Absolutely pure covers of sheaves

Let [Formula: see text] be a scheme and [Formula: see text] be the category of all sheaves of [Formula: see text]-modules. Enochs’ conjecture states that any covering class in [Formula: see text] is closed under directed colimits. In this work, if Abs[Formula: see text] is the class of all absolutely pure objects in [Formula: see text], then we prove the validity of the conjecture for Abs[Formula: see text]. This gives a characterization of locally coherent schemes. In addition, we study some homological properties of absolutely pure sheaves of [Formula: see text]-modules and answer a question raised in [P. Rothmaler, When are pure injective envelopes of flat modules flat? Comm. Algebra 30(6) (2002) 3077–3085].