Pure-direct-injective Sheaves

We study the notion of pure-direct-injective objects in categories of sheaves by introducing the class of pure-direct-injective sheaves with respect to the geometrical purity. We investigate whether pure-direct-injective objects are preserved or reflected by the three functors associated with an open subset: the restriction, the extension by zero, and the direct image. We also exhibit a few examples for the pure-direct-injective sheaves, which enable us to distinguish pure-injective sheaves from the pure-direct-injective sheaves. Finally, we show that any coherent sheaf is pure-injective in the subcategory of coherent sheaves over a Noetherian scheme [Formula: see text]. Also, over [Formula: see text], where [Formula: see text] is an Artinian principal ideal ring, every quasi-coherent sheaf is pure-direct-injective in the subcategory of quasi-coherent sheaves.