[Paper Review] A Course on Derived Categories

Amnon Yekutieli|arXiv (Cornell University)|Jun 28, 2012

Algebraic structures and combinatorial models16 references6 citations

TL;DR

This course notes provide a comprehensive introduction to derived categories and derived functors in homological algebra, focusing on foundational concepts like triangulated categories, derived functors, and dualizing complexes. The key contribution is a systematic development of rigid dualizing complexes for commutative and noncommutative algebras, with applications to algebraic geometry and perverse sheaves via descent and duality theory in derived categories.

ABSTRACT

These are notes for an advanced course given at Ben Gurion University in Spring 2012.

Motivation & Objective

• To provide a self-contained, accessible introduction to derived categories for graduate students and researchers in algebraic geometry and homological algebra.
• To clarify the role of derived functors and resolutions in extending classical homological constructions to derived settings.
• To establish the existence and uniqueness of rigid dualizing complexes over commutative and noncommutative algebras, and to explore their descent properties.
• To connect derived category theory to geometric objects such as perverse coherent sheaves on schemes via the derived Hom and duality functors.
• To demonstrate how rigid dualizing complexes unify duality in algebraic geometry and relate to modern concepts like Calabi-Yau algebras.

Proposed method

• Develops derived categories as localizations of homotopy categories of complexes at quasi-isomorphisms, using triangulated and localization techniques.
• Introduces K-injective and K-projective resolutions as tools for constructing right and left derived functors.
• Applies the theory of full triangulated subcategories and localization to define the derived category D(M) for an abelian category M.
• Uses the derived Hom functor RHom and derived tensor product to define duality in the derived category.
• Applies descent theory and the stack property of perverse sheaves to construct global rigid dualizing complexes on schemes from affine local data.
• Leverages rigidity conditions to ensure uniqueness and compatibility of dualizing complexes under base change.

Experimental results

Research questions

• RQ1 How can derived categories be systematically constructed from abelian categories using localization and triangulated structures?
• RQ2 What conditions ensure the existence and uniqueness of dualizing complexes in commutative and noncommutative settings?
• RQ3 How do rigid dualizing complexes behave under base change and descent in algebraic geometry?
• RQ4 In what sense do derived functors provide a natural generalization of classical homological functors?
• RQ5 How do perverse coherent sheaves arise as images of duality functors in derived categories of schemes?

Key findings

• The derived category D(Mod A) of a commutative ring A admits a contravariant triangulated duality functor RHom(−, A) that induces reflexivity for bounded complexes with finitely generated cohomology.
• Rigid dualizing complexes exist and are unique up to unique isomorphism in the derived category, providing a canonical duality for commutative and noncommutative algebras.
• The assignment of perverse coherent sheaves to open subsets of a scheme forms a stack, ensuring effective descent and justifying the term 'perverse sheaves'.
• For a finite type scheme over a field, the rigid dualizing complex can be constructed by gluing local rigid dualizing complexes on an affine cover via descent isomorphisms.
• The derived category construction allows a unified treatment of duality, where the natural transformation η: 1 → RDR becomes an isomorphism for complexes with finitely generated cohomology.
• Rigid dualizing complexes are deeply connected to Calabi-Yau algebras, where A[n] is a rigid dualizing complex if A is a Calabi-Yau algebra of dimension n.