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.