrm{/}{c}_J^p$; putting $a_J^{\mathrm{\prime }}=a_J{c}_J^{p-1}\in A_J$ gives $c_J^{p}f=a_J^{\mathrm{\prime }}{g}^p\in A_J$, since $g^p\in A^p\subseteq A_J$. The denominator $c_J$ may depend on $J$. Every coordinate derivation $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{X}_i$ or $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{Y}_j$ kills $A_J$, as does each coefficient derivation ${\delta }_i$ dual to $b_i$ for $i\in J$. Applying any such derivation $D$ to $c_J^{p}f=a_J^{\mathrm{\prime }}{g}^p$ yields $c_J^{p}D(f)=0$, so $D(f)=0$. For each coordinate derivation choose any $J$; for each ${\delta }_i$ choose $J$ containing $i$. Thus all coordinate derivatives of $f$ vanish, so only monomials with every $X,Y$ exponent divisible by $p$ occur. Also every coefficient of $f$ is killed by every ${\delta }_i$; expanding that coefficient in the finite-support $p$-monomial $k^p$-basis shows it lies in $k^p$. Hence $f\in k^p[\text{\hspace{0.17em}⁣}[X_1^p,\dots ,X_n^p]\text{\hspace{0.17em}⁣}][Y_1^p,\dots ,Y_m^p]=A^p$, and $x=f\mathrm{/}{g}^p\in K^p$.

For every finite field extension $L\mathrm{/}K$, ${\bigcap }_J{L}^p{K}_J=L^p$. We prove the more general assertion by induction on $[E:F]$: if $F$ has characteristic $p$ and $(F_{\alpha })$ is a downward-directed family of purely inseparable subfields of an extension field containing $F^p$, with ${\bigcap }_{\alpha }{F}_{\alpha }=F^p$, then ${\bigcap }_{\alpha }{E}^p{F}_{\alpha }=E^p$ for every finite extension $E\mathrm{/}F$. Steps 2.1 and 3.1 give these hypotheses for $F=K$ and $F_{\alpha }=K_J$. The base case $E=F$ is the intersection hypothesis. If $F⊊M⊊E$, induction gives ${\bigcap }_{\alpha }{M}^p{F}_{\alpha }=M^p$. The family $M_{\alpha }=M^p{F}_{\alpha }$ is downward directed, purely inseparable over $M^p$, and has intersection $M^p$; also $E^p{M}_{\alpha }=E^p{F}_{\alpha }$. Applying induction to $E\mathrm{/}M$ proves the claim. It remains to consider an extension with no proper intermediate field. Such an extension is simple; it is either separable or purely inseparable of degree $p$. In the separable case write $E=F(\theta )$ and let $d=[E:F]$. Then $E^p=F^p({\theta }^p)$ is separable of degree $d$ over $F^p$. It is linearly disjoint from each purely inseparable $F_{\alpha }\mathrm{/}{F}^p$, so $1,{\theta }^p,\dots ,({\theta }^p{)}^{d-1}$ remains a basis of $E^p{F}_{\alpha }\mathrm{/}{F}_{\alpha }$. If $z$ belongs to every $E^p{F}_{\alpha }$, its unique coordinates in this basis lie in every $F_{\alpha }$, hence in $F^p$; thus $z\in E^p$. In the purely inseparable case write $E=F(\theta )$ with ${\theta }^p=t\in F\setminus F^p$. Then $E^p=F^p(t)$. Since $t\mathrm{\notin }{\bigcap }_{\alpha }{F}_{\alpha }$, choose ${\alpha }_0$ with $t\mathrm{\notin }{F}_{{\alpha }_0}$ and restrict to the cofinal family $F_{\alpha }\subseteq F_{{\alpha }_0}$. For these indices, $t^p\in F^p\subseteq F_{\alpha }$ but $t\mathrm{\notin }{F}_{\alpha }$, so $1,t,\dots ,t^{p-1}$ is a basis of $E^p{F}_{\alpha }\mathrm{/}{F}_{\alpha }$. The same unique-coordinate argument puts every element of the intersection in $E^p$.

The Axiom of Choice and the Axiom of Dependent Choice are inherited from the Zorn and derivation suppliers; the only quoted input for the family $(K_J)$ is the cofinal subfamily of steps 2.1 and 4.1.