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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.

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1 Answer 1

up vote 8 down vote accepted

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.
It is also proved in Taylor's book: 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.

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Thanks ! –  Steven Gubkin Apr 24 '12 at 22:18
    
For the second statement, I guess you mean $H^i(U, \mathcal O)=0$, right? –  Henri Apr 25 '12 at 7:16
    
Dear @Henri: yes, it was a typo that I just corrected. Thanks a lot for alerting me to it. –  Georges Elencwajg Apr 25 '12 at 18:34

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