A Topology on Points on Stacks

2020/05/20 by Atticus Christensen, Christensen, Atticus · 2 citations
Computer Science · Mathematics · #11R56 #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Primary 14G20 #Secondary 14A20

paper · pdf · doi:10.48550/arxiv.2005.10231

openalex publication_date 2020/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

For a variety over certain topological rings R, like ℤ_p or ℂ, there is a well-studied way to topologize the R-points on the variety. In this paper, we generalize this definition to algebraic stacks. For an algebraic stack \mathfrakX over many topological rings R, we define a topology on the isomorphism classes of R-points of \mathfrakX. We prove expected properties of the resulting topological spaces including functoriality. Then, we extend the definition to the case when R is the ring of adeles of some global field. Finally, we use this last definition to strengthen the local-global compatibility for stacky curves of Bhargava--Poonen to a strong approximation result.

Cited by

• The integral Hasse principle for stacky curves associated to a family of generalized Fermat equations
• Approximation theorems for classifying stacks over number fields

Related

• Foundations of Topological Stacks I
• Characterizing algebraic stacks
• Étale Stacks as Prolongations
• The pro-étale topology for algebraic stacks
• Topological and Smooth Stacks