Serre functors and local duality for affine quotients

TL;DR

This paper investigates Serre functors for categories of quasicoherent sheaves on certain algebraic stacks, establishing their description via local cohomology and deriving duality theorems for local rings.

Contribution

It provides a new description of Serre functors on specific stacks and develops analogues of Matlis and local duality theorems in this context.

Findings

Serre functor is given by tensoring with local cohomology at the closed orbit.

Develops analogues of Matlis and local duality theorems for local rings.

Establishes a connection between Serre functors and local duality in algebraic stacks.

Abstract

The purpose of this short note is to study Serre functors of categories of quasicoherent sheaves on stacks of the form $\mathcal{Y} = \mathrm{Spec} A/G$ where $G$ is a reductive group acting on $\mathrm{Spec} A$ with a unique closed orbit. We show that the Serre functor is given by tensoring with the local cohomology of $\omega_\mathcal{Y}$ at the unique closed orbit. Using this description, we develop analogues of the Matlis and local duality theorems for local rings.