Computing Ext Groups

Executive summary

Resolve the easy variable and shift dimensions

Ext is rarely computed from the definition. The practical toolkit is small: pick whichever variable has the shorter resolution, use additivity to split direct sums, use the long exact sequences to relate an unknown Ext to known ones, and use dimension shifting to reduce higher Ext to Ext¹ of a syzygy. Over the integers everything reduces to one computation that is worth memorising.

Learning objectives

Choose the variable that minimises work.
Apply additivity across direct sums.
Use the long exact sequences to compute an unknown term.
Apply dimension shifting.
Reproduce the standard table of Ext groups over ℤ.

Section 01The standard table over ℤ

Ext<sup>1</sup><sub>ℤ</sub>(C, A) for the basic cases
C	A	Ext¹(C, A)	Reason
ℤ	any	0	ℤ is free, hence projective
ℤ/mℤ	ℤ	ℤ/mℤ	From the length-one resolution
ℤ/mℤ	ℤ/nℤ	ℤ/gcd(m, n)ℤ	A/mA with A = ℤ/nℤ
ℤ/mℤ	ℚ	0	ℚ is divisible, hence injective
ℚ	ℤ	ℝ (as a ℚ-vector space, uncountable)	ℚ is not projective — a genuinely infinite Ext
any C	injective A	0	Every extension splits

Ext1(ℚ, ℤ) is not zero

It is uncountable. ℚ is torsion-free but not free, and not projective, so extensions of ℚ by ℤ abound. This is the standard counterexample to the assumption that torsion-free behaves like free outside the finitely generated case.

Section 02Additivity and reduction

Ext converts finite direct sums into direct sums in either variable, and converts an arbitrary direct sum in the first variable into a product:

Extⁿ(⊕_i C_i, A) ≅ ∏_i Extⁿ(C_i, A)

Combined with the structure theorem, this computes Ext between any two finitely generated abelian groups: decompose both, apply the table entry by entry, reassemble.

AlgorithmExt between finitely generated abelian groupsin: finitely generated abelian C, A → out: Ext¹(C, A)

Decompose C and A into cyclic factors by the structure theorem.
Discard any free factor of C — it contributes 0 to Ext¹. Free is projective.
For each pair of cyclic factors, read the entry from the table.
Assemble the direct sum over all pairs.
Extⁿ = 0 for n ≥ 2, since ℤ is a PID.

Section 03Long exact sequences and dimension shifting

A short exact sequence in either variable produces a long exact sequence in Ext. Two unknowns and one known term usually determine the third.

… → Extⁿ(C, A) → Extⁿ(C, B) → Extⁿ(C, C′) → Extⁿ⁺¹(C, A) → …

AlgorithmDimension shiftingin: Extⁿ(C, A) for n > 1 → out: Ext¹ of a syzygy

Take a short exact sequence 0 → K → P → C → 0 with P projective.
The long exact sequence has Extⁿ(P, A) = 0 for n ≥ 1. P is projective, so its higher Ext vanishes.
Exactness then forces Extⁿ(K, A) ≅ Extⁿ⁺¹(C, A) for n ≥ 1.
Iterating reduces Extⁿ of C to Ext¹ of the (n−1)st syzygy.

Choose the variable deliberately

For Ext(ℤ/m, A) resolve the first variable — the resolution has length 1. For Ext(C, ℚ/ℤ) resolve the second — the target is already injective and the answer is immediate. Time spent choosing is repaid many times.

ReferenceFrequently asked questions

Is Ext1(C, A) = 0 enough to conclude C is projective?

Only if it vanishes for every A. Vanishing for one particular A says only that extensions by that A split. The projectivity criterion is universal quantification over the second variable.

How do I compute Ext over a non-commutative ring?

The same way, but the result is only an abelian group, not a module, unless extra structure is present. When Λ is an algebra over a commutative ring k, Ext is a k-module, which is the usual working situation in group and Lie algebra cohomology.

Why is Ext1(ℚ/ℤ, ℤ) interesting?

It is isomorphic to the profinite completion of ℤ, which appears in the universal coefficient theorem for cohomology with compact supports and in comparisons between algebraic and topological completions. It is a good illustration that Ext of large modules can be structurally rich.

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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 Computing Ext Groups. 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 Computing Ext Groups as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—long, exact, section, computing, groups—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 Computing Ext Groups?