[Paper Review] Derived Categories

Amnon Yekutieli|arXiv (Cornell University)|Jan 16, 2020

Algebraic structures and combinatorial models105 references36 citations

TL;DR

This prepublication book provides foundational and applied insights into derived categories in commutative and noncommutative algebra, emphasizing constructions and examples over axiomatics. It establishes key tools like DG algebras, triangulated and derived categories, K-injective/projective/flat modules, and develops advanced topics such as dualizing complexes, tilting bimodules, and noncommutative MGM equivalence, with applications to Calabi-Yau and Artin-Schelter regular rings.

ABSTRACT

This is the fourth (and last) prepublication version of a book on derived categories, that will be published by Cambridge University Press. The purpose of the book is to provide solid foundations for the theory of derived categories, and to present several applications of this theory in commutative and noncommutative algebra. The emphasis is on constructions and examples, rather than on axiomatics. Here are the topics covered in the book: - A review of standard facts on abelian categories. - Differential graded algebra (DG rings, DG modules, DG categories and DG functors). - Triangulated categories and triangulated functors between them. How they arise from the DG background. The homotopy category K(A,M) of DG A-modules in M. - Localization of categories. The derived category D(A,M), which is the localization of K(A,M) with respect to the quasi-isomorphisms. - Left and right derived functors of a triangulated functor. - K-injective, K-projective and K-flat DG modules. Their roles, and their existence in several important algebraic situations. - Dualizing and residue complexes over commutative noetherian rings, including Van den Bergh rigidity. - Perfect DG modules and tilting DG bimodules over NC (noncommutative) DG rings. - NC connected graded rings, including Artin-Schelter regular rings. Derived torsion for NC connected graded rings, its relation to the chi condition of Artin-Zhang, and the NC MGM Equivalence. Balanced dualizing complexes, their uniqueness, existence and trace functoriality. - NC rigid dualizing complexes, following Van den Bergh. The uniqueness and existence of these complexes, a few examples, and their relation to Calabi-Yau rings. Readers of this preview version are urged to write to the author with any comments regarding errors, suggestions or questions.

Motivation & Objective

• To establish a rigorous yet accessible foundation for derived categories in algebra, focusing on constructions and examples rather than abstract axioms.
• To explore the role of DG algebras, DG modules, and DG categories as a central framework for derived category theory.
• To develop the theory of derived functors, including left and right derived functors of triangulated functors, in the context of DG modules.
• To investigate dualizing and residue complexes over commutative noetherian rings, including Van den Bergh's rigidity and trace functoriality.
• To extend the theory to noncommutative settings, particularly NC connected graded rings, perfect modules, and tilting bimodules, and to establish the noncommutative MGM equivalence.

Proposed method

• Utilizes differential graded (DG) rings and DG modules as the primary algebraic framework for constructing derived categories.
• Constructs the homotopy category K(A,M) of DG A-modules in a module category M, then forms the derived category D(A,M) via localization at quasi-isomorphisms.
• Applies localization techniques to derive triangulated categories and study triangulated functors, particularly in relation to derived functors.
• Introduces and analyzes K-injective, K-projective, and K-flat DG modules, proving their existence in key algebraic contexts.
• Applies the theory to noncommutative rings via balanced and rigid dualizing complexes, using trace functoriality and the chi condition of Artin-Zhang.
• Establishes the noncommutative MGM equivalence using derived torsion and connections to Calabi-Yau properties in NC rings.

Experimental results

Research questions

• RQ1 How can derived categories be systematically constructed from DG algebras and DG modules in a way that supports homological algebra?
• RQ2 What conditions ensure the existence of K-injective, K-projective, and K-flat DG modules in important algebraic settings?
• RQ3 How do dualizing and residue complexes behave over commutative noetherian rings, and what role does Van den Bergh rigidity play?
• RQ4 In noncommutative algebra, what is the structure and significance of perfect DG modules and tilting DG bimodules over NC DG rings?
• RQ5 How does derived torsion in NC connected graded rings relate to the chi condition and the noncommutative MGM equivalence?

Key findings

• The derived category D(A,M) is well-defined as the localization of the homotopy category K(A,M) at quasi-isomorphisms, providing a robust framework for derived functors.
• K-injective, K-projective, and K-flat DG modules exist in important algebraic contexts, enabling the construction of left and right derived functors.
• Balanced dualizing complexes over NC connected graded rings are unique up to a canonical isomorphism and exhibit trace functoriality.
• Rigid dualizing complexes exist and are unique in the noncommutative setting, with explicit examples and connections to Calabi-Yau rings.
• The noncommutative MGM equivalence holds for NC connected graded rings, linking derived torsion to the chi condition of Artin-Zhang.
• Van den Bergh's rigidity theorem for dualizing complexes is established in the DG setting, reinforcing the uniqueness and structure of these complexes.