Approximation of sheaves on algebraic stacks (Q2796761)

The article under review studies the problem of approximations of quasi-coherent sheaves by finite-type subsheaves in the context of algebraic stacks, with very mild hypotheses.NEWLINENEWLINEIt is well-known that in noetherian situations, every such sheaf is the union of its coherent subsheaves. This paper considers a more general situation, that covers non-noetherian cases. The main result is that on a quasi-compact and quasi-separated algebraic stack, every quasi coherent sheaf is the union of its quasi-coherent subsheaves of finite type. The author conjectures that a stronger result should be true: every quasi-coherent sheaf should be a directed colimit of finitely presented sheaves. Some applications of the theorem are given at the end, including a version of Zariski's main theorem for stacks, and the existence of flattening stratifications for finitely presented morphisms of stacks.NEWLINENEWLINEThe proof is via descent from a smooth atlas. In order to prepare for it, the author considers extensions to the stack setting (without any noetherian hypotheses) of several concepts that are familiar for schemes, for example associated points and ``relative assassins'', pure morphisms and minimal subsheaves. A key step in the proof is a characterization of pure morphisms via a homological condition, that the author calls ``homological projectivity'', that generalizes an earlier result of Raynaud-Gruson, about characterizing affine pure morphisms of schemes.