Graph relabelling and skew-zigzag cohomology classes

The first cohomology class classifying a skew-zigzag parameter is natural under graph isomorphisms and under a change of coefficient group. Thus changing vertex labels transports the class by the induced isomorphism of graph cohomology, and a monoid homomorphism k → l carries the class of a parameter to the class of its image under the induced change of coefficients H¹(G, kˣ) → H¹(G, lˣ).

The change of coefficients on cohomology is injective as soon as the induced map on units is, so two parameters whose images under a monoid homomorphism have the same cohomology class were already gauge equivalent, and, for a graph with a finite vertex set over a commutative ring, their relation quotients were already isomorphic by an algebra isomorphism fixing every vertex idempotent. The map on units is the hypothesis under which this is true, and no statement of injectivity is made without it: the coefficient groups of the cohomology are the unit groups of the coefficient monoids.

The parameter classification follows C. Couture, Skew-Zigzag Algebras, Section 4. The cohomology transport uses the graph cohomology construction in TauCeti.Combinatorics.SimpleGraph.Cohomology.Basic, and the change of coefficients in TauCeti.Combinatorics.SimpleGraph.Cohomology.Map.

@[simp]

theorem TauCeti.SkewZigzagParameter.cohomologyClass_relabel {k : Type w} {V : Type u} {W : Type v} {G : SimpleGraph V} {H : SimpleGraph W} [CommMonoid k] (e : G ≃g H) (c : SkewZigzagParameter k G) : (cohomologyClass k H) (relabel e c) = (SimpleGraph.firstCohomologyRelabel kˣ e) ((cohomologyClass k G) c)

Relabelling a skew-zigzag parameter transports its first cohomology class along the same graph isomorphism.

Scalar extension of cohomology classes

@[simp]

theorem TauCeti.SkewZigzagParameter.cohomologyClass_map {k : Type w} {V : Type u} {G : SimpleGraph V} {l : Type z} [CommMonoid k] [CommMonoid l] (f : k →* l) (c : SkewZigzagParameter k G) : (cohomologyClass l G) (map f c) = (G.firstCohomologyMap (Units.map f)) ((cohomologyClass k G) c)

A coefficient homomorphism carries the cohomology class of a parameter to the cohomology class of its image: the class of c.map f is the class obtained from the class of c by the change of coefficients on units kˣ → lˣ.

theorem TauCeti.SkewZigzagParameter.cohomologyClass_map_eq_iff {k : Type w} {V : Type u} {G : SimpleGraph V} {l : Type z} [CommMonoid k] [CommMonoid l] (f : k →* l) (hf : Function.Injective ⇑(Units.map f)) {c c' : SkewZigzagParameter k G} : (cohomologyClass l G) (map f c) = (cohomologyClass l G) (map f c') ↔ c.IsGaugeEquivalent c'

Injectivity on units makes scalar extension preserve the classification: the parameters whose images along f have the same cohomology class are exactly the gauge equivalent ones, the easy direction because gauge equivalence is preserved by f and the other because the induced map on cohomology is injective on units.

Detecting the quotient classification after mapping coefficients

theorem TauCeti.SkewZigzagParameter.cohomologyClass_map_eq_iff_exists_vertexFixing_algEquiv {k : Type w} {V : Type u} {G : SimpleGraph V} {l : Type z} [CommRing k] [CommMonoid l] [Finite V] (f : k →* l) (hf : Function.Injective ⇑(Units.map f)) {c c' : SkewZigzagParameter k G} : (cohomologyClass l G) (map f c) = (cohomologyClass l G) (map f 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 is injective under a coefficient homomorphism that is injective on units: two parameters on a graph with a finite vertex set whose images along f have the same cohomology class, with f injective on units, have already isomorphic relation quotients, by an algebra isomorphism fixing every vertex idempotent.