Let $X$ be a compact complex n-fold . Then for every coherent sheaf $\mathfrak{F}$ on $X$ , and every holomorphic line bundle $L$ on $X$ , then the dimension of $H^0 (X,\mathfrak{F}\otimes\mathcal{O}\_X(L))$ does not depend on $L$ when dim Supp$\mathfrak{F}=0$ .



how to prove the following fact in sheaf cohomology ?
$\dim\mathrm{Supp}\\, \mathfrak F=0$ implies that $\mathfrak F\otimes \mathscr O\_X(L)\simeq \mathfrak F$ and hence its cohomology is independent of $L$.


