Homological Algebra

This section covers the foundations of homological algebra, providing tools for measuring the failure of exactness and computing invariants of algebraic structures.

Chain Complexes

Basic Definitions

• Chain complex
• Cochain complex
• Chain map
• Chain homotopy
• Exact complex

Homology and Cohomology

• Homology module
• Cohomology module

Resolutions

• Projective resolution
• Injective resolution

Derived Functors

General Theory

• Derived functor
• Long exact sequence of derived functors

Ext and Tor

• Ext
• Tor
• Ext and Tor as derived functors

Theorems

Exactness Properties

• Tensor product is right exact
• Hom functor is left exact

Long Exact Sequences

• Long exact sequence of Tor
• Long exact sequence of Ext

Lemmas

Diagram Lemmas

• Snake lemma
• Five lemma
• Four lemma
• Nine lemma (3×3 lemma)

Construction Lemmas

• Horseshoe lemma
• Connecting homomorphism lemma

Propositions

• Exactness of Hom and tensor in short exact sequences
• Projective resolutions exist
• Injective resolutions exist

Corollaries

• Snake lemma corollary
• Five lemma corollary
• Ext¹ classifies extensions