Projective Covers of Flat Contramodules

Abstract We show that a direct limit of projective contramodules (over right linear topological ring) is if it has cover. A similar result obtained for $\infty $-strictly flat dimension not exceeding $1$, using an argument based on the notion Jacobson radical. Covers and precovers limits more general classes objects, both in abelian categories with exact nonexact limits, are also discussed, eye towards Enochs conjecture about covers locally split (mono)morphisms as main technique. In particular, we offer simple elementary proof left class $n$-tilting cotorsion pair category limits.