what's the cohomological dimension of a Stein space? I want to know the "cohomological dimension" of a Stein space.
I know that:

*

*for $X$ differential manifold and for every sheaf $F$ of abelian
groups, I have $H_c^j(X,F)=H^j(X,F)=0$ for $j>\dim_{R}X$ (sheaves and
manifold, Kashiwara, Schapira)


*for $Y$ a noetherian scheme of dimension n (or, if it's false, just an algebraic variety over an algebraically closed field $k$) and $F$ a sheaf of group,
$H_c^j(X,F)=H^j(X,F)=0$ for $j>\dim Y$.
Does there exist an equivalent result for stein space? Something like:
for all sheaf $F$ of $C$-vector spaces in a stein space of complex dimension $d$,
$H_c^j(X,F)=H^j(X,F)=0$ for $j>d$?
Thanks for any answer! (and sorry for my  English)
 A: a) Using Morse theory  Hamm proved a theorem here  implying that every Stein complex space  $X$ of complex dimension $n$ is homotopy equivalent to a CW-complex of (real) dimension $\leq n$.
Notice that he does not assume that $X$ is a manifold: that space might have singularities.
He deduces that for any closed analytic subset  $A\subset X$ (with $X\setminus A$ still of dimension $n$) one has for all $i\gt n$ : $$H^i(X,A;\mathbb Z)=0$$ b) If a  topological space has Lebesgue dimension (also known as covering dimension) $\leq n$, then for all sheaves of abelian groups $\mathcal A$ and all $i\gt n$ its Čech cohomology groups vanish: $H_{Čech}^i(X,\mathcal A)=0$  .
If the space $X$ is paracompact, its genuine (=derived functor=Grothendieck)  cohomology is equal to its Čech cohomology and we thus also have $H^i(X,\mathcal A)=0$ .
 However Hamm's result does not imply directly anything about Lebesgue dimension: after all a point and an $n$-cell are homotopy equivalent but their Lebesgue dimensions are $0$ and $n$.
So Hamm's result does not seem to straightforwardly imply vanishing results for the cohomology of Stein spaces with values in non constant sheaves.  
c) However for the most important class of sheaves on a Stein space, the coherent sheaves, all positive dimensional cohomology groups vanish: for $i\gt 0$ $$H^i(X,\mathcal F)=0$$ This is due to Oka-Cartan-Serre, but it would be insulting  to assume that you didn't know  that very well :-)
