sheaf cohomology of a single point

Question

Is it true that $H^q$({$\star$}, $\mathscr{A}$)= $H^q$({$\star$}, $\mathscr{B}$), q$\geq$ 0? Where $\mathscr{A}$ and $\mathscr{B}$ are two different sheaves on {$\star$}, the single point space.

Answer 1

In general $$H^0(X,\mathscr{A})=\Gamma(\mathscr{A})$$ in the case of a one point space, $H^q(p,\mathscr{A})=0$ for $q\geq 1$.