Split complexes are homotopy equivalent to complexes with zero differential #
Let K be a homological complex, of any shape, in an abelian category. Write Zᵢ for its cycles
and Hᵢ for its homology. Suppose that in every degree the inclusion Zᵢ ⟶ Kᵢ of the cycles is
a split monomorphism and the projection Zᵢ ⟶ Hᵢ onto the homology is a split epimorphism. Then
K is chain homotopy equivalent to the complex with terms Hᵢ and zero differentials
(HomologicalComplex.exists_homotopyEquiv_d_eq_zero).
Both hypotheses hold for every complex of vector spaces, and more generally of semisimple modules
(see TauCeti.Algebra.Homology.Semisimple). The second holds whenever the homology objects are
projective (HomologicalComplex.isSplitEpi_homologyπ_of_projective). The first holds for a
complex of projective modules over a principal ideal domain, since the image of the differential
out of Kᵢ is then projective (HomologicalComplex.isSplitMono_iCycles_of_isPrincipalIdealRing);
mathematically, the same argument works over any hereditary ring. The result therefore reduces
statements about the homology of such complexes, such as the Künneth theorem, to complexes with
zero differential.
For a chain complex indexed by ℕ whose terms and homology objects are projective, both
hypotheses hold over any ring. The projection of the cycles Zₙ onto the projective homology
splits, so the boundaries Bₙ = ker (Zₙ ⟶ Hₙ) are a retract of Zₙ; and Kₙ₊₁ ⟶ Bₙ is the
cokernel of Zₙ₊₁ ⟶ Kₙ₊₁, so Zₙ₊₁ splits off Kₙ₊₁ once Bₙ is projective. Starting from
Z₀ = K₀, induction on n shows that every Zₙ is projective and splits off
(ChainComplex.isSplitMono_iCycles_of_projective_homology). This is what makes the Künneth and
universal coefficient theorems collapse over an arbitrary commutative ring for singular chains with
projective homology. The induction up to degree n only uses projectivity of the homology below
n (ChainComplex.isSplitMono_iCycles_of_projective_homology_lt), so it applies for instance to a
complex which is exact below n.
When only the cycles split off, the projection r ≫ π onto the complex K.homologyComplex of
homology objects (HomologicalComplex.toHomologyComplex) is still a chain map inducing
isomorphisms on homology (HomologicalComplex.isIso_homologyMap_toHomologyComplex), although it
need not be a homotopy equivalence. This is the situation for complexes of free abelian groups,
and it is what splits the Künneth short exact sequence. This part only needs a category with zero
morphisms and homology for K, rather than an abelian category.
The construction is the classical splitting. Choose a retraction r : Kᵢ ⟶ Zᵢ of the inclusion
ι : Zᵢ ⟶ Kᵢ and a section s : Hᵢ ⟶ Zᵢ of the projection π : Zᵢ ⟶ Hᵢ. The chain maps are
s ≫ ι : Hᵢ ⟶ Kᵢ and r ≫ π : Kᵢ ⟶ Hᵢ, and their composite on H is the identity. For a
relation j → i of the shape, the differential Kⱼ ⟶ Kᵢ corestricts to an epimorphism onto the
boundaries Bᵢ = ker π with kernel Zⱼ, so it is the cokernel of Zⱼ ⟶ Kⱼ. The idempotent
𝟙 - r ≫ ι of Kⱼ kills Zⱼ and therefore factors through Bᵢ; the resulting map Bᵢ ⟶ Kⱼ
inverts the differential on the complement of the cycles. Precomposing it with the
Bᵢ-component r ≫ (𝟙 - π ≫ s) of Kᵢ gives the homotopy h : Kᵢ ⟶ Kⱼ, with
𝟙 - (r ≫ π) ≫ (s ≫ ι) = d h + h d.
Main results #
HomologicalComplex.exists_homotopyEquiv_d_eq_zero: a complex whose cycles split off and whose homology splits off its cycles is homotopy equivalent to a complex with zero differentials.HomologicalComplex.isSplitEpi_homologyπ_of_projective: the projection onto a projective homology object splits.HomologicalComplex.isSplitMono_iCycles_of_projective: the cycles split off a term whose quotient by them is projective.HomologicalComplex.projective_kernel_homologyπandHomologicalComplex.isSplitMono_iCycles_of_projective_kernel_homologyπ: the boundaries are projective when the cycles and the homology are, and the cyclesZᵢsplit offKᵢwhen the boundariesBⱼof the next degree are projective.HomologicalComplex.Acyclic.isSplitMono_iCyclesandHomologicalComplex.Acyclic.nonempty_homotopy_id_zero_of_projective_cycles: in an acyclic complex the cycles split off once those of the next degree are projective, and an acyclic complex whose cycles are projective is contractible.ChainComplex.projective_cycles_of_projective_homology_ltandChainComplex.isSplitMono_iCycles_of_projective_homology_lt: for a chain complex indexed byℕwith projective terms whose homology is projective below degreen, the cycles in degreenare projective and split off;ChainComplex.projective_cycles_of_projective_homologyandChainComplex.isSplitMono_iCycles_of_projective_homologyare the case of projective homology in every degree.HomologicalComplex.homologyComplex,HomologicalComplex.toHomologyComplexandHomologicalComplex.isIso_homologyMap_toHomologyComplex: the complex of homology objects, and the projection onto it when the cycles split off, a quasi-isomorphism.
References #
- C. Weibel, An Introduction to Homological Algebra, Section 1.4 (split complexes).
The projection from the cycles onto a projective homology object is a split epimorphism.
This only requires homology in degree i, in a category with zero morphisms.
The cycles of a projective term are projective when they split off the term.
The complex of homology objects #
The complex with the homology objects of K as terms and zero differentials.
Instances For
The projection r ≫ π : Kᵢ ⟶ Hᵢ onto the complex of homology objects, where r is the
chosen retraction of the inclusion of the cycles; it is a chain map since boundaries have zero
homology class. It depends on the choice of r, so it need not be natural in K.
Instances For
The components of the projection onto the complex of homology objects.
On the cycles, the projection onto the complex of homology objects is the projection onto homology.
On the cycles, the projection onto the complex of homology objects is the projection onto homology.
The projection onto the complex of homology objects induces isomorphisms on homology.
The inclusion of the homology objects #
The inclusion of the cycles is a split monomorphism when the quotient of the term by its cycles
is projective. This only requires homology in degree i and the displayed cokernel in a
preadditive category.
The boundaries Bᵢ = ker (Zᵢ ⟶ Hᵢ) are projective when the cycles Zᵢ and the homology Hᵢ
are: the projection of the cycles onto projective homology splits, so the boundaries are a retract
of the cycles. This only requires homology in degree i and the displayed kernel in a
preadditive category.
The differential embeds the quotient of a term by its cycles into the next term.
For a relation i → j of the shape, the inclusion of the cycles Zᵢ ⟶ Kᵢ is a split
monomorphism when the boundaries Bⱼ = ker (Zⱼ ⟶ Hⱼ) are projective, since the differential
corestricts to a cokernel Kᵢ ⟶ Bⱼ of that inclusion.
Inverting the differential on the complement of the cycles #
The homotopy #
A complex in an abelian category whose cycles split off its terms and whose homology splits off
its cycles is chain homotopy equivalent to a complex with zero differentials. This applies to
every complex of semisimple modules, for instance of vector spaces over a division ring. The terms
of such a complex are its homology, so by HomotopyEquiv.toHomologyIso they are the homology of
K.
In an acyclic complex, the cycles Zᵢ split off the term Kᵢ once the cycles of the next
degree are projective.
An acyclic complex whose cycles are projective is contractible: the cycles split off the terms, and the complex is homotopy equivalent to its homology, which vanishes.
Chain complexes of projective objects with projective homology #
In a chain complex indexed by ℕ with projective terms, the cycles in degree n are
projective when the homology objects below degree n are projective.
In a chain complex indexed by ℕ with projective terms, the cycles in degree n split off
the term when the homology objects below degree n are projective.
In a chain complex indexed by ℕ whose terms and homology objects are all projective, the
cycles are projective in every degree.
In a chain complex indexed by ℕ whose terms and homology objects are all projective, the
cycles split off the terms in every degree. Since the homology then also splits off the cycles
(HomologicalComplex.isSplitEpi_homologyπ_of_projective), such a complex is homotopy equivalent
to its homology with zero differentials (HomologicalComplex.exists_homotopyEquiv_d_eq_zero).