In Vakil's notes (http://math.stanford.edu/~vakil/216blog/FOAGjan2915public.pdf), Theorem 22.3.10, he shows that, if $X\hookrightarrow Y$ is a closed embedding of smooth varieties over $k$, then ${\rm Bl}_X Y$ is smooth.
During the proof, he shows that ${\rm Bl}_X Y$ is regular at all the points on the exceptional divisor, and then immediately concludes smoothness.
I thought that we needed $k$ is perfect in order to get smoothness from regularity (12.2.10 in the same notes). What is special about this situation that I'm missing?
It's possible he is implicitly using that blowup commutes with flat base change (should be in stacks somewhere), and then working over an extension field...