Rigid Dualizing Complexes via Differential Graded Algebras (Survey)

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.

💡 Research Summary

This survey collects and clarifies recent advances concerning rigid dualizing complexes over commutative algebras, emphasizing the indispensable role of differential graded (DG) algebras in their construction. The paper begins with a concise recollection of Grothendieck’s notion of a dualizing complex: for a Noetherian commutative ring (A) of finite Krull dimension, a bounded complex (D) of (A)-modules with finite injective dimension and finite cohomology such that the functor (\mathbf{R}!\operatorname{Hom}_A(-,D)) yields a duality on the derived category of finitely generated modules. While dualizing complexes are central to the theory of Grothendieck duality, they lack a natural functorial behavior with respect to ring homomorphisms (f\colon A\to B).

To overcome this deficiency, the concept of a rigid complex is introduced. A rigid complex is a pair ((D,\eta)) where (D) is a dualizing complex and (\eta) is a “rigidifying isomorphism” \