Classical and scheme smoothness over a perfect field

(classical) implies (scheme). Suppose $X\to \mathrm{Spec}k$ is classical smooth and let $x\in X$. By [F1] the classical condition is the local-standard-smooth condition, so there are affine open neighbourhoods $x\in V=\mathrm{Spec}B$ and $U=\mathrm{Spec}k\subseteq \mathrm{Spec}k$ with the map $k\to B$ standard smooth at the prime of $x$: after shrinking, $B$ is a finitely presented $k$-algebra carrying a standard smooth presentation, in particular $X\to \mathrm{Spec}k$ is locally of finite presentation at $x$ by [F5] and the structural map on this chart is a standard smooth presentation. The chart therefore has an invertible Jacobian minor in the sense of the presentation, and since the structural morphism is locally of finite presentation, the "if" direction of the Jacobian criterion [F4] makes $X\to \mathrm{Spec}k$ smooth at $x$. As $x$ was arbitrary, (classical) implies (scheme).

(scheme) implies (classical). Suppose $X\to \mathrm{Spec}k$ is smooth in the scheme-theoretic sense and let $x\in X$. It is locally of finite presentation at $x$ by [F3], so the "only if" direction of the Jacobian criterion [F4] exhibits affine neighbourhoods of $x$ and of the image point and a localisation $C_h$ of the coordinate ring with a standard smooth presentation whose Jacobian minor is invertible in $C_h$. That is precisely the local-standard-smooth condition of [F1] at $x$; as $x$ was arbitrary, (scheme) implies (classical). This completes clause 1.

Irreducibility is a separate convention. Let $X=\mathrm{Spec}k[t]\bigsqcup \mathrm{Spec}k[t]$ be the disjoint union of two copies of the affine line over $k$. It is a finite-type reduced $k$-scheme, and it is not irreducible and not connected, its two components being disjoint nonempty open subschemes. It is classically smooth: every point lies in one of the two copies, and that copy carries the standard smooth presentation $k\to k[t]$, namely the case $n=1$, $c=0$ of [F5] with empty equation list, which is affine over $k$ with an invertible (empty) Jacobian minor. By step 1.1 the structure morphism is scheme-smooth. Hence irreducibility and connectedness are not implied by smoothness and must be imposed separately if the earlier convention requires them; this is clause 4.

Perfect base field. Assume now that $k$ is perfect. By [F2], $X$ is regular if and only if $X\to \mathrm{Spec}k$ is smooth in the local-standard-smooth convention, i.e. if and only if condition (classical) holds; by step 1.1 and step 2.1 condition (classical) is equivalent to condition (scheme). Hence all three conditions are equivalent, which is clause 2. No reducedness, irreducibility or separatedness enters, as [F2] imposes none.

The earlier variety convention. Let $k$ be perfect. If $X$ is classical smooth in the earlier convention, then by step 1.1 the structure morphism $X\to \mathrm{Spec}k$ is scheme-smooth, and by step 3.1 $X$ is regular, i.e. every local ring is a regular local ring. Conversely, let $X$ be a reduced finite-type $k$-scheme with scheme-smooth structure morphism; by step 2.1 it is classical smooth in the local-standard-smooth sense, and by step 3.1 it is regular, so it is classically smooth at every point in the pointwise regular-local reading used for varieties; reducedness is part of the earlier notion of a variety but is not needed for either implication. This is clause 3.

Assumption accounting. The Axiom of Choice [F6] is assumed in the Statement and is used exactly through the classical-to-scheme and regular-equals-smooth suppliers: the Jacobian criterion [F4] of steps 1.1 and 2.1 and the perfect-field equivalence [F2] of steps 3.1 and 4.1, together with the field-change characterization [F1]. No other selection is made. [F1, F2, F4, F6, step 4.1, step 2.2] $\mathrm{\square }$