Extensions of Modules and the Baer Sum

Executive summary

Extensions form a group before any resolution is chosen

An extension of C by A is a short exact sequence 0 → A → B → C → 0. Two are equivalent when an isomorphism of the middle terms commutes with both ends. The Baer sum — pull back along the diagonal of C, push out along the addition of A — makes the equivalence classes an abelian group with the split extension as zero. That group is Ext¹(C, A), defined here without a single resolution.

Learning objectives

Define an extension and equivalence of extensions.
Construct the Baer sum by pullback and pushout.
Identify the zero element and inverses.
Compute a small example by hand.
Explain why equivalence is finer than isomorphism of middle terms.

Section 01Extensions and equivalence

Two extensions are equivalent when there is φ: B → B′ making both squares commute with the identity on A and C. By the short five lemma such a φ is automatically an isomorphism.

Equivalence is finer than isomorphism of the middle term

Two extensions can have isomorphic middle modules and still be inequivalent, because the isomorphism may fail to respect the maps to and from A and C. Classification of extensions is a classification of sequences, not of modules.

The split extension, with B = A ⊕ C and the canonical maps, is always present. The content of the theory is how many others there are.

Small examples over ℤ
A	C	Extensions up to equivalence	Ext¹(C, A)
ℤ	ℤ/2ℤ	Two: split, and 0 → ℤ → ℤ → ℤ/2ℤ → 0	ℤ/2ℤ
ℤ/2ℤ	ℤ/2ℤ	Two: ℤ/2 ⊕ ℤ/2, and ℤ/4ℤ	ℤ/2ℤ
ℤ	ℤ	One: split only	0
ℤ/2ℤ	ℤ	One: split only	0
ℚ	any C	Split only — ℚ is injective	0

Section 02The Baer sum

AlgorithmBaer sum of two extensionsin: two extensions of C by A → out: their sum, well defined on classes

Given extensions E and E′ of C by A, form the direct sum extension of C ⊕ C by A ⊕ A.
Pull back along the diagonal Δ: C → C ⊕ C. Yields an extension of C by A ⊕ A.
Push out along the addition map ∇: A ⊕ A → A. Yields an extension of C by A.
The class of the result is E + E′. It is independent of the representatives chosen.
The zero element is the split extension; the inverse of E is E with the map on A negated.

Why this is the right addition

It is the unique operation natural in both variables that has the split extension as identity. It also agrees with the addition Ext¹ inherits as a derived functor, which is the theorem tying the two definitions together.

Section 03Functoriality

Ext¹ is contravariant in C by pullback and covariant in A by pushout. A map C′ → C pulls an extension back; a map A → A′ pushes it forward.

Contravariant in CPullback

Given γ: C′ → C, form the fibre product B ×_C C′. The result is an extension of C′ by the same A.

Covariant in APushout

Given α: A → A′, form the pushout of B and A′ over A. The result is an extension of the same C by A′.

The variance is forced

It matches Hom: contravariant in the first argument, covariant in the second. That is not a coincidence — Ext¹ is a derived functor of Hom, and derived functors inherit variance from the functor they derive.

ReferenceFrequently asked questions

Why is the split extension the zero element?

Because pulling back the split extension along the diagonal and pushing out along addition returns the original extension unchanged. It is the neutral element for the operation, which is what zero means here.

Can two inequivalent extensions have isomorphic middle terms?

Yes. Over ℤ/p²ℤ there are extensions of ℤ/p by ℤ/p that are inequivalent but whose middle terms are abstractly isomorphic. Equivalence tracks the maps, not just the objects.

Does the Baer sum require modules?

No — only pullbacks, pushouts and a zero object, so it works in any abelian category. That is why Ext¹ can be defined for sheaves and for representations without a resolution being available.

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Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.

Handbook application: from concept to controlled practice

Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Extensions of Modules and the Baer Sum. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.

The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.

Treat Extensions of Modules and the Baer Sum as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—extensions, baer, modules, equivalence, section—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.

For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.

A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.

Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.

Step-by-step operating method

Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.

Worked-example protocol

Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.

Stage	Record	Quality check
Input	Objects, domain, notation, assumptions	Every symbol is defined
Method	Permitted operation or cited result at each step	All hypotheses hold
Output	Exact result and representation	Correct type, domain and form
Verification	Substitution, invariant or alternative derivation	Independent agreement
Boundary test	Zero, identity, degenerate or failed hypothesis	Scope is understood

Common failure modes and recovery actions

1. Watch for

Using a theorem without checking every hypothesis.

Recovery: Return to the governing definition or requirement and restate the decision in one sentence.

2. Watch for

Treating a suggestive example as a proof of the general case.

Recovery: Separate evidence from assumption, assign an owner and set a date for validation.

3. Watch for

Changing notation or conventions part-way through an argument.

Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.

4. Watch for

Hiding a division-by-zero, convergence, finiteness or commutativity assumption.

Recovery: Record the consequence, decision and rationale, then update the controlled baseline.

5. Watch for

Reporting a computed result without a residual, substitution or structural check.

Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.

Review checklist

Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?

Questions for deeper application

What is the most important distinction a practitioner must preserve when applying Extensions of Modules and the Baer Sum?