Cartan's theorem B states that every coherent analytic sheaf on a Stein manifold is acyclic. Hartshorne claims that the converse to this theorem also holds, but doesn't supply a reference. Does anyone where to find a proof of this fact? 


Thank you kindly.



Reference for the converse of Cartan's Theorem B
Actually a more general result is true:  

If $X$ is a complex *space* (maybe not smooth) and if $H^1(X,\mathcal I)$ is zero for all coherent sheaves of ideals $\mathcal I\subset \mathcal O\_X$, then $X$ is a Stein space.  

This is Proposition 52.6 In L.Kaup-B.Kaup's [*Holomorphic Functions of Several Variables*](http://books.google.fr/books/about/Holomorphic_functions_of_several_variabl.html?id=nDgBsOurnAIC&redir_esc=y).  

It is also proved in [Taylor's book](http://books.google.fr/books?id=i8lUNpZ379MC&source=gbs_similarbooks_r&redir_esc=y): Proposition 11.4.5


An amusing variant is that an open subset $U\subset \mathbb C^n$ is Stein iff $H^i(U,\mathcal O)$ for all $1\leq i\leq n-1$. 
This is proved as Theorem 2, §5, Chapter V, page 159 of Grauert-Remmert's *Theory of Stein Spaces*.


