[Paper Review] Rigid Dualizing Complexes via Differential Graded Algebras (Survey)

Amnon Yekutieli|ArXiv.org|Sep 13, 2007

Graph theory and applicationsMathematics22 citations

TL;DR

This survey establishes a framework for rigid dualizing complexes over commutative noetherian rings using differential graded (DG) algebras to handle torsion in non-flat base changes. It introduces rigid complexes relative to a base ring, proves their existence and uniqueness up to unique isomorphism, and applies them to characterize Cohen-Macaulay and Gorenstein homomorphisms via a relative dualizing module that is rigid and functorial.

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.

Motivation & Objective

• To develop a functorial and canonical framework for dualizing complexes over commutative rings by eliminating automorphisms through rigidity.
• To address the failure of derived base change in non-flat settings by employing differential graded algebras to resolve torsion issues.
• To characterize Cohen-Macaulay and Gorenstein homomorphisms using a rigid complex that induces a relative dualizing module.
• To establish a base change theorem for rigid complexes in cartesian diagrams of rings, preserving rigidity and duality properties.
• To provide a geometrically meaningful construction of dualizing complexes on schemes via perverse coherent sheaves, building on the algebraic foundation.

Proposed method

• Use semi-free DG algebras over a base ring $A$ to construct resolutions of $B$-algebras, even when $B$ is not flat over $A$.
• Define a rigid complex over $B$ relative to $A$ via a quasi-isomorphism $\tilde{B} \to B$ and a rigidifying isomorphism in the derived category.
• Employ the derived tensor product $M \otimes^\mathrm{L}_A M$ in the DG setting to define rigidity, replacing the non-derived $M \otimes_A M$.
• Introduce the notion of a 'traction' $\mathbb{K} \to A$ for a tractable ring $A$, where $\mathbb{K}$ is regular and noetherian of finite Krull dimension.
• Construct the rigid dualizing complex $R_{B/A}$ as a relative version of $R_{A/\mathbb{K}}$, satisfying $R_{A/\mathbb{K}} \otimes^\mathrm{L}_A R_{B/A} \cong R_{B/\mathbb{K}}$.
• Use the rigid complex $R_{B/A}$ to define the relative dualizing module $\boldsymbol{\omega}_{B/A}$ as a direct sum of shifts of flat $B_i$-modules over connected components of $\operatorname{Spec}B$.

Experimental results

Research questions

• RQ1 How can dualizing complexes be made functorial and canonical in the presence of torsion, particularly when base change is not flat?
• RQ2 What is the role of differential graded algebras in constructing rigid complexes over non-flat algebras?
• RQ3 How does the rigid complex $R_{B/A}$ relate to the classical dualizing complex in the case where $A$ is Gorenstein?
• RQ4 Can Cohen-Macaulay and Gorenstein properties of ring homomorphisms be characterized algebraically using the rigid complex $R_{B/A}$?
• RQ5 Does the rigid complex $R_{B/A}$ satisfy a base change property in cartesian diagrams of rings, and if so, how is rigidity preserved?

Key findings

• Rigid dualizing complexes exist and are unique up to a unique rigid isomorphism for tractable noetherian rings, resolving the ambiguity of automorphisms in classical dualizing complexes.
• The rigid complex $R_{B/A}$ satisfies the base change property: $R_{A/\mathbb{K}} \otimes^\mathrm{L}_A R_{B/A} \cong R_{B/\mathbb{K}}$ for any traction $\mathbb{K} \to A$.
• When $A$ is Gorenstein, the rigid complex $R_{B/A}$ is a dualizing complex over $B$, and the relative dualizing module $\boldsymbol{\omega}_{B/A}$ is a shift of a flat $B$-module.
• A flat homomorphism $f^*: A \to B$ of essentially finite type is essentially Cohen-Macaulay if and only if $R_{B/A} \cong \bigoplus_i \boldsymbol{\omega}_{B_i/A}[n_i]$ for flat $B_i$-modules $\boldsymbol{\omega}_{B_i/A}$.
• The relative dualizing module $\boldsymbol{\omega}_{B/A}$ is invertible if and only if $f^*$ is essentially Gorenstein, providing a cohomological characterization.
• The base change theorem holds: for a cartesian diagram of rings, the relative dualizing module satisfies $\boldsymbol{\omega}_{B'/A'} \cong A' \otimes_A \boldsymbol{\omega}_{B/A}$, preserving rigidity.