Rigid Dualizing Complexes via Differential Graded Algebras (Survey)

TL;DR

This survey reviews recent developments in rigid dualizing complexes over commutative algebras, emphasizing the role of differential graded algebras in their construction and applications.

Contribution

It provides a comprehensive overview of rigid dualizing complexes, their functorial properties, and their applications to Cohen-Macaulay homomorphisms and dualizing sheaves.

Findings

Rigid dualizing complexes can be constructed using differential graded algebras.

Rigid complexes help understand Cohen-Macaulay homomorphisms.

The survey summarizes recent advances in the theory of dualizing complexes.

Abstract

In this article we survey recent results on rigid dualizing complexes over commutative algebras. We begin by recalling what are dualizing complexes. Next we define rigid complexes, and explain their functorial properties. Due to the possible presence of torsion, we must use differential graded algebras in the constructions. We then discuss rigid dualizing complexes. Finally we show how rigid complexes can be used to understand Cohen-Macaulay homomorphisms and relative dualizing sheaves.