(also nonabelian homological algebra)
Context
Basic definitions
Stable homotopy theory notions
Constructions
Lemmas
Homology theories
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
and see the Lab lecture notes
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
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
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
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
1.5.1 mapping cone
1.5.5 mapping cylinder
1.5.8 fiber sequence
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
derived functor in homological algebra
Definition 2.1.1 delta-functor
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
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
Application 2.5.4 global section functor, abelian sheaf cohomology
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
Tensor product of complexes 2.7.1 tensor product of chain complexes
Lemma 2.7.3 acyclic assembly lemma
Proposition 3.1.2-3.1.3 relation to torsion subgroups
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
Corollary 3.3.11 Localization for Ext
Vista 3.4.6 Yoneda extension group?
Definition 3.5.1 lim^1
Definition 3.5.6 Mittag-Leffler condition
Exercise 3.5.5 pullback
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
Exercise 3.6.2 hereditary ring?
homological dimension?
Last revised on July 29, 2021 at 18:43:52. See the history of this page for a list of all contributions to it.