nLab An Introduction to Homological Algebra

Context

Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

Theorems

This entry provides a hyperlinked index for the textbook

• An Introduction to Homological Algebra

Cambridge University Press (1994)

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

Contents

1 Chain complexes

1.1 Complexes of $R$-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

Exercise 1.1.5 exactness and weak nullity

Definition 1.1.2 quasi-isomorphism

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 1.2.3 (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

$R$Mod

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

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

presheaf

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 $\delta$-Functor

Definition 2.1.1 delta-functor

2.2 Projective resolutions

Definition 2.2.4 projective resolution (cofibrant replacement)

Horseshoe lemma 2.2.8 horseshoe lemma

2.3 Injective resolutions

Baer’s criterion 2.3.1 Baer's criterion

Definition 2.3.5 injective resolution (fibrant replacement)

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

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 $\mathrm{Tor}$ and $\mathrm{Ext}$

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 $\mathrm{Tor}$ for abelian groups

Proposition 3.1.2-3.1.3 relation to torsion subgroups

3.2 $\mathrm{Tor}$ 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 $\mathrm{Ext}$ for nice rings

Corollary 3.3.11 Localization for Ext

3.4 $\mathrm{Ext}$ and extensions

extension

group extension

Vista 3.4.6 Yoneda extension group?

3.5 Derived functors of the inverse limit

tower

(AB4)-category

directed limit

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?