Documentation

TauCeti.Combinatorics.SimpleGraph.Cohomology.Map

Changing the coefficients of the first cohomology of a graph #

A homomorphism of coefficient groups A →* B sends the 1-cochains of a simple graph to the 1-cochains and, since the coboundary of a vertex function φ : V → A is carried to the coboundary of f ∘ φ, the first cohomology H¹(G, A) to H¹(G, B). This is the map through which a change of coefficient group acts on the first cohomology, and it is the map induced on H¹(G, kˣ) by a monoid homomorphism k → l, the two coefficient groups being the units of the coefficient rings and the parameters being recorded by unit-valued ratios.

The map on cohomology is injective as soon as the homomorphism of coefficient groups is. The injectivity is the statement that a cochain whose image is a coboundary is itself a coboundary, and this is a statement about paths: the product of the values of a cochain along a path from a representative of the connected component of a vertex to that vertex is a vertex value whose image is, by the hypothesis, a coboundary of a vertex function, and injectivity then makes the coboundary of the path product the original cochain.

No surjectivity statement is developed in this file: the classification results which follow from the injectivity below are the reason the map is introduced.

Main definitions #

Main results #

References #

The first cohomology of a graph with coefficients in the units of a field is the group in which C. Couture, Skew-Zigzag Algebras, Section 4, https://arxiv.org/abs/1509.08405, classifies skew-zigzag parameters. The map above is the change of coefficients along a homomorphism of coefficient groups: the map induced on the first cohomology by that homomorphism, which carries the class of a parameter of coefficients k to the class of its image of coefficients l.

source
def SimpleGraph.oneCochainsMap {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) :
↥(G.oneCochains A) →* ↥(G.oneCochains B)

The homomorphism of 1-cochains induced by a homomorphism of coefficient groups.

Equations
Instances For
    source
    @[simp]
    theorem SimpleGraph.oneCochainsMap_apply {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) (σ : ↥(G.oneCochains A)) (d : G.Dart) :
    ↑((G.oneCochainsMap f) σ) d = f (↑σ d)

    The value of an image cochain on a dart is the image of the value of the cochain.

    source
    @[simp]
    theorem SimpleGraph.oneCochainsMap_coboundary {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) (φ : V → A) :
    (G.oneCochainsMap f) ((G.coboundary A) φ) = (G.coboundary B) (⇑f ∘ φ)

    A coboundary is carried to the coboundary of the composed vertex function.

    source
    theorem SimpleGraph.oneCochainsMap_injective {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) (hf : Function.Injective ⇑f) :
    Function.Injective ⇑(G.oneCochainsMap f)

    An injective homomorphism of coefficient groups induces an injective map on 1-cochains.

    source
    @[simp]
    theorem SimpleGraph.oneCochainsMap_id {V : Type u} (G : SimpleGraph V) {A : Type v} [CommGroup A] :
    G.oneCochainsMap (MonoidHom.id A) = MonoidHom.id ↥(G.oneCochains A)

    Mapping cochains along the identity homomorphism of coefficient groups changes nothing.

    source
    @[simp]
    theorem SimpleGraph.oneCochainsMap_comp {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] {C : Type t} [CommGroup C] (f : A →* B) (g : B →* C) :
    G.oneCochainsMap (g.comp f) = (G.oneCochainsMap g).comp (G.oneCochainsMap f)

    Mapping cochains along a composite of homomorphisms of coefficient groups is their successive mapping.

    source
    def SimpleGraph.firstCohomologyMap {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) :
    G.FirstCohomology A →* G.FirstCohomology B

    The map on the first cohomology induced by a homomorphism of coefficient groups.

    Equations
    Instances For
      source
      @[simp]
      theorem SimpleGraph.firstCohomologyMap_mk {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) (σ : ↥(G.oneCochains A)) :
      (G.firstCohomologyMap f) ((FirstCohomology.mk G A) σ) = (FirstCohomology.mk G B) ((G.oneCochainsMap f) σ)

      The induced map sends the cohomology class of a cochain to that of its image.

      source
      @[simp]
      theorem SimpleGraph.firstCohomologyMap_id {V : Type u} (G : SimpleGraph V) {A : Type v} [CommGroup A] :
      G.firstCohomologyMap (MonoidHom.id A) = MonoidHom.id (G.FirstCohomology A)

      Mapping cohomology classes along the identity homomorphism of coefficient groups changes nothing.

      source
      @[simp]
      theorem SimpleGraph.firstCohomologyMap_comp {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] {C : Type t} [CommGroup C] (f : A →* B) (g : B →* C) :
      G.firstCohomologyMap (g.comp f) = (G.firstCohomologyMap g).comp (G.firstCohomologyMap f)

      Mapping cohomology classes along a composite of homomorphisms of coefficient groups is their successive mapping.

      source
      theorem SimpleGraph.firstCohomologyMap_mk_eq_one_iff {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) (hf : Function.Injective ⇑f) (σ : ↥(G.oneCochains A)) :
      (G.firstCohomologyMap f) ((FirstCohomology.mk G A) σ) = 1 ↔ ∃ (φ : V → A), (G.coboundary A) φ = σ

      A 1-cochain is a coboundary exactly when its image is a coboundary, for an injective homomorphism of coefficient groups.

      source
      theorem SimpleGraph.firstCohomologyMap_injective {V : Type u} (G : SimpleGraph V) {A : Type v} {B : Type z} [CommGroup A] [CommGroup B] (f : A →* B) (hf : Function.Injective ⇑f) :
      Function.Injective ⇑(G.firstCohomologyMap f)

      An injective homomorphism of coefficient groups induces an injective map on the first cohomology.