Sayrafi, Mahrud2024-07-242024-07-242024-05https://hdl.handle.net/11299/264360University of Minnesota Ph.D. dissertation. May 2024. Major: Mathematics. Advisor: Christine Berkesch. 1 computer file (PDF); viii, 110 pages.This thesis concerns the interplay of algebraic invariants like multigraded syzygies and the geometry of toric varieties. An active program in commutative algebra seeks to construct virtual resolutions of ideals and module over multigraded polynomial rings known as Cox rings in order to study algebraic geometry on toric varieties. We solve several problems in different aspects of this program: uniqueness of virtual resolutions on products of projective spaces; existence of short virtual resolutions and Orlov's conjecture in Picard rank 2; Horrocks' splitting criterion for vector bundles on smooth projective toric varieties; Castelnuovo--Mumford regularity and truncations of multigraded modules; bounds on Castelnuovo--Mumford regularity of modules and powers of ideals; and computing direct summand decompositions of multigraded modules and sheaves.enalgebraic geometrycommutative algebraderived categoriestoric varietiesThesis or Dissertation