A Horrocks' theorem for reflexive sheaves

TL;DR

This paper introduces $m$-tail reflexive sheaves on projective spaces, establishing a rank lower bound, characterizing minimal rank cases, and constructing extensive families of higher rank sheaves.

Contribution

It defines $m$-tail reflexive sheaves, proves a rank inequality, and provides classifications and constructions for these sheaves on projective spaces.

Findings

Rank of $m$-tail reflexive sheaves is at least $nm - m$.

Complete classification of minimal rank $m$-tail reflexive sheaves.

Construction of large families of higher rank $m$-tail reflexive sheaves.

Abstract

In this paper, we define $m$-tail reflexive sheaves as reflexive sheaves on projective spaces with the simplest possible cohomology. We prove that the rank of any $m$-tail reflexive sheaf $\mathcal{E}$ on $\mathcal{P}^n$ is greater or equal to $ nm-m$. We completely describe $m$-tail reflexive sheaves on $\mathcal{P}^n$ of minimal rank and we construct huge families of $m$-tail reflexive sheaves of higher rank.