Exact Sequences and Diagram Chasing
Handbook guide to exact sequences and diagram chasing with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Exact Sequences and Diagram Chasing
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.
Definition
Suppose that the R-module M is the direct sum of the submodules A and B. Let f be the inclusion or injection map of A into M (simply the identity function on A), and let g be the natural projection of M on B, given by g(a + b) = b, a ∈A, b ∈B. The image of f, namely A, coincides with the kernel of g, and call the sequence f g A → M → B (1) is exact at M. A longer (possibly infinite) sequence of homomorphisms is called exact if it is exact at each junction, that is, everywhere except at the left and right endpoints, if they exist.
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.
The Five Lemma Consider the following commutative diagram with exact rows.
The Five Lemma Consider the following commutative diagram with exact rows. e f g h D → A → M → B → C s ↓ t ↓ u ↓ v ↓ w ↓ D′ → A′ → M ′ → B′ → C′ e′ f ′ g′ h′ If s, t, v and w are isomorphisms, so is u.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Corollary: The Short Five Lemma Consider the following commutative diagram
Corollary: The Short Five Lemma Consider the following commutative diagram with exact rows. f g 0 → A → M → B → 0 t ↓ u ↓ v ↓ 0 → A′ → M ′ → B′ → 0 f ′ g′ If t and v are isomorphisms, so is u.
Proof / verification strategy: Construct the natural homomorphism, identify its kernel and image, then apply the relevant quotient/isomorphism result. Check well-definedness before claiming injectivity or surjectivity.
Proposition
Let f g 0 → A → M → B → 0 be a short exact sequence. The following conditions are equivalent, and define a split exact sequence. (i) The sequence splits on the right. (ii) The sequence splits on the left.
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.
Corollary
If the sequence f g 0 → A → M → B → 0 is split exact with splitting maps e and h as in (3), then the “backwards” sequence e h 0 ← A ← M ← B ← 0 is also split exact, with splitting maps g and f.
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.
Quick-reference relationships
Problem-solving workflow
State the objects precisely
Write the sets, operations, maps and hypotheses before manipulating them.
Identify the governing definition
Most abstract arguments become shorter once the relevant universal or structural definition is explicit.
Apply results only under their hypotheses
Check finiteness, commutativity, injectivity, surjectivity and independence assumptions.
Keep notation consistent
Distinguish elements, subsets, equivalence classes, maps and quotient objects.
Use a small example as a check
A concrete model can expose a mistaken direction or missing condition.
Return to the structural meaning
State what the calculation proves about the original algebraic object.
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
- Skipping a hypothesis because it seems automatic.
- Using notation before defining the ambient structure.
- Treating an illustrative calculation as a universal result.
- Relying on intuition without verifying closure, well-definedness or exactness.
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 |
|---|---|---|
| 4.7 | Exact Sequences and Diagram Chasing | 80–84 |
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.