Let's call an Artin stack $X$ concentrated iff it is quasi-compact and quasi-separated (the latter usually being included in the definition of an Artin stack). The category of quasi-coherent sheaves $\mathrm{Qcoh}(X)$ is a locally presentable abelian Grothendieck category.

Question. Is $\mathrm{Qcoh}(X)$ locally finitely presentable? In other words, is every quasi-coherent sheaf a directed colimit of quasi-coherent sheaves of finite presentation?

This is known if $X$ is a concentrated scheme (EGA I, 6.9.12). It is also known when $X$ is noetherian (Lurie's Tannaka duality, Lemma 3.9). Is anything known about the general case?

Added: In his work about noetherian approximation (Theorem A + Prop. 2.7), David Rydh has shown that this is true for concentrated Deligne-Mumford stacks.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.