most recent 30 from http://mathoverflow.net 2013-05-23T11:22:32Z http://mathoverflow.net/feeds/question/107273 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107273/two-definitions-of-smoothness LMN 2012-09-15T17:48:14Z 2012-09-25T12:14:59Z

<p>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 $k$.</p> <p>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.)</p> <p>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 $\dim X' = \dim Y' + n$.</p> <p>In the situation of Hartshorne's definition, there is the extra hypothesis (*). Is this always satisfied, or are there just two definitions?</p>