Geometrically regular algebras and geometrically regular fibres

Let $k$ be a field. A $k$-algebra $A$ of finite type (Finitely presented modules and finitely presented algebras) is geometrically regular over $k$ when for every finitely generated field extension $K\mathrm{/}k$ (Finitely generated field extensions $F(a_1,\dots ,a_r)$) the $K$-algebra $A{\otimes }_{k}K$ (Universal mapping property of the tensor product of commutative algebras) is a regular Noetherian ring (regular noetherian ring). Since a finitely generated extension $K\mathrm{/}k$ makes $A{\otimes }_{k}K$ a finite-type $K$-algebra, the Noetherian condition in that clause is automatic; the regularity is the content. The quantifier is over the finitely generated extensions only, and the later field-test lemma shows that regularity for those is equivalent to regularity after every field extension of $k$.

Now let $R\to S$ be a homomorphism of commutative rings with $S$ finitely presented as an $R$-algebra (Finitely presented modules and finitely presented algebras), let $\mathfrak{q}\in \mathrm{Spec}S$ and let $\mathfrak{p}=\mathfrak{q}\cap R$. The fibre of $R\to S$ over $\mathfrak{p}$ is $S{\otimes }_R\kappa (\mathfrak{p})$, a $\kappa (\mathfrak{p})$-algebra, and it is geometrically regular at $\mathfrak{q}$ when for every field extension $K\mathrm{/}\kappa (\mathfrak{p})$ and every prime $\mathfrak{Q}$ of $(S{\otimes }_R\kappa (\mathfrak{p})){\otimes }_{\kappa (\mathfrak{p})}K$ lying over the image of $\mathfrak{q}$ in $S{\otimes }_R\kappa (\mathfrak{p})$, the local ring $((S{\otimes }_R\kappa (\mathfrak{p})){\otimes }_{\kappa (\mathfrak{p})}K{)}_{\mathfrak{Q}}$ is regular. A fibre is geometrically regular when it is geometrically regular at each of its points, and the condition is imposed on every field extension $K\mathrm{/}\kappa (\mathfrak{p})$, not only on the finitely generated ones; an empty fibre satisfies the pointwise condition vacuously, and the affine line over the residue field is the model case.

Three conventions belong to the definition. First, geometric regularity of a finite-type $k$-algebra is a statement about all scalar extensions $A{\otimes }_{k}K$, so it is strictly stronger than regularity of $A$ and is defined without using smoothness of the structural morphism; the equivalence with local standard smoothness is a theorem, not part of the definition. Second, the fibre condition is pointwise at a prime $\mathfrak{q}$ and ranges over all field extensions of the residue field $\kappa (\mathfrak{p})$, so that a rational point over an algebraic closure of $\kappa (\mathfrak{p})$ is covered without appealing to any later theorem. Third, the hypothesis that $S$ is finitely presented over $R$ is part of the definition of the fibre condition, since it is what lets the standard smooth presentations and the local presentation theorem apply to it.

The field-case equivalence with local standard smoothness is proved in Locally standard smooth iff flat with geometrically regular fibres, clause 3, under the Axiom of Choice assumed there. Standard smooth presentations and locally standard smooth maps supplies the presentation terminology, rather than the equivalence theorem.