Long Exact Homology Sequences and Module Resolutions
Handbook guide to long exact homology sequences and module resolutions with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
The Long Exact Homology Sequence
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Projective and Injective Resolutions
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
The Connecting Homomorphism
The Connecting Homomorphism We will now connect E′ to C′ in the snake diagram while preserving exactness. The idea is to zig-zag through the diagram along the path E′EBDCC′. Let z ∈E′ ⊆E; Since s is surjective, there exists y ∈B such that z = sy. Then tey = hsy = hz = 0 since E′ = ker h.
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Snake Lemma The sequence
Snake Lemma The sequence f s ∂ g t A′ → B′ → E′ → C′ → D′ → F ′ is exact.
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Definition We say that
Call f g 0 → C∗ → D∗ → E∗ → 0 where f and g are chain maps, is a short exact sequence of chain complexes if for each n, the corresponding sequence formed by the component maps fn : Cn →Dn and gn : Dn →En, is short exact. We will construct connecting homomorphisms ∂n : Hn(E∗) →Hn−1(C∗) such that the sequence g ∂ f g ∂ f · · · → Hn+1(E∗) → Hn(C∗) → Hn(D∗) → Hn(E∗) → Hn−1(C∗) → · · · is exact. [One has taken some liberties with the notation. In the second diagram, f stands for the map induced by fn on homology, namely, Hn(f); similarly for g.] The second diagram is the long exact homology sequence, and the result may be summarized as follows.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Theorem
A short exact sequence of chain complexes induces a long exact sequence of homology modules.
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
The connecting homomorphism explicitly If z ∈Hn(E∗), then z = zn +Bn(E∗)
The connecting homomorphism explicitly If z ∈Hn(E∗), then z = zn +Bn(E∗) for some zn ∈Zn(E∗). We apply (S2.4) to compute ∂z. One has zn + Bn(E∗) = gn(yn + Bn(D∗)) for some yn ∈Dn. Then dyn ∈Zn−1(D∗) and dyn = fn−1(xn−1) for some xn−1 ∈Zn−1(C∗).
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Naturality Suppose that we have a commutative diagram of short exact sequences
Naturality Suppose that one has a commutative diagram of short exact sequences of chain complexes, as shown below. 0 → C → D → E → 0 ↓ ↓ ↓ 0 → C′ → D′ → E′ → 0 Then there is a corresponding commutative diagram of long exact sequences: ∂ ∂ · · · → Hn(C∗) → Hn(D∗) → Hn(E∗) → Hn−1(C∗) → · · · ↓ ↓ ↓ ↓ · · · → Hn(C′ ∗) → Hn(D′ ∗) → Hn(E′ ∗) → Hn−1(C′ ∗) → · · · ∂ ∂
Proof / verification strategy: Write the relevant maps explicitly, compute kernels and images at each position, and use commutativity or an induced-map argument to preserve exactness.
Definition
A left resolution of a module M is an exact sequence · · · P2 →P1 →P0 →M →0. A left resolution is a projective resolution if every Pi is projective, a free resolution if every Pi is free. By the first isomorphism theorem, M is isomorphic to the cokernel of the map P1 →P0, so in a sense no information is lost if M is removed. A deleted projective resolution is of the form · · · P2 → P1 → P0 → 0 ↓ M and the deleted version turns out to be more convenient in computations.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Proposition
Every module M has a free (hence projective) resolution.
Proof / verification strategy: Use the universal or lifting property that defines the construction, then check naturality and the effect on exact sequences.
Quick-reference relationships
Problem-solving workflow
Fix the coefficient ring and variance
State whether modules are left/right modules and whether a functor is covariant or contravariant.
Write the maps, not just the objects
Kernels, images, exactness and universal properties depend on the actual homomorphisms.
Use the appropriate universal property
Direct sums, products, tensor products, projectives, injectives and limits are best handled by their mapping property.
Check exactness at each position
Verify image equals kernel rather than relying on the appearance of a diagram.
Choose a resolution only when needed
Derived constructions should be tied to projective or injective resolutions and independence from the chosen resolution.
Test naturality and compatibility
For induced maps, ensure compositions and commutative squares behave as required.
Worked-solution emphasis from the supplied source
The source exercises reinforce definition checking, construction of examples and direct use of the formal results in this article.
Common mistakes and boundary conditions
- Forgetting the coefficient ring when comparing modules.
- Assuming tensor product preserves every exact sequence.
- Confusing direct sum with direct product for infinite families.
- Reading exactness from a diagram without checking image equals kernel.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| S3 | The Long Exact Homology Sequence | 229–230 |
| S4 | Projective and Injective Resolutions | 231–231 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.