David Kruml, Masaryk University (Brno, Czech Republic)
Flat precovers in varieties

In 2000,Bican, El Bashir and Enochs proved a conjecture that every module has a flat (pre)cover. Flat objects can be defined in any variety V as directed colimits of finitely presented projective objects and then we can ask for an existence of flat precovers in V. We will start with an example of a variety where the flat covers conjecture (FCC) is false. It seems that there is a correspondence between FCC and a special condition of cosimplicial objects in V. We will discuss dependence on set-theory. I will show that Johnstone's naturally Maltsev varieties satisfy the condition at least in its simplest case.