On the structure of rigid semistable sheaves on algebraic surfaces

Let $S$ be a smooth projective surface, let $K$ be the canonical class of $S$ and let $H$ be an ample divisor such that $H\cdot K<0$. We prove that for any rigid sheaf $F$ ($\operatorname{Ext}^1(F,F)=0$) that is Mumford–Takemoto semistable with respect to $H$ there exists an exceptional set $(E_1,\dots,E_n)$ of sheaves on $S$ such that $F$ can be constructed from $\{E_i\}$ by means of a finite sequence of extensions.