Equalizers and Flatness Properties of Acts

Abstract

In 1971, Stenström proved that the strongly flat right acts A _S over a monoid S (that is, the acts that are directed colimits of finitely generated free acts) are those for which the functor A _S ⊗ - (from the category of left S-acts into the category of sets) preserves pullbacks and equalizers. In 1991, it was shown by Bulman-Fleming that in fact pullback preservation alone was sufficient. Recently, the present authors, together with Valdis Laan, published two papers (Comm. Algebra 29(2) (2001), 829-850, 851-878) giving a spectrum of properties based on preservation of particular types of pullbacks that is sufficiently broad to capture all known notions of flatness, and to give new ones besides. The present paper initiates an analogous study for equalizer preservation properties. Again it turns out that the “standard” properties of flatness, weak flatness, principal weak flatness, and torsion freeness can all be described in terms of equalizer preservation, and that new properties arise as well.