Smooth vs regular vs non-singular This is a very basic question, but I can't find a clean answer anywhere.
In introductory algebraic geometry books working over the complex numbers, it's usual to use these three words interchangeably.  A point on a variety $X$ is smooth/regular/nonsingular if the dimension of the tangent space at the point is equal to the dimension of the variety.
On the other hand, I know that people sometimes find it important to distinguish between these terms, maybe when defining smoothness of morphisms, or working over non-closed fields,...
I want to make sure I know the right definitions of these terms in current use.  In what contexts should each be defined?  What implies what?  How should I think of them?
Edit: See for example https://en.wikipedia.org/wiki/Regular_scheme , which says there are regular schemes that aren't smooth.  There are also these notes of Vakil, where he has crossed out "smooth" and replaced it with "nonsingular": https://math.stanford.edu/~vakil/0708-216/216class21.pdf  The notes seem to suggest it's because "smooth" is reserved as a property of morphisms.  Is there a reason Wiki is happy to say "smooth scheme" but Ravi isn't?  Is "nonsingular" the same as "regular"?  
 A: In the general context, "regular" is a property of a scheme (or a ring, or local ring), and "smooth" is a property of a morphism of schemes.
"Regular" means exactly that at every point, the dimension of the (Zariski) tangent space is equal to the (Krull) dimension (of the local ring at that point).
A map $f: X \to Y$ is smooth if the fibers over geometric points of $Y$ are regular, and $f$ is locally of finite presentation and flat.
We also use the relative point of view, so a scheme $X$ over $S$ is a smooth scheme over $S$ if the map $X \to S$ is smooth.
The first potential source of confusion is that a map between two regular schemes can fail to be smooth. This is not hard to see once you realize that there's no need for the fibers to be regular - for instance $xy$ defines a map $\mathbb A^2 \to \mathbb A^1$.
The second potential source of confusion is that a regular scheme over a perfect field is necessarily smooth over that field, but for an imperfect field this fails. See some examples. Usually over an imperfect field you want to consider smooth schemes and not regular ones as they are better-behaved.
Going in the other direction, smooth schemes over non-regular bases can fail to be regular, but smooth schemes over regular bases will be regular.
I think people rarely use "nonsingular" when they are trying to be careful about this distinction, but I think it's more likely to mean "regular".
