I am trying to learn about etale morphisms, and one definition that is given often is a smooth morphism of relative dimension $0$. I know that relative dimension can be defined generally in the case of flat morphisms, but there seems to be some confusion in the literature about the definition. Can anyone tell me definitively which of the following is actually meant by "relative dimension $n$":
Every non-empty fiber has dimension $n$, every non-empty fiber has pure dimension $n$, every fiber has dimension $n$, or every fiber has pure dimension $n$.
I have noticed some sources do say literally every fiber, but then a corollary of this is that any flat morphism is surjective, since otherwise some fibers would be empty. So it seems more likey that it is only referring to non-empty fibers.
At risk of inviting Cunningham's law down on myself, the definition I'm used to, have seen in the literature, and think is the "usual one" is the statement that a morphism of schemes $f:X\to Y$ is of relative dimension $d$ iff for all $y\in Y$, the fiber $X_y$ is equidimensional of dimension $d$. That is, all irreducible components of $X_y$ are of dimension $d$ for any $y$. In particular, we allow empty fibers since they have no irreducible components (an irreducible topological space is nonempty).