The Hilbert Stack

Abstract

Let π: X -> S be a morphism of algebraic stacks that is locally of finite presentation with affine stabilizers. We prove that there is an algebraic S-stack, the Hilbert stack, parameterizing proper algebraic stacks mapping quasi-finitely to X. This was previously unknown, even for a morphism of schemes.

29 pages; major revision; no longer uses derived algebraic geometry; introduced Generalized Stein factorization; greatly simplified dévissage argument; two new appendices on coherent cohomology on formal schemes and henselian pairs

References in corpus (4)

• Quotients by finite equivalence relations
• Artin's axioms, composition and moduli spaces
• Cohomology and base change for algebraic stacks
• Openness of versality via coherent functors

Cited by in corpus (11)

• Coherent Tannaka duality and algebraicity of Hom-stacks
• The étale local structure of algebraic stacks
• General Hilbert stacks and Quot schemes
• Autoequivalences of twisted K3 surfaces
• Remarks about bubbles
• Mayer-Vietoris squares in algebraic geometry
• Generalizing the GAGA Principle
• Addendum: Étale dévissage, descent and pushouts of stacks
• Nori fundamental gerbe of essentially finite covers and Galois closure of towers of torsors
• The moduli space of cyclic covers in positive characteristic
• Good Hilbert functors