  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, Cortes-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.
