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 $n$??
 A: One should read the comments by Matt E before reading the following. I am keeping this just as a  record. 
Let $X = \hbox{Proj} \, (K [x,y])$ and $Y = \hbox{Proj} \, (K[x,y]/(x^2))$. Write $i: Y  \hookrightarrow X$ and $F = i_* \mathcal{O}_Y$.  Then $\dim \hbox{Supp} \, F = 0$. One has $\Gamma(X, F) = 1$, but $\Gamma(X, F(i)) = 2$ for $i \ge 1$. 
I belive that this has to do with Hilbert-polynomial. Since $\dim F = 0$, the degree of the Hilbert polynomial is $0$, i.e., it is a constant. Since degree of $F$ is $2$, one has $\Gamma(X, F(i)) = 2$ for $i \gg 0$. However, this does not tell us the behavior of the Hilbert-funtion when $i$ is "small".

In the above $\Gamma(X, F) = 1$ is incorrect. The exact sequence of graded $S= K[x,y]$-modules
$$
0 \to (x^2) \to S \to S/(x^2) \to 0 
$$ induces an exact sequence of $\mathcal{O}_X$-modules
$$
0 \to \mathcal{O}_X (-2) \to \mathcal{O}_X \to F \to 0.
$$
Taking $\Gamma(X,-)$, we obtain an exact sequence
$$
H^0(X, \mathcal{O}_X (-2)) \to H^0(X, \mathcal{O}_X) \to H^0(X, F) \to H^1(X, \mathcal{O}_X (-2)) \to H^1(X, \mathcal{O}_X ).
$$
Since $H^0(X, \mathcal{O}_X (-2)) =  H^1(X, \mathcal{O}_X ) = 0$ and $H^0(X, \mathcal{O}_X) = H^1(X, \mathcal{O}_X (-2)) = 1$, we see that $H^0(X, F) = \Gamma(X,F) = 2$. 
A: 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.
