Universal Coefficients and Kunneth Theorems — Examples

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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.

Universal-coefficient homology with cyclic coefficients

Statement

For $C = (0 \to Z \overset{m}{\to} Z \to 0)$ in degrees $1, 0$ and coefficients $Z / r$ , $H_{1} (C \otimes Z / r) = {\overset{ˉ}{x} : m \overset{ˉ}{x} = 0}$ and $H_{0} (C \otimes Z / r) = (Z / r) / m (Z / r)$ .

Verification

Given: the tensor complex $0 \to Z / r \overset{m}{\to} Z / r \to 0$ .

1.1

Its degree-one homology is the kernel and its degree-zero homology the cokernel of multiplication by $m$ .

given

2.1

These are respectively the $m$ -torsion subgroup and quotient displayed in the statement, exactly as the homological UCT predicts.

step 1.1algebra∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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.

Universal-coefficient cohomology of a two-term free complex

Statement

For a nonzero integer $m$ , let $C = (0 \to Z \overset{m}{\to} Z \to 0)$ . With coefficients $Z$ , one has $H^{0} Hom (C, Z) = 0$ and $H^{1} Hom (C, Z) ≅ Z / m$ .

Verification

Given: the cochain complex $0 \leftarrow Z \overset{m}{\leftarrow} Z \leftarrow 0$ .

1.1

Multiplication by nonzero $m$ on $Z$ has kernel $0$ and cokernel $Z / m$ .

given

2.1

Thus the cochain calculation has the displayed cohomology; it agrees with ${Ext}_{Z}^{1} (Z / m, Z) ≅ Z / m$ .

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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 nonzero Tor correction in universal coefficients

Statement

For the complex $0 \to Z \overset{2}{\to} Z \to 0$ and coefficients $Z / 2$ , the degree-one UCT correction is ${Tor}_{1}^{Z} (Z / 2, Z / 2) ≅ Z / 2$ .

Verification

Given: $H_{0} C ≅ Z / 2$ , $H_{1} C = 0$ , and coefficients $Z / 2$ .

1.1

Tensoring produces $0 \to Z / 2 \overset{0}{\to} Z / 2 \to 0$ , so $H_{1} (C \otimes Z / 2) ≅ Z / 2$ .

given

2.1

Since $H_{1} C \otimes Z / 2 = 0$ , the degree-one UCT sequence identifies this nonzero group with the stated Tor correction.

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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.

Kunneth for two cyclic two-term complexes

Statement

For the two complexes $0 \to Z \overset{2}{\to} Z \to 0$ , the Kunneth sequence has a nonzero degree-one term ${Tor}_{1}^{Z} (Z / 2, Z / 2) ≅ Z / 2$ .

Verification

Given: $H_{0} C = H_{0} D ≅ Z / 2$ and $H_{i} C = H_{i} D = 0$ for $i > 0$ .

1.1

In degree one, the tensor-product side of the Kunneth sequence is zero because one of the two homology degrees would be positive.

given

2.1

The quotient is ${Tor}_{1} (Z / 2, Z / 2) ≅ Z / 2$ , hence $H_{1} (C \otimes D) ≅ Z / 2$ .

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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.

Kunneth over a field

Statement

For $C = (0 \to k \overset{0}{\to} k \to 0)$ and $D = (0 \to k \to 0)$ concentrated in degree $0$ , cross product identifies $H_{*} (C \otimes_{k} D)$ with $H_{*} (C) \otimes_{k} H_{*} (D)$ .

Verification

Given: the displayed complexes of $k$ -vector spaces.

1.1

The zero differential makes $H_{0} C ≅ H_{1} C ≅ k$ and $H_{0} D ≅ k$ .

given

2.1

The tensor complex has $k$ in total degrees zero and one with zero differential, matching the two cross-product summands; no Tor term occurs over $k$ .

step 1.1∎

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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 nonnatural choice of universal-coefficient splitting

Statement

Choosing a complement in a free cycle-boundary decomposition gives a UCT splitting that is changed by a chain automorphism moving that complement; it is therefore not natural.

Counterexample

Given: the free complex $C_{1} = Z e \oplus Z f$ , $C_{0} = Z g$ , with $d (e) = 2 g$ and $d (f) = 0$ , and the coefficient group $G = Z / 2$ .

1.1

Here $H_{1} (C) = Z f$ , $H_{0} (C) = Z / 2$ , and the differential of $C \otimes G$ is zero. Thus the degree-one UCT sequence is $0 ⟶ (Z / 2) f ⟶ (Z / 2) e \oplus (Z / 2) f \overset{q}{\to} Z / 2 ⟶ 0$ , where $q (a e + b f) = a$ .

givenalgebra

2.1

The maps $u_{1} (e) = e + f$ , $u_{1} (f) = f$ , and $u_{0} (g) = g$ define a chain automorphism inducing the identity on $H_{1} (C)$ and $H_{0} (C)$ , but $u_{1} \otimes 1$ sends $(a, b)$ to $(a, a + b)$ on the middle UCT term.

step 1.1algebra

3.1

Any section has $s (1) = (1, t)$ . Naturality with respect to $u$ would require $(u_{1} \otimes 1) s (1) = s (1)$ because $u$ acts identically on the two outer terms, but $(1, t + 1) \neq (1, t)$ . Hence no section for this complex and coefficient group is invariant under all chain automorphisms.

step 2.1contradiction∎

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ 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.

Euler characteristic of a tensor-product complex

Statement

For $C = (0 \to Z^{2} \to Z^{3} \to 0)$ and $D = (0 \to Z \to Z^{4} \to 0)$ with zero differentials, $χ (C \otimes D) = χ (C) χ (D)$ .

Verification

Given: the displayed finite free complexes with zero differentials.

1.1

Their Euler characteristics are $3 - 2 = 1$ and $4 - 1 = 3$ .

given

2.1

Expanding the finite total complex gives the product of the two alternating rank sums, hence $χ (C \otimes D) = 3 = 1 \cdot 3$ .

step 1.1algebra∎

Universal Coefficients and Kunneth Theorems — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

Universal-coefficient homology with cyclic coefficients

Statement

Verification

Universal-coefficient cohomology of a two-term free complex

Statement

Verification

A nonzero Tor correction in universal coefficients

Statement

Verification

Kunneth for two cyclic two-term complexes

Statement

Verification

Kunneth over a field

Statement

Verification

A nonnatural choice of universal-coefficient splitting

Statement

Counterexample

Euler characteristic of a tensor-product complex

Statement

Verification

Sources