Does vanishing of cohomology of locally free sheaves imply affiness of scheme We have Serre criterion of affiness of a scheme which states that if a quasi compact scheme has higher cohomology vanishing for all the quasi coherent sheaves,then the scheme is affine.
I wonder whether we have similar statement for locally free sheaves as following:
Let $X$ be a noetherian scheme,let $F$ be arbitrary locally free sheaf on $X$,if higher cohomology of $F$ vanishing(for $i\geq 1$),then $X$ is affine scheme.Is this statement true?
For $X$ be quasi compact scheme,I think it is not true,but for noetherian scheme,I do not know 
Maybe it is a stupid question.
 A: This is an incomplete answer, which shows that finding a counter-example would be quite difficult: the result holds  if your scheme is divisorial and of finite dimension. Indeed the first hypothesis guarantees that for any coherent $\mathcal{F}$, there is an exact sequence  $0\rightarrow \mathcal{K}\rightarrow E\rightarrow \mathcal{F}$ with $E$ locally free. Then $H^{i}(X,\mathcal{F})\cong H^{i+1}(X,\mathcal{K})$; applying this again to $\mathcal{K}$ and going on, you'll arrive eventually to $H^{i}(X,\mathcal{F})\cong H^{N}(X,\mathcal{G})$ with $\mathcal{G}$ coherent and $N>\dim(X)$, hence $H^{i}(X,\mathcal{F})=0$. 
A: There are many versions of Serre's criterion for affineness.  One version states that for every quasi-compact, quasi-separated scheme $X$, $X$ is affine if and only if $H^1(X,\mathcal{F})$ vanishes for every quasi-coherent sheaf $\mathcal{F}$ that is locally finitely generated.  In particular, if $X$ is Noetherian, then $X$ is affine if and only if $H^1(X,\mathcal{F})$ vanishes for every coherent sheaf.  
For this criterion, it does not suffice to consider only locally free sheaves (of finite rank).  Let $n>1$ be any integer.  Let $X$ be the quasi-compact, quasi-separated, yet non-separated scheme obtained by glueing two copies, $X_1$ and $X_2$, of $\mathbb{A}^n$ along the common open $X_{1,2} = \mathbb{A}^n\setminus\{0\}$.  There is a unique morphism of schemes, $f:X\to \mathbb{A}^n$ that restricts to the identity on $X_1$ and $X_2$.  Using the S2 property, every locally free sheaf on $X$ is of the form $f^*E$.  Thus, by the Quillen-Suslin theorem, every locally free sheaf on $X$ is a direct sum of copies of the structure sheaf.  By straightforward computation, $H^1(X,\mathcal{O}_X)$ vanishes.  Yet $X$ is not affine, since $X$ is not separated. 
