Direct products of Gorenstein flat modules over coherent rings

Abstract

We construct a commutative coherent local algebra of Krull dimension zero and a countably generated projectively coresolved Gorenstein flat module whose countable power has infinite Gorenstein flat dimension. This answers negatively the question of whether Gorenstein flat modules over a right coherent ring are closed under direct products. As a consequence, we obtain a pure-injective Ding injective module whose character module has infinite Gorenstein flat dimension and whose character bidual has infinite Gorenstein injective dimension. We also give equivalent conditions for Gorenstein flat modules over a right coherent ring to be closed under direct products, in terms of relative Mittag--Leffler conditions, stationarity, and Gorenstein flat and injective dimensions of character biduals. For the constructed ring, we exhibit a decreasing countable sequence of strong submodules of a free module whose intersection has infinite Gorenstein flat dimension. Thus the abstract elementary class of Gorenstein flat modules, with the strong submodule relation, has finite character but does not admit intersections.