Pure resolutions of vector bundles on complex projective spaces

TL;DR

This paper investigates pure resolutions of vector bundles on complex projective spaces, demonstrating the existence of simple vector bundles with arbitrary homological dimensions and analyzing specific pure resolutions related to Koszul complexes and Gorenstein algebras.

Contribution

It introduces new constructions of simple vector bundles with arbitrary homological dimensions and analyzes their pure resolutions using quiver representations.

Findings

Existence of simple vector bundles of rank n on Pn with any homological dimension.

Pure resolutions from Koszul complexes and Gorenstein algebras have simple vector bundle syzygies.

Alternative proof of a key result using quiver representations.

Abstract

We prove three results on pure resolutions of vector bundles on projective spaces. First, we show that there are simple vector bundles of rank n on Pn with arbitrary homological dimension. We then analyze the pure resolutions given by the sheafification of the Koszul complex of a certain algebra and by the sheafification of the minimal free resolution of a compressed Gorenstein Artinian graded algebra, proving that their syzygies are simple vector bundles. Our main tool is a result originally established by Brambilla, for which we give an alternative proof using representations of quivers.