The Diagram Lemmas in an Abelian Category — Examples

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The published module five lemma as an instance

Example

Take the ambient abelian category to be $R - M o d$ . Then the categorical five lemma is the isomorphism clause of the already-published module five lemma.

Facts & Assumptions

Given: A commutative five-term diagram with exact rows in $R - M o d$ .

[L1]

The categorical five lemma holds in any abelian category (Five lemma in an abelian category).

[L2]

The module case is already published under the expected name (The Five Lemma for modules).

Verification

1.1

The hypotheses of [L1] specialize verbatim to a commutative exact-row diagram of modules.

L1

2.1

The conclusion of [L1] is exactly the final, isomorphism clause of [L2]. The separate injective and surjective clauses of [L2] are sharper module statements, while its final clause is the module-valued instance of the categorical five lemma.

L2step 1.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The published module snake lemma as an instance

Example

Inside $R - M o d$ , the categorical snake lemma becomes the published module snake lemma.

Facts & Assumptions

Given: Snake data in the category of modules.

[L1]

The categorical snake lemma holds in every abelian category (Snake lemma in an abelian category).

[L2]

The module snake lemma is already on disk (The Snake Lemma for modules).

Verification

1.1

Module categories are abelian, so the module diagram satisfies the hypotheses of [L1].

L1

2.1

The kernel, cokernel, and connecting-map terms in [L1] are exactly the ones named in [L2]. Thus the published module theorem is the specialization of the categorical snake lemma to modules.

L2step 1.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The published module four lemma as an instance

Example

When the ambient abelian category is a module category, the categorical four lemma is exactly the published module four lemma.

Facts & Assumptions

Given: A commutative exact-row four-term diagram of modules.

[L1]

The categorical four lemma applies in any abelian category (Four lemma in an abelian category).

[L2]

The module four lemma is already published (The injective and surjective Four Lemmas).

Verification

1.1

The module diagram is one instance of the abelian-category diagram in [L1].

L1

2.1

Both the monic and epic conclusions then coincide with the two halves stated in [L2]. Hence the published module theorem is the module instance of the categorical four lemma.

L2step 1.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The connecting morphism computed for a short exact sequence of abelian groups

Example

In $A b$ , consider

diagram failed to render:
0 \arrow[r] & \mathbb Z \arrow[r, "\times 2"] \arrow[d, "\times 2"'] & \mathbb Z \arrow[r] \arrow[d, "\times 2"'] & \mathbb Z/2 \arrow[r] \arrow[d, "0"'] & 0 \\
0 \arrow[r] & \mathbb Z \arrow[r, "\times 2"'] & \mathbb Z \arrow[r] & \mathbb Z/2 \arrow[r] & 0.

The connecting morphism $δ : \ker (0) ≅ Z / 2 ⟶ coker (\times 2) ≅ Z / 2$ is the identity map.

Facts & Assumptions

Given: The diagram above in $A b$ .

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

[L2]

The connecting morphism exists, and the snake sequence is exact (The connecting morphism exists and is unique, Snake lemma in an abelian category).

Verification

1.1

By [L1], the displayed diagram is valid snake data. Its kernel and cokernel terms are $\ker (\times 2) = 0, \ker (0) = Z / 2, coker (\times 2) = Z / 2.$

L1algebra

2.1

The snake sequence from [L2] therefore reduces to $0 \to 0 \to 0 \to Z / 2 \overset{δ}{\to} Z / 2 \to Z / 2 \to Z / 2 \to 0.$ Exactness at the source and target of $δ$ forces $δ$ to be both injective and surjective, hence the identity automorphism of $Z / 2$ .

L2step 1.1algebra

3.1

So in this concrete diagram the connecting morphism is the identity on $Z / 2$ .

step 2.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The snake lemma applied to multiplication by an integer

Example

Fix $n \geq 1$ and apply multiplication by $n$ to the short exact sequence $0 \to Z \overset{\times n}{\to} Z \to Z / n \to 0.$ The snake lemma produces $0 \to 0 \to 0 \to Z / n \overset{δ}{\to} Z / n \overset{0}{\to} Z / n \overset{1}{\to} Z / n \to 0,$ so the connecting morphism is an isomorphism, and under the standard identifications it is the identity.

Facts & Assumptions

Given: The multiplication-by- $n$ endomorphism of the short exact sequence above.

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

[L2]

The snake lemma gives the exact sequence attached to that ladder (Snake lemma in an abelian category).

Verification

1.1

Multiplication by $n$ on $Z$ has zero kernel and cokernel $Z / n$ , while the induced map on the quotient term $Z / n$ is zero. The induced map $coker (\times n) \to coker (\times n)$ is multiplication by $n$ on $Z / n$ , hence is $0$ , and the induced map $coker (\times n) \to coker (0)$ is the identity of $Z / n$ .

L1algebra

2.1

Substituting those terms into [L2] gives the displayed exact sequence. Exactness at the first copy of $Z / n$ forces $δ$ to be an isomorphism. In the standard snake construction, the class of $1$ in $\ker (0) = Z / n$ lifts to $1 \in Z$ and then maps to the class of $1$ in $coker (\times n) = Z / n$ , so under these standard identifications $δ$ is the identity.

L2step 1.1algebra

3.1

This is the concrete snake sequence for multiplication by an integer.

step 2.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The nine lemma verified on a diagram of cyclic groups

Example

In $A b$ , take the commutative $3 \times 3$ diagram whose top row is the zero short exact sequence $0 \to 0 \to 0 \to 0 \to 0,$ whose middle and bottom rows are both $0 \to Z / 2 \to Z / 4 \to Z / 2 \to 0,$ whose vertical maps from the top row to the middle row are zero, and whose vertical maps from the middle row to the bottom row are identities. Then each column is short exact, and the nine lemma says that the top row is short exact if and only if the bottom row is.

Facts & Assumptions

Given: The cyclic-group $3 \times 3$ diagram just described.

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

[L2]

The nine lemma applies to any such $3 \times 3$ diagram (Nine lemma in an abelian category).

Verification

1.1

In this diagram the middle and bottom rows are the standard short exact sequence $0 \to Z / 2 \to Z / 4 \to Z / 2 \to 0,$ and each column is one of the short exact sequences $0 \to 0 \to G \overset{1_{G}}{\to} G \to 0$ with $G \in {Z / 2, Z / 4}$ or the zero sequence. Hence all three columns and the middle row are short exact in $A b$ .

L1algebra

2.1

The top row is the zero short exact sequence and the bottom row is the standard short exact sequence above, so both outer rows are short exact. This agrees with [L2], which predicts that under the hypotheses verified in step 1.1 the two outer rows stand or fall together.

L2step 1.1algebra

3.1

Thus this cyclic-group diagram is a concrete instance of the nine lemma.

step 2.1∎

CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

A snake configuration whose kernel row is not short exact

Statement refuted

In every snake configuration, the induced kernel row is already short exact.

Facts & Assumptions

Given: The multiplication-by-two snake configuration from The kernel row of a morphism of short exact sequences need not be short exact.

[L1]

That published example already shows the kernel row can fail to be short exact (The kernel row of a morphism of short exact sequences need not be short exact).

[L2]

The snake lemma repairs the failure by adding the connecting morphism (Snake lemma in an abelian category).

Counterexample

1.1

By [L1], the chosen diagram is a valid snake configuration in $A b$ whose kernel row is $0 \to 0 \to 0 \to Z / 2,$ and that row is not short exact.

L1

2.1

Nevertheless [L2] adds the connecting morphism $Z / 2 \to Z / 2,$ after which the full snake sequence is exact. So the kernel row alone is not the whole story.

L2step 1.1∎

ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ rests on unproved material (inherited)Open item page →

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The short five lemma chased with members

Example

In $A b$ , the identity morphism between the short exact sequence $0 \to Z \overset{\times 2}{\to} Z \to Z / 2 \to 0$ and itself is the simplest concrete member chase for the short five lemma.

Facts & Assumptions

Given: The identity ladder on the displayed short exact sequence.

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

[L2]

The short five lemma holds in every abelian category (Short five lemma in an abelian category).

Verification

1.1

Every member of the source sequence is carried to the identical member in the target sequence, so the outer comparison maps are isomorphisms in $A b$ .

L1algebra

2.1

The proof of [L2] then specializes to the tautological chase that the middle identity map is both monic and epic. This example is trivial on purpose: it shows the member language in the easiest possible concrete case.

L2step 1.1

3.1

Hence the identity ladder on a short exact sequence of abelian groups is a concrete instance of the short five lemma.

step 2.1∎

The Diagram Lemmas in an Abelian Category — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

The published module five lemma as an instance

Example

Facts & Assumptions

Verification

The published module snake lemma as an instance

Example

Facts & Assumptions

Verification

The published module four lemma as an instance

Example

Facts & Assumptions

Verification

The connecting morphism computed for a short exact sequence of abelian groups

Example

Facts & Assumptions

Verification

The snake lemma applied to multiplication by an integer

Example

Facts & Assumptions

Verification

The nine lemma verified on a diagram of cyclic groups

Example

Facts & Assumptions

Verification

A snake configuration whose kernel row is not short exact

Statement refuted

Facts & Assumptions

Counterexample

The short five lemma chased with members

Example

Facts & Assumptions

Verification

Sources