Any algebraic stack X can be represented by a groupoid object in the category of schemes: that is, a pair of schemes Ob, Mor and morphisms source, target: Mor → Ob, inversion: Mor → Mor, composition: Mor ×_{Ob} Mor → Mor and identity: Ob → Mor that satisfy certain axioms. Yet this description of the stack X might be misleading.
Namely, given a field F which is not algebraically closed, we have a natural functor between the groupoid (Ob(F),Mor(F)) and the groupoid X(F). While this functor is fully faithful, it is often not essentially surjective.
In joint work with Nir Avni (in progress) we show that any algebraic groupoid has a presentation such that this functor will be essentially surjective for many fields (and under some assumptions on the stack, for any field). The results are also extended to Henselian rings.
Despite the title, the talk will be about usual stacks and not infinity-stacks, yet in some of the proofs it is more convenient to use the language of higher categories and I’ll try to explain why.
No prior knowledge of infinity stacks will be assumed, but a superficial acquaintance with usual stacks will be helpful.
- Séminaire Géométrie et théorie des modèles