The cohomology class of a skew-zigzag parameter

A skew-zigzag parameter c of a simple graph G labels each ordered pair of incident edges by a unit-valued ratio between the two backtracks they carry. This file identifies the gauge classes of such parameters with the first cohomology H¹(G, kˣ) of the graph.

The identification goes through local edge coordinates. Every parameter can be written as ratio(h, h') = τ h / τ h' for a unit τ h on each oriented edge, and the 1-cochain h ↦ τ h / τ h.symm has a cohomology class which does not depend on the choice of τ; this is the cohomology class of c. Parameters form a group under the ratio-by-ratio product and the class is a group homomorphism. Its kernel consists of the gauge-trivial parameters, and every class is attained: the edge coordinate τ can be put on one orientation of every edge, so no square roots of units are needed. Hence:

gauge classes of skew-zigzag parameters of G  ≃  H¹(G, kˣ),

over any commutative monoid k. Over a commutative ring and for a finite graph, the gauge classes are the vertex-fixing isomorphism classes of the skew-zigzag relation quotients, so the cohomology class classifies those quotients up to vertex-fixing isomorphism.

Main definitions

• TauCeti.SkewZigzagParameter.cohomologyClass: the cohomology class of a skew-zigzag parameter, as a group homomorphism to H¹(G, kˣ).
• TauCeti.SkewZigzagParameter.gaugeClassEquivFirstCohomology: the gauge classes of skew-zigzag parameters are in bijection with H¹(G, kˣ).

Main results

• TauCeti.SkewZigzagParameter.cohomologyClass_apply: the cohomology class of a parameter is the class of its 1-cochain of transition factors.
• TauCeti.SkewZigzagParameter.cohomologyClass_eq_mk_of_ratio_eq_div: the cohomology class of a parameter written in edge coordinates τ is the class of h ↦ τ h / τ h.symm.
• TauCeti.SkewZigzagParameter.cohomologyClass_eq_one_iff: a parameter has trivial class exactly when it is gauge trivial.
• TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff: two parameters have the same class exactly when they are gauge equivalent.
• TauCeti.SkewZigzagParameter.cohomologyClass_surjective: every class in H¹(G, kˣ) is the class of a parameter.
• TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff_exists_vertexFixing_algEquiv: over a commutative ring and for a finite graph, two parameters have the same class exactly when their relation quotients are isomorphic by an isomorphism fixing every vertex idempotent.

References

C. Couture, Skew-Zigzag Algebras, Section 4, Theorem 4.8, https://arxiv.org/abs/1509.08405, which identifies the vertex-fixing graded isomorphism classes of skew-zigzag algebras with the graph cohomology H¹(G, kˣ).

Parameters from edge coordinates

def TauCeti.SkewZigzagParameter.ofEdgeCoordinate {k : Type w} {V : Type u} {G : SimpleGraph V} [Monoid k] (τ : ⦃i j : V⦄ → G.Adj i j → kˣ) : SkewZigzagParameter k G

The parameter with prescribed edge coordinates τ, whose ratios are τ h / τ h'.

@[simp]

theorem TauCeti.SkewZigzagParameter.ofEdgeCoordinate_ratio {k : Type w} {V : Type u} {G : SimpleGraph V} [Monoid k] (τ : ⦃i j : V⦄ → G.Adj i j → kˣ) {i j j' : V} (h : G.Adj i j) (h' : G.Adj i j') : (ofEdgeCoordinate τ).ratio h h' = τ h / τ h'

The ratio of the parameter constructed from edge coordinates is their quotient.

The cohomology class

noncomputable def TauCeti.SkewZigzagParameter.cohomologyClass (k : Type w) {V : Type u} (G : SimpleGraph V) [CommMonoid k] : SkewZigzagParameter k G →* G.FirstCohomology kˣ

The cohomology class of a skew-zigzag parameter in the first cohomology H¹(G, kˣ) of the graph: the class of the 1-cochain h ↦ τ h / τ h.symm for any edge coordinates τ with ratio(h, h') = τ h / τ h' (see cohomologyClass_eq_mk_of_ratio_eq_div).

theorem TauCeti.SkewZigzagParameter.cohomologyClass_apply {k : Type w} {V : Type u} {G : SimpleGraph V} [CommMonoid k] (c : SkewZigzagParameter k G) : (cohomologyClass k G) c = (SimpleGraph.FirstCohomology.mk G kˣ) ⟨fun (d : G.Dart) => c.transition ⋯, ⋯⟩

The cohomology class of a parameter is the class of its 1-cochain of transition factors h ↦ transition c h.

theorem TauCeti.SkewZigzagParameter.cohomologyClass_eq_mk_of_ratio_eq_div {k : Type w} {V : Type u} {G : SimpleGraph V} [CommMonoid k] (c : SkewZigzagParameter k G) (τ : ⦃i j : V⦄ → G.Adj i j → kˣ) (hτ : ∀ ⦃i j j' : V⦄ (h : G.Adj i j) (h' : G.Adj i j'), c.ratio h h' = τ h / τ h') : (cohomologyClass k G) c = (SimpleGraph.FirstCohomology.mk G kˣ) ⟨fun (d : G.Dart) => τ ⋯ / τ ⋯, ⋯⟩

The cohomology class of a parameter in edge coordinates: if ratio(h, h') = τ h / τ h' for every two edges at a common vertex, the class of the parameter is the class of the 1-cochain h ↦ τ h / τ h.symm.

theorem TauCeti.SkewZigzagParameter.cohomologyClass_eq_one_iff {k : Type w} {V : Type u} {G : SimpleGraph V} [CommMonoid k] {c : SkewZigzagParameter k G} : (cohomologyClass k G) c = 1 ↔ IsGaugeEquivalent 1 c

A parameter has trivial cohomology class exactly when it is gauge trivial.

theorem TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff {k : Type w} {V : Type u} {G : SimpleGraph V} [CommMonoid k] {c c' : SkewZigzagParameter k G} : (cohomologyClass k G) c = (cohomologyClass k G) c' ↔ c.IsGaugeEquivalent c'

Two parameters have the same cohomology class exactly when they are gauge equivalent.

@[simp]

theorem TauCeti.SkewZigzagParameter.cohomologyClass_gauge {k : Type w} {V : Type u} {G : SimpleGraph V} [CommMonoid k] (c : SkewZigzagParameter k G) (u : ⦃x y : DoubledQuiver G⦄ → (x ⟶ y) → kˣ) : (cohomologyClass k G) (c.gauge u) = (cohomologyClass k G) c

A gauge transform does not change the cohomology class.

Every class is attained

theorem TauCeti.SkewZigzagParameter.cohomologyClass_surjective {k : Type w} {V : Type u} {G : SimpleGraph V} [CommMonoid k] : Function.Surjective ⇑(cohomologyClass k G)

Every class in H¹(G, kˣ) is the cohomology class of a skew-zigzag parameter.

Gauge classes and vertex-fixing isomorphism classes

noncomputable def TauCeti.SkewZigzagParameter.gaugeClassEquivFirstCohomology (k : Type w) {V : Type u} (G : SimpleGraph V) [CommMonoid k] : MulAction.orbitRel.Quotient (⦃x y : DoubledQuiver G⦄ → (x ⟶ y) → kˣ) (SkewZigzagParameter k G) ≃ G.FirstCohomology kˣ

The gauge classes of skew-zigzag parameters are in bijection with H¹(G, kˣ), a class being sent to the cohomology class of any of its parameters.

@[simp]

theorem TauCeti.SkewZigzagParameter.gaugeClassEquivFirstCohomology_mk {k : Type w} {V : Type u} {G : SimpleGraph V} [CommMonoid k] (c : SkewZigzagParameter k G) : (gaugeClassEquivFirstCohomology k G) ⟦c⟧ = (cohomologyClass k G) c

The bijection sends the gauge class of a parameter to its cohomology class.

theorem TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff_exists_vertexFixing_algEquiv {V : Type u} {G : SimpleGraph V} {k : Type w} [CommRing k] [Finite V] {c c' : SkewZigzagParameter k G} : (cohomologyClass k G) c = (cohomologyClass k G) c' ↔ ∃ (φ : skewZigzagQuotient k G c ≃ₐ[k] skewZigzagQuotient k G c'), ∀ (i : V), φ ((skewZigzagMk k G c) (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G i))) = (skewZigzagMk k G c') (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G i))

Couture's classification: over a commutative ring, two skew-zigzag parameters of a finite graph have the same cohomology class in H¹(G, kˣ) exactly when their relation quotients are isomorphic by an algebra isomorphism fixing every vertex idempotent.