Derived categories for the working mathematician

TL;DR

This paper introduces derived categories and related concepts in algebraic geometry, providing intuitive motivation and connecting them to familiar topological ideas to simplify understanding of complex homological algebra tools.

Contribution

It offers a natural, simplified perspective on derived categories and homological algebra, making these advanced topics more accessible to mathematicians and physicists.

Findings

Provides motivation for derived categories from algebraic geometry and topology

Connects complex homological algebra concepts to familiar topological ideas

Simplifies understanding of cohomology, Ext, Tor, and spectral sequences

Abstract

It is becoming increasingly difficult for geometers and even physicists to avoid papers containing phrases like `triangulated category', not to mention derived functors. I will give some motivation for such things from algebraic geometry, and show how the concepts are already familiar from topology. This gives a natural and simple way to look at cohomology and other scary concepts in homological algebra like Ext, Tor, hypercohomology and spectral sequences.