[Paper Review] Hom stacks

Masao Aoki|arXiv (Cornell University)|Mar 17, 2005

Algebraic Geometry and Number Theory19 citations

TL;DR

This paper establishes that the Hom 2-functor parameterizing 1-morphisms between algebraic stacks is representable by an algebraic stack under suitable conditions, using Artin's criterion for algebraizability. As a key application, it proves the Picard 2-functor, which classifies line bundles on algebraic stacks, is also representable by an algebraic stack.

ABSTRACT

We study Hom 2-functors parameterizing 1-morphisms of algebraic stacks, and prove that it is representable by an algebraic stack under certain conditions, using Artin's criterion. As an application we study Picard 2-functors which parameterizes line bundles on algebraic stacks.

Motivation & Objective

• To investigate the representability of the Hom 2-functor that parameterizes 1-morphisms between algebraic stacks.
• To determine conditions under which this Hom 2-functor is representable by an algebraic stack.
• To apply the representability result to the Picard 2-functor, which classifies line bundles on algebraic stacks.
• To extend classical moduli theory to higher categorical structures using algebraic stacks.
• To provide a foundational framework for studying line bundles and morphisms in algebraic geometry via 2-categorical methods.

Proposed method

• Applies Artin's criterion for algebraizability of functors to establish representability of the Hom 2-functor.
• Analyzes the 2-functor of 1-morphisms between algebraic stacks as a stack over the big fppf site.
• Uses deformation theory and infinitesimal lifting properties to verify the conditions of Artin's criterion.
• Establishes that the Hom 2-functor satisfies the necessary local finiteness and openness of versal families conditions.
• Applies the representability result to the Picard 2-functor by showing it satisfies the same criteria.
• Demonstrates that the Picard 2-functor is representable by an algebraic stack under the same conditions as the Hom 2-functor.

Experimental results

Research questions

• RQ1 Under what conditions is the Hom 2-functor of 1-morphisms between algebraic stacks representable by an algebraic stack?
• RQ2 Can Artin's criterion be applied to prove representability of the Hom 2-functor in the context of algebraic stacks?
• RQ3 Is the Picard 2-functor, which classifies line bundles on algebraic stacks, representable by an algebraic stack?
• RQ4 How do deformation-theoretic properties of morphisms influence the representability of the Hom 2-functor?
• RQ5 What structural properties of algebraic stacks ensure the representability of moduli 2-functors like Hom and Picard?

Key findings

• The Hom 2-functor parameterizing 1-morphisms between algebraic stacks is representable by an algebraic stack when the target stack is locally of finite presentation and has a diagonal that is locally of finite type.
• The representability of the Hom 2-functor is established via Artin's criterion, relying on deformation and obstruction theory.
• The Picard 2-functor, which assigns to each scheme the groupoid of line bundles on the pullback of the stack, is representable by an algebraic stack under the same conditions.
• The proof shows that the Picard 2-functor satisfies the necessary conditions for algebraizability, including openness of versal families and effective formal gluing.
• The results extend classical representability theorems to the 2-categorical setting of algebraic stacks.
• The framework provides a systematic method to construct moduli spaces for morphisms and line bundles in higher algebraic geometry.