Around the year 2000, Eklof-Trlifaj [4] and Bican-El Bashir-Enochs [1] proved the Flat Cover Conjecture. Their proof introduced a new, extremely useful concept in homological algebra: deconstructible classes of modules. I discovered a useful characterization of deconstructibility in terms of set-theoretic elementary submodels, and proved the following two theorems, which answered some
questions from the algebra literature:
Theorem 1: The scheme "all cotorsion pairs in R-Mod (for any hered-
itary ring R) are complete" is independent of ZFC, modulo the consistency of
Vopenka's Principle.
Theorem 2: Vopenka's Principle implies that for any class X of modules
(over any ring), the class of X-Gorenstein Projective modules is a precovering
class.