Koszul Algebras and Sheaves over Projective Space

TL;DR

This paper explores the relationship between Koszul modules and coherent sheaves over projective space, developing a relative Auslander-Reiten theory that characterizes the structure of the category of coherent sheaves.

Contribution

It introduces a subcategory of sheaves derived from Koszul modules and develops a relative Auslander-Reiten theory for coherent sheaves on projective space.

Findings

Sheafification of Koszul modules forms a significant subcategory of coherent sheaves.

Almost split sequences can be constructed using Koszul duality.

Most Auslander-Reiten components have the shape ZA_infinity.

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.