The Hilbert Stack

Details The Hilbert Stack The Hilbert Stack

2010-11-24

arxiv.org/abs/1011.5484v1

Mathematics

Algebraic Geometry

48 pages

Scientific paper

We show that for any locally finitely presented morphism of algebraic stacks X -> S with quasi-compact and separated diagonal, there is an algebraic stack HS(X/S), the Hilbert stack, parameterizing proper algebraic stacks with finite diagonal mapping quasi-finitely to X. The technical heart of this is a generalization of formal GAGA to a non-separated morphism of algebraic stacks, something that was previously unknown for a morphism of schemes. We also employ derived algebraic geometry, in an essential way, to prove the algebraicity of the stack HS(X/S). The Hilbert stack, for a morphism of algebraic spaces, was claimed to exist by M. Artin in [Appendix 1, MR0399094], but was left unproved due to a lack of foundational results for non-separated algebraic spaces.

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