Surface geometric regularity field test and generic spread

In characteristic $p>0$, choose a transcendence basis $y$ for $K\mathrm{/}k$ and let $M$ be the maximal separable subextension of the finite extension $K\mathrm{/}k(y)$. If $K\mathrm{\ne }M$, choose $\beta \in K\setminus M$ with $\alpha ={\beta }^p\in M$. Adjoin to $k$ the $p$th roots of the finitely many coefficients occurring in the numerator and denominator polynomials of the coefficients of the separable minimal polynomial of $\alpha$ over $k(y)$, and adjoin $y^{1\mathrm{/}p}$ to the upper field. This gives finite purely inseparable enlargements. In the separable compositum, that polynomial has $p$th-power coefficients; taking their roots gives a separable polynomial for a root whose $p$th power is $\alpha$. Uniqueness of a $p$th root in a field puts $\beta$ in the enlarged separable subfield. Thus the remaining inseparable degree strictly decreases. Induction makes the upper extension separably generated; characteristic zero needs no enlargement.

A separably generated field extension $E\mathrm{/}k$ is finite separable over $k(y)$ for a separating transcendence basis $y$. By the primitive-element theorem write $E=k(y)(\theta )$. Clear the coefficients of its monic minimal polynomial by localizing $k[y]$, and invert its nonzero derivative at $\theta$; this gives a standard smooth domain with fraction field $E$. For a finite-type domain map $R\to C$ whose fraction-field extension is separably generated, the same construction is over $R$ after inverting finitely many nonzero base elements. The resulting smooth algebra and $C$ have the same fraction field and agree after further localization: express each finite generating family rationally in the other and invert all the denominators. This supplies a nonempty standard smooth source open over a localized base.

Suppose $T{\otimes }_k{k}^{\mathrm{\prime }}$ is regular for every finite purely inseparable $k^{\mathrm{\prime }}\mathrm{/}k$. Given a finitely generated $E\mathrm{/}k$, step 1.1 gives finite purely inseparable $k^{\mathrm{\prime }}\mathrm{/}k$ and $E^{\mathrm{\prime }}\mathrm{/}E$ with $E^{\mathrm{\prime }}\mathrm{/}{k}^{\mathrm{\prime }}$ separably generated. Choose the standard smooth model of $E^{\mathrm{\prime }}\mathrm{/}{k}^{\mathrm{\prime }}$ from step 1.2. Its base change to the regular ring $T{\otimes }_k{k}^{\mathrm{\prime }}$ is flat with regular geometric fibres by [F3] and [F4]; [F5] makes it regular locally. Localizing to its fraction field gives regularity of $T{\otimes }_k{E}^{\mathrm{\prime }}$.

The map $T{\otimes }_{k}E\to T{\otimes }_k{E}^{\mathrm{\prime }}$ is faithfully flat. At every prime of the source choose a prime above it; [F5] descends regularity of the corresponding target local ring, so $T{\otimes }_{k}E$ is regular. This proves sufficiency of the purely inseparable test; necessity follows because such extensions are finitely generated.

If $T$ is geometrically regular and $E\mathrm{/}k$ is finitely generated, then every finitely generated extension $F\mathrm{/}E$ is finitely generated over $k$. Hence $(T{\otimes }_{k}E){\otimes }_{E}F=T{\otimes }_{k}F$ is regular, proving stability under the stated field extensions. The generic-spread assertion is step 1.2.

AC is inherited from the cited regularity and basis suppliers; the finite inseparable induction and the denominator choices introduce no additional choice hypothesis.