[Paper Review] A note on group actions on algebraic stacks

Matthieu Romagny|ArXiv.org|May 16, 2003

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper establishes foundational definitions for group actions on algebraic stacks and proves the algebraicity of fixed point stacks under proper flat group schemes and quotient stacks under separated flat group schemes. It shows that the quotient stack 𝒪/M/G is isomorphic to the stack of G-torsors, and is algebraic when G is flat, separated, and of finite presentation over the base scheme S.

ABSTRACT

We give the basic definitions of group actions on (algebraic) stacks, and prove the existence of fixed points and quotients as (algebraic) stacks.

Motivation & Objective

• To formalize the theory of group actions on algebraic stacks, which had not been systematically treated in the literature despite frequent use in moduli problems.
• To address the need for a rigorous framework in applications such as orbifold Gromov-Witten theory and Hurwitz stacks parameterizing Galois covers.
• To establish the existence and algebraicity of quotient stacks and fixed point stacks under natural geometric assumptions on the group scheme.
• To provide a precise characterization of the quotient stack as the stack of G-torsors, ensuring compatibility with descent and fppf topology.
• To extend classical results on quotients of schemes to the setting of algebraic stacks, particularly in mixed characteristic and for finite flat group schemes.

Proposed method

• Define group actions on algebraic stacks via 2-commutative diagrams in the 2-category of stacks, using 1-morphisms and 2-isomorphisms to encode equivariance.
• Construct the quotient stack 𝒪/M/G as the stack associated to a prestack of G-equivariant maps, ensuring it represents the functor of G-torsors.
• Prove that the quotient stack is isomorphic to the stack of G-torsors by constructing a fully faithful and locally essentially surjective morphism.
• Establish algebraicity of the quotient using the diagonal morphism and representability of Isom-schemes, leveraging fppf descent and base change properties.
• Use the canonical morphism 𝒪/M → 𝒪/M/G to show that the quotient is fppf-locally representable via an atlas from 𝒪/M.
• Apply the criterion of algebraicity from Laumon-Moret-Bailly to verify that the diagonal is representable, separated, and quasi-compact.

Experimental results

Research questions

• RQ1 Under what conditions does the fixed point stack of a group action on an algebraic stack exist and remain algebraic?
• RQ2 When is the quotient stack of an algebraic stack by a group action itself algebraic?
• RQ3 How can the quotient stack be characterized as the stack of torsors for a given group scheme action?
• RQ4 What is the relationship between the stack of G-linearized line bundles on 𝒪/M and the Picard stack of the quotient stack?
• RQ5 How does the formation of the quotient stack commute with base change in the base scheme S?

Key findings

• The quotient stack 𝒪/M/G is isomorphic to the stack of G-torsors over 𝒪/M, denoted (𝒪/M/G)*, and this identification is canonical.
• If G is flat, separated, and of finite presentation over S, then the quotient stack 𝒪/M/G is algebraic, hence an object in the 2-category of algebraic stacks.
• The canonical projection 𝒪/M → 𝒪/M/G is an fppf morphism and serves as the universal G-torsor over the quotient stack.
• The formation of the quotient stack commutes with base change on the base scheme S, ensuring functoriality.
• The diagonal morphism of 𝒪/M/G is representable, separated, and quasi-compact, which implies algebraicity via standard criteria.
• For a G-linearized line bundle F on 𝒪/M, the stack of G-linearized line bundles is isomorphic to the Picard stack of the quotient, i.e., 𝒫ic(𝒪/M/G) ≅ 𝒫ic^G(𝒪/M) ≅ 𝒫ic(𝒪/M)^G.