Precovering status of Gorenstein projective modules over arbitrary rings (ZFC)

Determine whether for every ring R the class 𝔾ℙ_R of Gorenstein projective left R-modules is a precovering class in ZFC (equivalently, whether every R-module admits a 𝔾ℙ_R-precover).

Background

The paper recalls a longstanding problem in Gorenstein homological algebra: whether the class of Gorenstein projective modules forms a precovering class over arbitrary rings. For Gorenstein flat modules the answer is known to be affirmative in ZFC, but for Gorenstein projectives it is only known for certain classes of rings and under large-cardinal assumptions.

Resolving this would align the Gorenstein projective theory with the classical flat/projective approximation framework and clarify the extent to which large-cardinal methods are necessary.