This is confusing, there appear to be possibly two definitions of smoothness in algebraic geometry for a morphism $f: X \rightarrow Y$ of schemes of finite type over an arbitrary field $k$.

Definition 1 (Milne, Etale Cohomology pp.30-31): Let $f: X \rightarrow Y$ be locally of finite type. Then we say $f$ is smooth if it is flat and $\Omega_{X/Y}$ is locally free of rank the relative dimension. (For reference, this is the same definition as in the stacks project.)

Definition 2: (Hartshorne, III.10, p268). A morphism $f: X \rightarrow Y$ of schemes of finite type over a field $k$ is smooth of relative dimension $n$ if it is flat, the sheaf of relative differentials $\Omega_{X/Y}$ is locally free of rank $n$, and if: (*) If $X' \subset X$ and $Y' \subset Y$ are irreducible components such that $f(X') \subset Y'$ then $\dim X' = \dim Y' + n$.

In the situation of Hartshorne's definition, there is the extra hypothesis (*). Is this always satisfied, or are there just two definitions?

somecondition to relate n to fiber dimensions. Hartshorne's $(\ast)$ is focusing exclusively on dimensions of generic fibers, so some work is needed to show that Definition 2 forcesallfibers to have pure dimension $n$. Definition 1 is conceptually superior. – grp Sep 15 '12 at 21:31puredimension $n$. To prove the same for allnon-emptyfibers, by base change to dvr's over $Y$ and the purity hypothesis for the generic fibers we can pass to irreducible components (after the dvr base change, using the easy nature of flatness over a dvr!) to reduce to the case of anirreduciblescheme fppf over a dvr. Now apply results about flatness and dimension theory (or EGA IV$_3$, 14.3.10). – grp Sep 15 '12 at 23:59