Note on Motivic Semiorthogonal Decompositions for Elementary Abelian 2-Group Actions

TL;DR

This paper establishes a semiorthogonal decomposition for the derived category of certain smooth Deligne-Mumford stacks with elementary Abelian 2-group stabilizers, linking it to the derived categories of inertia stack components.

Contribution

It introduces a motivic semiorthogonal decomposition framework for stacks with elementary Abelian 2-group actions, extending previous results to a broader class of stacks.

Findings

Semiorthogonal decomposition exists for the derived category of the stack.

Decomposition components are equivalent to derived categories of inertia stack parts.

Applicable to stacks with smooth coarse moduli of inertia stack.

Abstract

Let $\mathcal{X}$ be a smooth Deligne-Mumford stack which is generically a scheme and has quasi-projective coarse moduli. If $\mathcal{X}$ has elementary Abelian 2-group stabilizers and the coarse moduli of the inertia stack is smooth, we show there exists a semiorthogonal decomposition of the derived category of $\mathcal{X}$ where the pieces are equivalent to the derived category of the components of the coarse moduli of the inertia stack.