This question is ***almost*** a duplicate of [that question](https://mathoverflow.net/q/375368/4149),
which has a [good answer](https://mathoverflow.net/a/375415/4149).
The difference is that I ask for ***references*** rather than proofs.
By a reference I mean a reference to a book, or to a paper, or to an arXiv preprint.
The answerer does not know references; see
[his comment](https://mathoverflow.net/questions/375368/smooth-morphism-algebraic-geometry-vs-submersion-differential-geo-ehresma/375415#comment997469_375415).


Let $f\colon X\to Y$ be a surjective morphism of algebraic varieties over $\Bbb C$.
Consider the following assertions:



> 
> **Assertion 1.** If the morphism $f$ is smooth, then it is a *submersion*,
> that is, for any $\Bbb C$-point $x\in X({\Bbb C})$,
> the linear map of the tangent spaces $d\_x\colon T\_x(X)\to T\_{f(x)} Y$ is surjective.
> 
> 
> 



> 
> **Assertion 2.** If the varieties $X$ and $Y$ are smooth,
> then the morphism $f$ is smooth if and only if it is a submersion.
> 
> 
> 


I am asking for *references* for these two assertions.
Many thanks in advance!



Smooth morphisms vs. submersions
Assertion 2 is stated as Proposition 10.4 page 270 in Hartshorne's book. In fact, assertions 1 is equally proved in Hartshorne proposition 10.4 (though it is not in the statement of the proposition). It suffices to notice that the kernel of:
$$ \Omega\_{Y/\mathbb{C}} \longrightarrow \Omega\_{X/ \mathbb{C}} \longrightarrow \Omega\_{X/Y} \longrightarrow 0$$
is a torsion-sheaf as:


$\bullet$ $\Omega\_{X/Y}$ is locally free of rank $\dim X - \dim Y$ (definition of smoothness)


$\bullet$ $\Omega\_{X/\mathbb{C}}$ (resp. $\Omega\_{Y/\mathbb{C}}$) is generically of rank $\dim X$ (resp. $\dim Y$) because $X$ is integral (resp. $Y$ is integral).


Then dualize this exact sequence an oberve that dual of a torsion sheaf is zero.


