A constructive approach to the module of twisted global sections on relative projective spaces

TL;DR

This paper presents a new, efficient constructive method for computing twisted global sections on relative projective spaces, improving upon traditional approaches by replacing complex resolutions with simpler linear regularity techniques.

Contribution

It introduces a novel description based on the relative BGG-correspondence that avoids the full BGG process, using linear saturation and regularity for more efficient computation.

Findings

The new method accurately computes modules of twisted global sections.

Elementary proofs confirm the equivalence of the two descriptions.

The approach simplifies computations by replacing Tate resolutions with linear regularity.

Abstract

The ideal transform of a graded module $M$ is known to compute the module of twisted global sections of the sheafification of $M$ over a relative projective space. We introduce a second description motivated by the relative BGG-correspondence. However, our approach avoids the full BGG-correspondence by replacing the Tate resolution with the computationally more efficient purely linear saturation and the Castelnuovo-Mumford regularity with the often enough much smaller linear regularity. This paper provides elementary, constructive, and unified proofs that these two descriptions compute the (truncated) modules of twisted global sections. The main argument relies on an established characterization of Gabriel monads.