Koszul Algebras and Sheaves over Projective Space

We are going to show that the sheafication of graded Koszul modules KΓ over Γn = K [x0, x1...xn] form an important subcategory ∧ KΓ of the coherents sheaves on projective space, Coh(P n). One reason is that any coherent sheave over P n belongs to ∧ KΓup to shift. More importantly, the category KΓ allows a concept of almost split sequence obtained by exploiting Koszul duality between graded Koszul modules over Γ and over the exterior algebra Λ. This is then used to develop a kind of relative Auslander-Reiten theory for the category Coh(Pn), with respect to this theory, all but finitely many Auslander-Reiten components for Coh(Pn) have the shape ZA∞. We also describe the remaining ones.