The existence of flat covers over Noetherian rings of finite Krull dimension

Abstract

Bass characterized the rings R with the property that every left R-module has a projective cover. These are the left perfect rings. A ring is left perfect if and only if the class of projective R-modules coincides with the class of flat R-modules, so the projective covers over these rings are flat covers. This prompts the conjecture that over any ring R, every left R-module has a flat cover. Known classes of rings for which the conjecture holds include Von Neumann regular rings (trivially), the left perfect rings (Bass), Prufer domains (Enochs), and then more generally, all right coherent rings of finite weak global dimension (Belshoff, Enochs, Xu). In this paper we show that the conjecture holds for all commutative Noetherian rings of finite Krull dimension and so for all local rings and all coordinate rings of affine algebraic varieties.