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 $(^\perp\big({\rm Abs}(X)\‎ ,‎\ {\rm Abs}(X)\big)$ is a cotorsion theory‎. ‎The existence of a duality between absolutely pure envelopes and flat covers is proved as expected‎.