Relation between Supp and sheaves Let $\mathcal{F}$ be a cohorent sheaf on projective scheme $X$.
My question is simple... If $\operatorname{dim}\operatorname{Supp}\mathcal{F}$ is zero, then $\mathcal{F}(n) =\mathcal{F}$ for any integer $n$??

Let $Z$ be the support of $F$.  Then $\mathcal F(n) := \mathcal F \otimes \mathcal O(n) = \mathcal F \otimes (\mathcal O(n)_{| Z})$ (i.e. we can restrict $\mathcal O(n)$ to $Z$ before computing the tensor product).
Because $Z$ is zero-dimensional, it is a union of Specs of Artinian local rings,
and so (exercise) any invertible sheaf on $Z$ is trivial.  Thus $\mathcal O(n)_{|Z} \cong \mathcal O_Z$, and so $\mathcal F(n) \cong \mathcal F$, 
as required.
