Flat Quasi-coherent Sheaves as Directed Colimits, and Quasi-coherent Cotorsion Periodicity

Abstract

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably quasi-compact, countably quasi-separated scheme. Moreover, for three categories of complexes of flat quasi-coherent sheaves, we show that all complexes in the category can be obtained as directed colimits of complexes of locally countably presentable flat quasi-coherent sheaves from the same category. In particular, on a quasi-compact semi-separated scheme, every flat quasi-coherent sheaf is a directed colimit of flat quasi-coherent sheaves of finite projective dimension. In the second part of the paper, we discuss cotorsion periodicity in category-theoretic context, generalizing an argument of Bazzoni, Cortés-Izurdiaga, and Estrada. As the main application, we deduce the assertion that any cotorsion-periodic quasi-coherent sheaf on a quasi-compact semi-separated scheme is cotorsion.