Mathematics•Derived Functors
Ext via Projective and Injective Resolutions
The balance theorem: two entirely different computations, one answer.
Resolve whichever variable is easier
Extn(C, A) can be computed as the n-th right derived functor of Hom(−, A) applied to C, or of Hom(C, −) applied to A. These are different computations with different inputs, and they agree. The proof compares both against a double complex built from a projective resolution of C and an injective resolution of A, and the practical consequence is complete freedom in choosing which side to resolve.
Learning objectives
- State the balance theorem.
- Outline the double complex proof.
- Choose the cheaper resolution in a given computation.
- Explain why Tor is symmetric while Ext is only balanced.
Section 01The statement
The isomorphisms are natural in both variables. So Ext is well defined without specifying which side is resolved, and the two long exact sequences — one for each variable — belong to the same bifunctor.
Tor is symmetric: Tor(M, N) ≅ Tor(N, M), because both variables are covariant and both resolved projectively. Ext is only balanced: the two variables have different variance and are resolved by different kinds of object, so no exchange isomorphism exists.
Section 02The double complex argument
- Take a projective resolution P• of C and an injective resolution I• of A.
- Form the double complex Hom(Pp, Iq).
- Filter by rows: since each Iq is injective, Hom(−, Iq) is exact, so taking homology in the p direction leaves only the column Hom(C, I•). This computes Ext by resolving A.
- Filter by columns: since each Pp is projective, Hom(Pp, −) is exact, so only the row Hom(P•, A) survives. This computes Ext by resolving C.
- Both filtrations compute the cohomology of the total complex, so the two answers agree.
Each filtration gives a spectral sequence that degenerates at the first page, because the exactness kills all but one row or column. The comparison is therefore a direct isomorphism rather than a filtration argument — which is why balance can be proved before spectral sequences are introduced.
Section 03Choosing a side
- Which variable has the better resolution?
- C is cyclic or f.g. over a PID Resolve C — the projective resolution has length 1.
- A is already injective or divisible Resolve A — the resolution is trivial and Ext vanishes above degree 0.
- C is a trivial module over a group ring Resolve C — the bar resolution is standard and explicit.
- Working with sheaves Resolve the second variable — sheaf categories have enough injectives but generally no projectives.
In a category without enough projectives, the projective computation simply does not exist. Balance then says nothing — it is an agreement between two computations, and with only one available the choice is made for you. This is the everyday situation in sheaf theory.
ReferenceFrequently asked questions
Does balance hold in every abelian category?
It holds whenever both kinds of resolution exist. With only one kind, Ext is defined by that one and balance is vacuous. The Yoneda definition by extensions agrees with either whenever they exist, which makes it the most robust definition.
Is there a balance theorem for Tor?
Yes, and it is stronger: resolving either variable projectively gives the same answer, and additionally Tor is symmetric under exchanging the variables. The double complex used is symmetric, which is what yields the extra conclusion.
What is a Cartan-Eilenberg resolution?
A resolution of an entire complex, rather than a module, by a double complex whose rows resolve the terms and whose associated cycle and homology complexes are also resolved. It is the tool for defining hyperhomology and for deriving functors on complexes.
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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 Ext via Projective and Injective Resolutions. 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 Ext via Projective and Injective Resolutions as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—double, complex, section, projective, injective—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 Ext via Projective and Injective Resolutions?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about double would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
- MIT OpenCourseWare — Number Theory I — Massachusetts Institute of Technology. Used for algebraic and analytic number theory. Accessed 2026-08-13.
- MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
- Page ID
- KV-MATH-0131
- Taxonomy
- ENG-MATH — Engineering / Mathematics
- Collection
- COL-HOMALG-001
- Topic stream
- HA-DERIVED
- Version
- 1.1.0 / content 2026.08
- Last reviewed
- 2026-08-06