Rigid dualizing complexes via differential graded algebras (survey)

Summary

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.

Introduction

This short article is based on a lecture I gave at the “Workshop on Triangulated Categories”, Leeds, August 2006. It is a survey of recent results on rigid dualizing complexes over commutative rings. Most of these results are joint work of mine with James Zhang. The idea of rigid dualizing complex is due to Michel Van den Bergh.

By default all rings considered in this article are commutative. We begin by recalling the notion of dualizing complex over a noetherian ring A. Next let B be a noetherian A-algebra. We define what is a rigid complex of B-modules relative to A. In making this definition we must use differential graded algebras (when B is not flat over A). The functorial properties of rigid complexes are explained. We then discuss rigid dualizing complexes, which by definition are complexes that are both rigid and dualizing. Finally we show how rigid complexes can be used to understand Cohen-Macaulay homomorphisms and relative dualizing sheaves.