nLab An Introduction to Homological Algebra



Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories


This entry provides a hyperlinked index for the textbook

which gives a first exposition to central concepts in homological algebra.

For a more comprehensive account of the theory see also chapters 8 and 12-18 of

and see the nnLab lecture notes


1 Chain complexes

1.1 Complexes of RR-modules

Definition 1.1.1 chain complex

Exercise 1.1.2 homology is functorial

Exercise 1.1.3 exact sequences of chain complexes are split

Exercise 1.1.4 internal hom of chain complexes

Definition 1.1.2 quasi-isomorphism

cochain complex, bounded chain complex

Exercise 1.1.5 exactness and weak nullity

Application 1.1.3 chain on a simplicial set, simplicial homology

Exercise 1.17 simplicial homology of the tetrahedron

Application 1.1.4 singular homology

1.2 Operations on chain complexes

Exercise 1.2.1 homology respects direct product

Definition 1.2.1 kernel, cokernel

Exercise 1.2.2 in an abelian category kernels/cokernels are the monos/epis

Exercise 1.2.3 (co)kernels of chain maps are degreewise (co)kernels

Definition 1.2.2 abelian category, abelian subcategory

Theorem 1.2.3 a category of chain complexes is itself abelian

Exercise 1.2.4 exact sequence of chain complexes is degreewise exact


Example 1.2.4 double complex

Sing trick 1.2.5 double complex with commuting/anti-commuting differentials

Total complex 1.2.6 total complex

Exercise 1.2.5 total complex of a bounded degreewise exact double complex is itself exact

Example 1.2.4 double complex

Truncations 1.2.7 truncation of a chain complex

Translation 1.2.8 suspension of a chain complex

Exercise 1.2.8 mapping cone

1.3 Long exact sequences

Theorem 1.3.1 connecting homomorphism, long exact sequences in homology

Exercise 1.3.1 3x3 lemma,

Snake lemma 1.3.2 snake lemma

Exercise 1.3.3 5 lemma

Remark 1.3.5 exact triangle

1.4 Chain homotopies

Definition 1.4.1 split exact sequence

Exercise 1.4.1 splitness of exact sequences of free modules

Definition 1.4.3 null homotopy

Exercise 1.4.3 split exact means identity is null homotopic

Definition 1.4.4 chain homotopy

Lemma 1.4.5 chain homotopy respects homology

Exercise 1.4.5 homotopy category of chain complexes

1.5 Mapping cones and cyclinders

1.5.1 mapping cone

1.5.5 mapping cylinder

1.5.8 fiber sequence

1.6 More on abelian categories

Theorem 1.6.1 Freyd-Mitchell embedding theorem

Functor categories 1.6.4 functor category


Definition 1.6.5 abelian sheaf

Definition 1.6.6 left/right exact functor

Yoneda embedding 1.6.10 Yoneda embedding

Yoneda lemma 1.6.11 Yoneda lemma

proof of the Freyd-Mitchell embedding theorem

2 Derived functors

derived functor in homological algebra

2.1 δ\delta-Functor

Definition 2.1.1 delta-functor

2.2 Projective resolutions

projective module (cofibrant object in the model structure on chain complexes)

Definition 2.2.4 projective resolution (cofibrant replacement)

Horseshoe lemma 2.2.8 horseshoe lemma

2.3 Injective resolutions

injective module (fibrant object in the other model structure on chain complexes)

Baer’s criterion 2.3.1 Baer's criterion

Definition 2.3.5 injective resolution (fibrant replacement)

Definition 2.3.9 adjoint functor

2.4 Left derived functors

left derived functor

2.5 Right derived functors

right derived functor

Application 2.5.4 global section functor, abelian sheaf cohomology

2.6 Adjoint functors and left/right exactness

adjoint functor

Definition 2.6.4 Tor

Application 2.6.5 sheafification

Application 2.6.6 direct image, inverse image

Application 2.6.7 colimit

Variation 2.6.9 limit

Definition 2.6.13 filtered category, filtered colimit

2.7 Balancing TorTor and ExtExt

Tensor product of complexes 2.7.1 tensor product of chain complexes

Lemma 2.7.3 acyclic assembly lemma

3 Tor and Ext

Tor and Ext

3.1 TorTor for abelian groups

Proposition 3.1.2-3.1.3 relation to torsion subgroups

3.2 TorTor and flatness

Definition 3.2.1 flat module

Definition 3.2.3 Pontrjagin duality

Flat resolution lemma 3.2.8 flat resolution lemma

Corollary 3.2.13 Localization for Tor

3.3 ExtExt for nice rings

Corollary 3.3.11 Localization for Ext

3.4 ExtExt and extensions


group extension

Vista 3.4.6 Yoneda extension group?

3.5 Derived functors of the inverse limit



directed limit

Definition 3.5.1 lim^1

Definition 3.5.6 Mittag-Leffler condition

Exercise 3.5.5 pullback

3.6 Universal coefficient theorem

Theorem 2.6.1 Künneth formula

Universal cofficient theorem for homology 3.6.2 universal coefficient theorem in homology

Theorem 3.6.3 Künneth formula for complexes?

Application 3.6.4 universal coefficient theorem in topology

Universal coefficient theorem in cohomology 3.6.5 universal coefficient theorem in cohomology

Eilenberg-Zilber theorem

Exercise 3.6.2 hereditary ring?

4 Homological dimension

4.1 Dimensions

4.2 Rings of Small Dimension

4.3 Change of Rings Theorem

4.4 Local rings

4.5 Koszul Complexes

4.6 Local Cohomology

5 Spectral sequences

5.1 Introduction

5.2 Terminology

5.3 Leray-Serre Spectral Sequence

5.4 Spectral sequence of a filtration

5.5 Convergence

5.6 Spectral sequence of a double complex

5.7 Hypercohomology

5.8 Grothendieck spectral sequence

5.9 Exact couples

6 Group homology and cohomology

7 Lie algebra homology and cohomology

8 Simplicial methods in homological algebra

8.1 Simplicial object

8.2 Operations on simplicial objects

8.3 Simplicial homotopy groups

8.4 The Dold-Kan correspondence

8.5 The Eilenberg-Zilber theorem

8.6 Canonical resolutions

8.7 Cotriple homology

8.8 Andre-Quillen Homology and Cohomology

9 Hochschild and cyclic homology

9.1 Hochschild Homology and Cohomology of Algebras

9.2 Derivations, Differentials and Separable Algebra

9.3 H 2H^2, Extensions, Smooth Algebras

9.4 Hochschild products

9.5 Morita Invariance

9.6 Cyclic Homology

9.7 Group Rings

9.8 Mixed Complexes

9.9 Graded Algebras

9.10 Lie Algebras of Matrices

10 The derived category

A Category Theory Language

A.1 Categories

A.2 Functors

A.3 Natural transformations

A.4 Abelian categories

A.5 Limits and colimits

A.6 Adjoint functors

category: reference

Last revised on July 29, 2021 at 18:43:52. See the history of this page for a list of all contributions to it.