GORENSTEIN PROJECTIVE PRECOVERS AND FINITELY PRESENTED MODULES

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. It is known that if the ring has finite Gorenstein global dimension, then every module has a Gorenstein projective precover. We prove here a “reduction” property — we show that, over any ring, it suffices to consider finitely presented modules: if there exists a nonnegative integer n such that every finitely presented module has Gorenstein projective dimension ≤ n, then the class of Gorenstein projective modules is special precovering.