Koszul Algebras and Sheaves over Projective Space

Abstract

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 $\overset{\wedge}{K}_{\Gamma}$ of the coherents sheaves on projective space, $Coh(P^{n}).$ One reason is that any coherent sheave over $P^{n}$ belongs to $\overset{\wedge}{K}_{\Gamma}$up to shift. More importantly, the category $K_{\Gamma}$ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over $\Gamma $ and over the exterior algebra $\Lambda .$ This is then used to develop a kind of relative Auslander-Reiten theory for the category $\mathit{Coh(P}^{n})$, with respect to this theory, all but finitely many Auslander-Reiten components for $\mathit{Coh(P}^{n})$ have the shape \textit{ZA}$_{\infty}.$ We also describe the remaining ones.