Fixed point stacks under groups of multiplicative type

TL;DR

This paper proves that fixed point stacks under groups of multiplicative type are algebraic, extending classical theorems to algebraic stacks with affine, finitely presented diagonals, thereby broadening the understanding of group actions in algebraic geometry.

Contribution

It extends two fundamental theorems of SGA3 to the setting of algebraic stacks, establishing algebraicity of fixed point stacks under multiplicative type group actions.

Findings

Fixed point stacks are algebraic under certain conditions.

Extended classical theorems to algebraic stacks.

Provided new tools for studying group actions on stacks.

Abstract

We prove that if a group scheme of multiplicative type acts on an algebraic stack with affine, finitely presented diagonal then the stack of fixed points is algebraic. For this, we extend two theorems of [SGA3.2] on functors of subgroups of multiplicative type, and functors of homomorphisms from a group of multiplicative type.