The universal coefficient theorem for singular cohomology #
Let k be a hereditary commutative ring, expressed by requiring every k-module to have
projective dimension at most one. This holds for principal ideal domains, for instance ℤ
(ModuleCat.hasProjectiveDimensionLE_one_of_isPrincipalIdealRing). Let R be a projective
k-module, for instance k itself, and let M be any k-module. For a topological space X,
the singular chain modules Cₙ(X; R) = ⨁ R are projective (SSet.projective_chainComplex_X).
Their cycles are then projective and split off the terms
(TauCeti.Algebra.Homology.HomologicalComplex.Projective.Dimension). These are the hypotheses
of the algebraic universal coefficient sequence
(TauCeti.Algebra.Homology.UniversalCoefficient.Basic), so
0 ⟶ Ext¹(Hₙ(X; R), M) ⟶ Hⁿ⁺¹(X; R, M) ⟶ Hom(Hₙ₊₁(X; R), M) ⟶ 0
is exact, where the second map is the Kronecker map TopCat.singularKronecker evaluating
cohomology classes on homology classes. The sequence splits, and the Kronecker map is surjective
in every degree; in degree 0 it is bijective (TopCat.singularKronecker_bijective_zero). Both
maps are natural in X. The splitting TopCat.singularKroneckerSection, and the resulting
decomposition TopCat.singularUniversalCoefficientEquiv, depend on a chosen retraction of the
cycles in each degree and need not be natural in X.
With k = R = ℤ this is the classical sequence
0 ⟶ Ext(Hₙ(X; ℤ), M) ⟶ Hⁿ⁺¹(X; M) ⟶ Hom(Hₙ₊₁(X; ℤ), M) ⟶ 0 for an abelian group M. For an
injective coefficient module the Ext¹ term vanishes and the Kronecker map is an isomorphism over
any ring (TopCat.singularKroneckerEquiv).
Over an arbitrary commutative ring k the cycles of the singular chain complex need not split off,
and there is no such short exact sequence in general. It survives, with vanishing Ext¹ term,
when the homology is projective: if R and every Hₙ(X; R) are projective, the singular chain
complex is a complex of projective modules with projective homology, so its cycles are projective
and split off (ChainComplex.isSplitMono_iCycles_of_projective_homology), and the Kronecker map
is an isomorphism Hⁿ(X; R, M) ≃ₗ[k] Hom(Hₙ(X; R), M) in every degree
(TopCat.singularKroneckerEquivOfProjectiveHomology).
Over a hereditary ring, a single projective homology module already kills the Ext¹ term next
to it: if Hₙ(X; R) is projective, the Kronecker map is an isomorphism in degree n + 1
(TopCat.singularKronecker_bijective_succ_of_projective_homology). So if Hₙ(X; R) is projective
and Hₙ₊₁(X; R) = 0, then Hⁿ⁺¹(X; R, M) = 0
(TopCat.isZero_singularCohomology_succ_of_projective_homology), and likewise for pairs. For
instance, integral cohomology vanishes above the top degree of a space whose integral homology is
free in that degree and zero above it, while in the top degree it need not be torsion-free.
Main declarations #
TopCat.singularKronecker_bijective_of_projective_homologyandTopCat.singularKroneckerEquivOfProjectiveHomology: over any commutative ring, the Kronecker map is an isomorphism when the coefficients and the homology are projective.TopPair.singularKronecker_bijective_of_projective_homologyandTopPair.singularKroneckerEquivOfProjectiveHomology: the same for relative cohomology of a pair whose relative homology is projective.TopCat.singularKronecker_bijective_of_projective_homology_ltandTopCat.injective_homologyMap_linearYonedaObjMap_of_projective_homology_lt: the Kronecker map is bijective in degreenwhen the homology is projective belown, so that an injective map of coefficient modules is injective on cohomology in degreen.TopPair.singularKronecker_bijective_of_projective_homology_lt: the relative Kronecker map is bijective in degreenwhen the relative homology is projective belown, for instance when it vanishes there.TopPair.isZero_singularCohomology_of_isZero_singularHomology: relative cohomology vanishes in degreenwhen relative homology vanishes up to degreen.TopCat.singularKronecker_bijective_succ_of_projective_homology,TopCat.isZero_singularCohomology_succ_of_projective_homologyand theirTopPairversions: over a hereditary ring, the Kronecker map is bijective in degreen + 1whenHₙ(X; R)is projective, and cohomology vanishes in degreen + 1when moreoverHₙ₊₁(X; R) = 0.TopCat.singularExtToCohomology: the mapExt¹(Hₙ(X; R), M) →ₗ[k] Hⁿ⁺¹(X; R, M), computed on extension classes byTopCat.singularExtToCohomology_extClass_comp_mk₀.TopCat.singularExtToCohomology_injectiveandTopCat.exact_singularExtToCohomology_singularKronecker: exactness at the first two places.TopCat.singularKroneckerSectionandTopCat.singularKronecker_singularKroneckerSection: ak-linear right inverse of the Kronecker map, so thatTopCat.singularKronecker_surjectivegives exactness at the last place and the sequence splits.TopCat.singularUniversalCoefficientEquiv: the resulting decompositionHⁿ⁺¹(X; R, M) ≃ₗ[k] Ext¹(Hₙ(X; R), M) × Hom(Hₙ₊₁(X; R), M).TopCat.singularExtToCohomology_naturality: naturality of the first map inX; the naturality of the Kronecker map isTopCat.singularKronecker_naturality.
References #
- A. Hatcher, Algebraic Topology, Section 3.1, Theorem 3.2.
- C. Weibel, An Introduction to Homological Algebra, Section 3.6.
The universal coefficient theorem below the first non-projective homology, over any
commutative ring k: for a projective coefficient module R, if the singular homology modules
Hⱼ(X; R) are projective for j < n, the Kronecker map Hⁿ(X; R, M) →ₗ[k] (Hₙ(X; R) ⟶ M) is
bijective, for every module M. For n = 1 the hypothesis is that H₀(X; R) is projective, which
holds for free R (ModuleCat.free_singularHomology_zero).
Injective coefficient maps are injective on cohomology below the first non-projective
homology: for a projective coefficient module R, if Hⱼ(X; R) is projective for j < n, an
injective map g : M ⟶ M' of coefficient modules induces an injective map
Hⁿ(X; R, M) → Hⁿ(X; R, M'), because both are Hom(Hₙ(X; R), -). For n = 1 and R = ℤ this
makes H¹(X; ℤ) → H¹(X; ℝ) injective, so H¹(X; ℤ) is torsion-free.
The universal coefficient theorem with projective homology, over any commutative ring k:
for a projective coefficient module R, if the singular homology modules Hₙ(X; R) are all
projective, the Kronecker map Hⁿ(X; R, M) →ₗ[k] (Hₙ(X; R) ⟶ M) is bijective in every degree,
for every module M. The singular chain modules are projective, so the cycles split off and the
Ext¹ term of the universal coefficient sequence vanishes.
The universal coefficient isomorphism with projective homology: over any commutative ring,
for a projective coefficient module R such that every Hₙ(X; R) is projective, the Kronecker
map is a k-linear equivalence Hⁿ(X; R, M) ≃ₗ[k] (Hₙ(X; R) ⟶ M). This applies to R = k
whenever every Hₙ(X; k) is a free k-module, as for spheres.
Instances For
The equivalence TopCat.singularKroneckerEquivOfProjectiveHomology is the Kronecker map.
The Ext term of the universal coefficient sequence of singular cohomology over a
hereditary commutative ring, the k-linear map Ext¹(Hₙ(X; R), M) →ₗ[k] Hⁿ⁺¹(X; R, M). It is the
algebraic map TauCeti.ChainComplex.extToHomology of the singular chain complex: an extension
class represented by a morphism β : Bₙ ⟶ M out of the singular boundaries goes to the class of
the cocycle Cₙ₊₁(X; R) ⟶ Bₙ ⟶ M.
Instances For
The Ext term of the universal coefficient sequence of singular cohomology is the algebraic
map TauCeti.ChainComplex.extToHomology of the singular chain complex.
The characterization of TopCat.singularExtToCohomology: the class in
Ext¹(Hₙ(X; R), M) of a morphism β : Bₙ ⟶ M out of the singular boundaries
Bₙ = ker(Zₙ ⟶ Hₙ(X; R)) goes to the class of the cocycle Cₙ₊₁(X; R) ⟶ Bₙ ⟶ M.
Injectivity in the universal coefficient sequence: over a hereditary commutative ring,
the map
Ext¹(Hₙ(X; R), M) →ₗ[k] Hⁿ⁺¹(X; R, M) is injective.
Exactness in the middle of the universal coefficient sequence: over a hereditary commutative
ring, a class in Hⁿ⁺¹(X; R, M) evaluates to zero on Hₙ₊₁(X; R) exactly when it comes from
Ext¹(Hₙ(X; R), M).
The splitting of the universal coefficient sequence of singular cohomology over a
hereditary commutative ring: a k-linear right inverse Hom(Hₙ(X; R), M) →ₗ[k] Hⁿ(X; R, M) of the
Kronecker map. It sends g to the class of the cocycle Cₙ(X; R) ⟶ Zₙ ⟶ Hₙ(X; R) ⟶ M, through
a chosen retraction of the inclusion of the singular cycles, and need not be natural in X.
Instances For
TopCat.singularKroneckerSection is a right inverse of the Kronecker map.
Surjectivity in the universal coefficient sequence: over a hereditary commutative ring,
every
morphism Hₙ(X; R) ⟶ M is the evaluation of a class in Hⁿ(X; R, M).
The splitting of the universal coefficient sequence of singular cohomology over a
hereditary commutative ring: a k-linear equivalence
Hⁿ⁺¹(X; R, M) ≃ₗ[k] Ext¹(Hₙ(X; R), M) × Hom(Hₙ₊₁(X; R), M) whose second component is the
Kronecker map (TopCat.singularUniversalCoefficientEquiv_apply_snd) and whose inverse restricts
to TopCat.singularExtToCohomology on the first factor
(TopCat.singularUniversalCoefficientEquiv_symm_inl). It is built from the section
TopCat.singularKroneckerSection and is not natural in X.
Instances For
The inverse of the splitting TopCat.singularUniversalCoefficientEquiv restricts to the
inclusion TopCat.singularExtToCohomology of the Ext term on the first factor.
The second component of the splitting TopCat.singularUniversalCoefficientEquiv is the
Kronecker map.
Naturality of the universal coefficient sequence in the space: for a continuous map
f : X ⟶ Y, pulling an extension back along f_* : Hₙ(X; R) ⟶ Hₙ(Y; R) and then including it in
the cohomology of X is including it in the cohomology of Y and then pulling back along f.
The universal coefficient theorem above a projective homology module, over a hereditary
commutative ring: for a projective coefficient module R, if Hₙ(X; R) is projective, the
Kronecker map Hⁿ⁺¹(X; R, M) →ₗ[k] (Hₙ₊₁(X; R) ⟶ M) is bijective, for every module M. The
Ext¹(Hₙ(X; R), M) term of the universal coefficient sequence vanishes.
Cohomology vanishes above a projective homology module followed by a zero one, over a
hereditary commutative ring: for a projective coefficient module R, if Hₙ(X; R) is projective
and Hₙ₊₁(X; R) = 0, then Hⁿ⁺¹(X; R, M) = 0 for every module M. With k = R = ℤ: if
Hₙ(X; ℤ) is free abelian and Hₙ₊₁(X; ℤ) = 0, then Hⁿ⁺¹(X; M) = 0.
The relative universal coefficient theorem below the first non-projective homology, over
any commutative ring k: for a projective coefficient module R, if the relative singular
homology modules Hⱼ(X, A; R) of a pair are projective for j < n, for instance zero, the
relative Kronecker map Hⁿ(X, A; R, M) →ₗ[k] (Hₙ(X, A; R) ⟶ M) is bijective, for every module
M.
Relative cohomology vanishes up to the degree where relative homology does: for a
projective coefficient module R, if Hⱼ(X, A; R) = 0 for j ≤ n, then
Hⁿ(X, A; R, M) = 0 for every module M.
The relative universal coefficient theorem above a projective homology module, over a
hereditary commutative ring: for a projective coefficient module R, if Hₙ(X, A; R) is
projective, the relative Kronecker map Hⁿ⁺¹(X, A; R, M) →ₗ[k] (Hₙ₊₁(X, A; R) ⟶ M) is bijective,
for every module M. Nothing is assumed about the relative homology in the other degrees.
Relative cohomology vanishes above a projective homology module followed by a zero one,
over a hereditary commutative ring: for a projective coefficient module R, if Hₙ(X, A; R) is
projective and Hₙ₊₁(X, A; R) = 0, then Hⁿ⁺¹(X, A; R, M) = 0 for every module M.
The relative universal coefficient theorem with projective homology, over any commutative
ring k: for a projective coefficient module R, if the relative singular homology modules
Hₙ(X, A; R) of a pair are all projective, the relative Kronecker map
Hⁿ(X, A; R, M) →ₗ[k] (Hₙ(X, A; R) ⟶ M) is bijective in every degree, for every module M.
The relative universal coefficient isomorphism with projective homology: over any
commutative ring, for a projective coefficient module R such that every relative homology module
Hₙ(X, A; R) is projective, the relative Kronecker map is a k-linear equivalence
Hⁿ(X, A; R, M) ≃ₗ[k] (Hₙ(X, A; R) ⟶ M).
Instances For
The equivalence TopPair.singularKroneckerEquivOfProjectiveHomology is the relative Kronecker
map.