The Ext Functor

Executive summary

Balance: either variable may be resolved

Extⁿ(C, A) can be computed by taking a projective resolution of C and applying Hom(−, A), or by taking an injective resolution of A and applying Hom(C, −). The two agree, and for n = 1 both agree with the group of extensions under the Baer sum. That triple agreement — the balance of Ext — is what makes the functor computable in practice, because one variable is usually far easier to resolve than the other.

Learning objectives

Compute Ext from a projective resolution.
Compute Ext from an injective resolution.
State the balance theorem and why it matters.
Apply the standard vanishing criteria.
Interpret Ext¹ as classifying extensions.

Section 01The three definitions

Via extensionsYoneda / Baer

Ext¹(C, A) = equivalence classes of extensions under the Baer sum. Needs no resolutions; extends to Extⁿ via n-fold extensions.

Via projectivesResolve C

Take P_• ↠ C projective, apply Hom(−, A), take cohomology of the resulting cochain complex.

AlgorithmExt from a projective resolutionin: modules C, A → out: Extⁿ(C, A) for all n

Choose a projective resolution … → P₁ → P₀ → C → 0.
Delete C to obtain the deleted resolution P_•.
Apply Hom(−, A), giving a cochain complex 0 → Hom(P₀, A) → Hom(P₁, A) → … Contravariance reverses the arrows.
Extⁿ(C, A) is the n-th cohomology of that complex.
Ext⁰(C, A) = Hom(C, A), recovering the original functor in degree 0.

The balance theorem

Resolving C by projectives and resolving A by injectives give canonically isomorphic answers. The proof compares both against a double complex built from a projective resolution of C and an injective resolution of A simultaneously. In practice this means: resolve whichever variable is easier.

Section 02Vanishing criteria

When Ext vanishes
Condition	Consequence
C projective	Extⁿ(C, A) = 0 for all n ≥ 1 and all A
A injective	Extⁿ(C, A) = 0 for all n ≥ 1 and all C
proj dim C ≤ d	Extⁿ(C, −) = 0 for n > d
inj dim A ≤ d	Extⁿ(−, A) = 0 for n > d
Λ a PID	Extⁿ = 0 for n ≥ 2 — global dimension 1
Λ semisimple	Extⁿ = 0 for n ≥ 1 — every sequence splits

Ext<sup>1</sup> = 0 means every extension splits

This is the working interpretation. Semisimplicity of a ring is exactly the statement that Ext¹ vanishes identically, and Maschke's theorem — that k[G] is semisimple when the characteristic does not divide |G| — is the reason ordinary representation theory has no higher cohomology.

Section 03A worked computation

Compute Ext^*(ℤ/mℤ, A) over ℤ. The resolution is short:

0 → ℤ →^×m ℤ → ℤ/mℤ → 0

Applying Hom(−, A) gives A →^×m A, so

Ext⁰ = A[m], Ext¹ = A/mA, Extⁿ = 0 for n ≥ 2

where A[m] is the m-torsion. So Ext¹(ℤ/m, ℤ) = ℤ/mℤ, matching the extension count: the m extensions of ℤ/m by ℤ correspond to the m elements of that group.

The pattern to remember

Ext¹ against ℤ converts a torsion module into its dual torsion module. This single computation drives the universal coefficient theorem in topology and the classification of finitely generated abelian group extensions.

ReferenceFrequently asked questions

Why does the resolution not matter?

Because any two projective resolutions of the same module are chain homotopy equivalent, by the comparison theorem, and homotopic chain maps induce equal maps on homology. The resulting isomorphism is canonical, so the answer is well defined.

Is Ext<sup>1</sup> always the extension group?

Yes, in any abelian category with enough projectives or enough injectives. Without either, the Yoneda definition by extensions still works and is taken as the definition — which is one reason it is worth knowing independently.

What if the category has neither enough projectives nor injectives?

Yoneda Ext remains defined and is a group. Higher Ext is defined by n-fold extensions modulo an equivalence relation. It agrees with the derived-functor definition whenever both exist.

NavigateContinue in this stream

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 The Ext Functor. 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 The Ext Functor as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—functor, section, extensions, projective, resolutions—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.