Computing weighted sheaf cohomology using noncommutative differential modules
Michael K. Brown, Daniel Erman
TL;DR
The paper develops a comprehensive framework to compute sheaf cohomology for coherent sheaves on weighted projective stacks and spaces by translating to differential modules over the exterior algebra via a nonstandard BGG correspondence, and then to Tate resolutions to extract cohomology data. It introduces differential $E$-modules and minimal free flag resolutions to handle weighted gradings, and proves that the Tate resolution of a sheaf encodes all twists $\mathcal F(j)$ through Betti numbers, enabling a finite, computable procedure. Two主要 algorithmic streams are provided: one for weighted projective stacks and one for weighted projective spaces/varieties, including a method to reconcile twists that do not commute with tilde via a finite module $M'$ and a feeding of stack computations into the variety setting. The approach promises computational advantages by working over $E$ rather than $S$, and lays the groundwork for practical implementations in Macaulay2, with important implications for cohomology computations in toric and weighted settings. Overall, the paper delivers a theoretically solid, algorithmically explicit pathway to cohomology tables in the weighted setting, expanding the toolkit for toric and stacky geometry.
Abstract
We describe a novel method for computing sheaf cohomology over weighted projective spaces and stacks using exterior algebra and differential module techniques, generalizing an algorithm due to Eisenbud-Fløystad-Schreyer over projective space.