2024/12/26 by Sean Cox, Cox, Sean
Computer Science · #Category Theory (math.CT) #Commutative Algebra (math.AC) #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2412.19380
openalex publication_date 2024/12/26 · openalex created_date 2024/12/31 · openalex updated_date 2026/07/31
Deconstructibility is an often-used sufficient condition on a class C of modules that allows one to carry out homological algebra \emphrelative to C. The principle Maximum Deconstructibility (MD) asserts that a certain necessary condition for a class to be deconstructible is also sufficient. MD implies, for example, that the classes of Gorenstein Projective modules, Ding Projective modules, their relativized variants, and all torsion classes are deconstructible over any ring. MD was known to follow from Vopěnka's Principle and imply the existence of an ω1-strongly compact cardinal. We prove that MD is equivalent to Vopěnka's Principle, and to the assertion that each torsion class of abelian groups is generated by a single group within the class (yielding the converse of a theorem of Göbel and Shelah).