On the complex projective n-space $/mathbb{P}^n$ there is a renowned description by Beilinson of the bounded derived category of coherent sheaves. It is generated by $/mathcal{O}(-i)$, $0/leq i /leq n$, such that, for all $0/leq i,j/leq n$, $r/in /mathbb{N}$,
$$Ext^r_{/mathbb{P}^n}(/mathcal{O}(-i),/mathcal{O}(-j))/cong /begin{cases} 0, /text{if} r /geq 1 ///mathbb{C}, /text{if} r = 0 /text{and} i = j, /end{cases}$$ and that Mod_{/mathbb{P}^n}(/mathcal{O}(-i),/mathcal{O}(-j))/neq 0$ iff $i/geq j$, and hence is triangulatedly equivalent to the bounded derived category of right modules of finite type over the endomorphism ring of the direct sum of those $/mathcal{O}(-i)$'s. The same carrying over, of course, to positive characteristicp, Hashimoto, Rumynin and I have recently observed that all $/mathcal{O}(-i)$, $0/leq i/leq n$, appear as direct summands of the Frobenius direct image of the structure sheaf of $/mathbb{P}^n$ iff $p/geq n+1$. A similar phenomenon is known also to hold on the complex quadrics due to Kapranovnd in positive characteristic by a recent work of Langer. We will try to explain the results free of characteristic using representation theory.