Rigidity, Residues and Duality: Overview and Recent Progress

TL;DR

This paper reviews the theory of rigid residue complexes in algebraic geometry, highlighting recent advances in duality theorems for schemes and stacks, emphasizing the rigid approach's robustness and geometric applications.

Contribution

It introduces a novel rigid residue complex framework for Grothendieck duality, extending classical approaches to schemes and stacks with new theorems and geometric methods.

Findings

Proved the Rigid Residue Theorem for schemes.

Established the Rigid Duality Theorem for proper maps.

Extended duality results to Deligne-Mumford stacks.

Abstract

In this article we explain the theory of rigid residue complexes in commutative algebra and algebraic geometry, summarizing the background, recent results and anticipated future results. Unlike all previous approaches to Grothendiec Duality, the rigid approach concentrates on the construction of rigid residue complexes over rings, and their intricate yet robust properties. The geometrization, i.e. the passage to rigid residue complexes on schemes and Deligne-Mumford (DM) stacks, by gluing, is fairly easy. In the geometric part of the theory, the main results are the Rigid Residue Theorem and the Rigid Duality Theorem for proper maps between schemes, and for tame proper maps between DM stacks.