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"? 



Smooth vs regular vs non-singular
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](https://en.wikipedia.org/wiki/Geometrically_regular_ring). 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".


