The Hom Functor and Left Exactness

Executive summary

Left exact in both variables, exact in neither

Hom(M, −) is covariant and Hom(−, N) is contravariant, and both are left exact: they carry a short exact sequence to a sequence exact except at the right-hand end. The obstruction at that end is a lifting or extension problem, and it is precisely what Ext¹ records. Recognising which variable is being varied, and which direction the arrows then run, is the single most common source of confusion in early homological algebra.

Learning objectives

State left exactness of Hom in each variable.
Produce an example where the last map fails to be surjective.
Explain the arrow reversal in the contravariant variable.
Identify additivity and why it is needed.

Section 01The two variables

0 → A → B → C → 0

gives two induced sequences, and the direction of the arrows differs:

0 → Hom(M,A) → Hom(M,B) → Hom(M,C) (covariant)

0 → Hom(C,N) → Hom(B,N) → Hom(A,N) (contravariant)

In both cases exactness holds at the two left positions and can fail at the right. The contravariant version reverses the sequence, so it is the map out of the submodule that may fail to be hit: not every homomorphism A → N extends to B.

Two different obstructions, one functor

Covariantly the question is lifting: does a map into C lift to B? Contravariantly it is extension: does a map out of A extend over B? Both are measured by Ext¹, in different variables, which is why Ext is a functor of two arguments.

Section 02A concrete failure

Take the short exact sequence of abelian groups

0 → ℤ →^×2 ℤ → ℤ/2ℤ → 0

and apply Hom(ℤ/2ℤ, −). Since Hom(ℤ/2ℤ, ℤ) = 0 but Hom(ℤ/2ℤ, ℤ/2ℤ) = ℤ/2ℤ, the induced sequence is

0 → 0 → 0 → ℤ/2ℤ

and the final map is visibly not surjective. The identity map of ℤ/2ℤ does not lift to ℤ. The cokernel ℤ/2ℤ is Ext¹(ℤ/2ℤ, ℤ), and it is non-zero for exactly the reason the sequence does not split.

Keep this example

It is the smallest non-trivial computation in the subject and it recurs constantly — in the universal coefficient theorem, in the classification of abelian group extensions, and as the first entry in every table of Ext groups.

Section 03Additivity and exactness vocabulary

Exactness properties of functors
Property	Meaning	Examples
Additive	Preserves finite direct sums and addition of morphisms	Hom, tensor, all derived functors
Left exact	Carries 0 → A → B → C to an exact 0 → FA → FB → FC	Hom in either variable; inverse limits
Right exact	Carries A → B → C → 0 to FA → FB → FC → 0 exact	Tensor product; direct limits over directed sets
Exact	Both	Localisation; Hom(P, −) for P projective; direct sums

Exactness is a property of the functor, not the sequence

Hom(P, −) is exact for every short exact sequence precisely when P is projective. Saying that a particular sequence stays exact under Hom is a much weaker statement, and conflating the two produces false general claims.

ReferenceFrequently asked questions

Which variable is contravariant?

The first. Hom(−, N) reverses arrows because a map A → B lets you pull a homomorphism out of B back to one out of A. The second variable is covariant: maps compose forward.

Is Hom(M, −) ever exact?

Exactly when M is projective — that is the definition, restated. Dually Hom(−, N) is exact exactly when N is injective. These two conditions are what make resolutions by projectives and injectives useful.

Does Hom preserve infinite direct sums?

In the second variable it preserves products, not sums; in the first it converts sums into products. Hom(⊕M_i, N) = ∏Hom(M_i, N). This asymmetry matters whenever infinite families appear, notably in universal coefficient arguments.

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 Hom Functor and Left Exactness. 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 Hom Functor and Left Exactness as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—functor, left, exact, exactness, variables—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 The Hom Functor and Left Exactness?