[Paper Review] Koszul Algebras and Sheaves over Projective Space

Roberto Martínez-Villa|ArXiv.org|May 28, 2004

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper establishes a relative Auslander-Reiten theory for coherent sheaves on projective space via Koszul duality between graded Koszul modules over polynomial and exterior algebras. It shows that all but finitely many Auslander-Reiten components in $\operatorname{Coh}(\mathbb{P}^n)$ have the shape $\mathbb{Z}\Delta$, and provides a characterization of locally free sheaves via the sheafification of Koszul modules, proving the existence of indecomposable vector bundles of arbitrary high rank.

ABSTRACT

We are going to show that the sheafication of graded Koszul modules $% K_Γ$ over $Γ_{n}=K[ x_{0},x_{1}...x_{n}] $ form an important subcategory $\overset{\wedge}{K}_Γ$ of the coherents sheaves on projective space, $Coh(P^{n}).$ One reason is that any coherent sheave over $P^{n}$ belongs to $\overset{\wedge}{K}_Γ$up to shift. More importantly, the category $K_Γ$ 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 $\mathit{Coh(P}^{n})$, with respect to this theory, all but finitely many Auslander-Reiten components for $\mathit{Coh(P}^{n})$ have the shape extit{ZA}$_{\infty}.$ We also describe the remaining ones.

Motivation & Objective

• To develop a relative Auslander-Reiten theory for the category $\operatorname{Coh}(\mathbb{P}^n)$ using Koszul duality.
• To characterize coherent sheaves on $\mathbb{P}^n$ via the sheafification of graded Koszul modules over the polynomial algebra $\Gamma_n = K[x_0, \dots, x_n]$.
• To determine the structure of Auslander-Reiten components in $\operatorname{Coh}(\mathbb{P}^n)$, showing that all but finitely many have the shape $\mathbb{Z}\Delta$.
• To provide a new characterization of locally free sheaves (vector bundles) on $\mathbb{P}^n$ as sheafifications of specific Koszul modules.

Proposed method

• Utilizes Koszul duality between the polynomial algebra $\Gamma_n$ and the exterior algebra $\Lambda_n$, establishing a derived equivalence via the BGG correspondence.
• Defines the category $\widehat{K}_\Gamma$ as the full subcategory of $\operatorname{Coh}(\mathbb{P}^n)$ formed by sheafifications of graded Koszul $\Gamma_n$-modules.
• Applies relative almost split sequences in $\widehat{K}_\Gamma$ by exploiting the duality between $K_\Lambda$ and $K_{\Gamma^{op}}$, where $\Lambda$ is the Yoneda algebra of $\Gamma_n$.
• Uses the quotient category $QGr\Gamma$ and the functor $\sim$ to relate graded modules to coherent sheaves on $\mathbb{P}^n$, with $\widetilde{M} = (\Gamma[f^{-1}] \otimes M)_0$ for homogeneous $f$.
• Applies Serre's theorem to establish the equivalence $\Gamma_* : \operatorname{Coh}(X) \to QGr\Gamma$ for $X = \mathbb{P}^n$ defined by a homogeneous ideal $I$, with inverse $\sim$.
• Analyzes the structure of Auslander-Reiten components via the action of the shift functor $\sigma$ on Koszul modules and their sheafifications, showing that components are either $\mathbb{Z}\Delta$ or of a specific infinite zigzag form.

Experimental results

Research questions

• RQ1 What is the structure of Auslander-Reiten components in the category $\operatorname{Coh}(\mathbb{P}^n)$, and how do they relate to Koszul duality?
• RQ2 Which graded modules over the polynomial algebra $\Gamma_n$ correspond to locally free sheaves on $\mathbb{P}^n$?
• RQ3 Can relative almost split sequences be defined in $\operatorname{Coh}(\mathbb{P}^n)$ using Koszul duality, and what do they reveal about the category’s structure?
• RQ4 Do there exist indecomposable vector bundles on $\mathbb{P}^n$ of arbitrarily high rank, and how can they be constructed?

Key findings

• All but finitely many Auslander-Reiten components in $\operatorname{Coh}(\mathbb{P}^n)$ have the shape $\mathbb{Z}\Delta$, where $\Delta$ is the separated quiver of $\Gamma_n/J^2$, with $J$ the Jacobson radical.
• The sheafification of Koszul modules over $\Gamma_n$ forms a full subcategory $\widehat{K}_\Gamma \subset \operatorname{Coh}(\mathbb{P}^n)$, and any coherent sheaf on $\mathbb{P}^n$ is isomorphic to an object in $\widehat{K}_\Gamma$ up to shift.
• For any indecomposable Koszul $\Lambda$-module $M$, the relative Auslander-Reiten component of $\pi F(M)$ is either contained in the non-negative part of $\mathbb{Z}\Delta$ or has a specific infinite zigzag structure.
• If $\pi F(M)$ is locally free, then all sheaves in its Auslander-Reiten component are locally free, and such components arise precisely when $M$ lies in the non-negative part of $\mathbb{Z}\Delta$.
• The rank of sheafifications $\pi F(\sigma^j M_i)$ grows strictly with $j$ and $i$, implying the existence of indecomposable vector bundles of arbitrary high rank on $\mathbb{P}^n$ for $n > 1$.
• The construction of such bundles relies on exact sequences of the form $0 \to X \to Y \to Z \to 0$ with $\operatorname{rk}(Y) = \operatorname{rk}(X) + \operatorname{rk}(Z)$, and the recursive rank growth via $\sigma$-action on modules.