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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