Introduction to Derived Categories

TL;DR

This paper introduces derived categories, a modern framework in homological algebra that extends classical concepts, offering richer tools for algebraic geometry and related fields.

Contribution

It provides an accessible overview of derived categories, including foundational theory and applications to noncommutative algebraic geometry, based on a minicourse.

Findings

Derived categories extend classical homological algebra.

Introduction of dualizing and tilting complexes.

Application to noncommutative algebraic geometry.

Abstract

Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the "old" homological algebra (of derived functors between abelian categories) was established. This "new" homological algebra, of derived categories and derived functors between them, provides a significantly richer and more flexible machinery than the "old" homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the "old" homological algebra. This paper is an edited version of the notes for a two-lecture minicourse given at MSRI in January 2013. Sections 1-5 are about the general theory of derived categories, and the material is taken from my manuscript "A Course on Derived Categories" (available online). Sections 6-9 are on more specialized topics, leaning towards noncommutative algebraic geometry.