Countably generated flat modules are quite flat

Abstract

We prove that if R is a commutative Noetherian ring, then every countably generated flat R-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of R in countable multiplicative subsets. We also show that if the spectrum of R is of cardinality less than κ, where κ is an uncountable regular cardinal, then every flat R-module is a transfinite extension of flat modules with less than κ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. Over a non-Noetherian commutative ring, countably presented flat modules do not need to be quite flat; still over von Neumann regular rings, S-almost perfect rings, and over strongly discrete valuation domains they are.