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arXiv · 2212.09639

Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity

Leonid Positselski·Jan Stovicek
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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, 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.

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Leonid Positselski, Jan Stovicek. 2024-12-10. Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. https://doi.org/10.1007/s10468-024-10296-4

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