Lemma 10.166.5. Let $k$ be a field. Let $A$ be an algebra over $k$. Let $k = \mathop{\mathrm{colim}}\nolimits k_ i$ be a directed colimit of subfields. If $A$ is geometrically regular over each $k_ i$, then $A$ is geometrically regular over $k$.
Proof. Let $k'/k$ be a finite purely inseparable field extension. We can get $k'$ by adjoining finitely many variables to $k$ and imposing finitely many polynomial relations. Hence we see that there exists an $i$ and a finite purely inseparable field extension $k_ i'/k_ i$ such that $k_ i = k \otimes _{k_ i} k_ i'$. Thus $A \otimes _ k k' = A \otimes _{k_ i} k_ i'$ and the lemma is clear. $\square$
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