Koszul Algebras and Sheaves over Projective Space

We are going to show that the sheafication of graded Koszul modules $% K_{\Gamma}$over$\Gamma_{n}=K[ x_{0},x_{1}…x_{n}] $ form an important subcategory ${}_{\mathrm{\Gamma }}$ of the coherents sheaves on projective space, $Coh(P^n)\mathrm{.}$ One reason is that any coherent sheave over $P^n$ belongs to ${}_{\mathrm{\Gamma }}$up to shift. More importantly, the category $K_{\mathrm{\Gamma }}$ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over $\mathrm{\Gamma }$ and over the exterior algebra $\mathrm{\Lambda }\mathrm{.}$ This is then used to develop a kind of relative Auslander-Reiten theory for the category ${Coh(P}^n)$, with respect to this theory, all but finitely many Auslander-Reiten components for ${Coh(P}^n)$ have the shape $\mathit{\text{ZA}}$${}_{\mathrm{\infty }}\mathrm{.}$ We also describe the remaining ones.