Absolute purity in the category of quasi coherent sheaves

Abstract

In this paper we consider the class of absolutely pure and Fp-injective quasi-coherent sheaves. We show that these two classes of quasi-coherent sheaves over a locally coherent scheme are equivalent. As a corollary we will show that the class of absolutely pure quasi-coherent sheaves over such a scheme is an enveloping and a covering class. It is proved that over a locally coherent scheme, the pair (?(Abs(X),Abs(X)) is a cotorsion theory. The existence of a duality between absolutely pure envelopes and flat covers is proved as expected.