The path-length grading of a skew-zigzag relation quotient

Every skew-zigzag relator of a simple graph is homogeneous for the path-length grading of the path algebra of the doubled quiver: a non-returning length-two path and the scalar relation backtrack(h) - ratio(h, h') • backtrack(h') between two backtracks at one vertex sit in degree two, and each long generator is a single path. So the skew relation ideal is homogeneous, and the skew-zigzag relation quotient Z_k(G, c) carries the induced grading, in which the vertex idempotents have degree 0, the arrows degree 1, and the backtracks degree 2.

The comparison isomorphisms between skew-zigzag quotients are graded: the gauge isomorphism rescales arrows, the relabelling along a graph isomorphism preserves path length, and the comparison of the constant parameter with the ordinary zigzag relation quotient is the identity on representatives. Consequently the vertex-fixing classification from Couture's setting extends here to finite graphs over commutative rings, with vertex-fixing graded isomorphisms: two parameters are gauge equivalent, equivalently have the same class in H¹(G, kˣ), exactly when their relation quotients are isomorphic by a graded isomorphism fixing every vertex idempotent. A graded isomorphism φ : Z_k(G, c) ≃ₐ[k] Z_k(G, c') is spelled out as an algebra isomorphism such that φ x lies in the degree-n piece of Z_k(G, c') exactly when x lies in the degree-n piece of Z_k(G, c), for every n.

Main definitions

• TauCeti.skewZigzagGrade: the induced degree-n piece of the skew-zigzag relation quotient, the descent of TauCeti.PathAlgebra.grade along TauCeti.skewZigzagMk.
• TauCeti.skewZigzagGradedAlgebra: the skew-zigzag relation quotient is a graded algebra for the induced path-length grading.

Main results

• TauCeti.isHomogeneous_skewZigzagIdeal: the skew relation ideal is homogeneous.
• TauCeti.isInternal_skewZigzagGrade: the quotient is the internal direct sum of its graded pieces.
• TauCeti.skewZigzagGrade_zero_eq_span_range_vertexIdempotent, TauCeti.skewZigzagGrade_one_eq_span_range_ofArrow, TauCeti.skewZigzagGrade_two_eq_span_range_backtrackElem and TauCeti.skewZigzagGrade_eq_bot_of_three_le: the pieces are spanned by the vertex idempotents, the arrows, and the backtracks, and vanish from degree three on.
• TauCeti.skewZigzagQuotientGaugeEquiv_mem_skewZigzagGrade_iff, TauCeti.skewZigzagQuotientEquiv_mem_skewZigzagGrade_iff and TauCeti.skewZigzagQuotientOneEquiv_mem_zigzagGrade_iff: the gauge, relabelling and constant-parameter isomorphisms are graded.
• TauCeti.SkewZigzagParameter.isGaugeEquivalent_iff_exists_graded_vertexFixing_algEquiv and TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff_exists_graded_vertexFixing_algEquiv: the vertex-fixing graded isomorphism classes are the gauge classes, and hence the classes in H¹(G, kˣ).

References

C. Couture, Skew-Zigzag Algebras, Section 3 for the grading and Section 4, Theorem 4.8, for the vertex-fixing classification of connected graphs over fields containing square roots, https://arxiv.org/abs/1509.08405. The graded classification here extends that result to finite graphs over commutative rings.

Homogeneity of the skew relators

theorem TauCeti.IsSkewZigzagRelator.isHomogeneousElem (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) (c : SkewZigzagParameter k G) {x : pathAlgebra k (DoubledQuiver G)} (hx : IsSkewZigzagRelator k G c x) : SetLike.IsHomogeneousElem (PathAlgebra.grade k (DoubledQuiver G)) x

The skew-zigzag relators are homogeneous: the non-returning quadratic paths and the scalar backtrack relators in degree two, and each long generator in the degree its own length names.

theorem TauCeti.isHomogeneous_skewZigzagIdeal (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) (c : SkewZigzagParameter k G) [Finite V] : Ideal.IsHomogeneous (PathAlgebra.grade k (DoubledQuiver G)) (TwoSidedIdeal.asIdeal (skewZigzagIdeal k G c))

The skew relation ideal is homogeneous for the path-length grading. This is the condition needed to descend the grading to the skew-zigzag quotient.

The induced grading on the skew relation quotient

noncomputable def TauCeti.skewZigzagGrade (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) (n : ℕ) : Submodule k (skewZigzagQuotient k G c)

The induced grading on the skew-zigzag relation quotient: the degree-n piece is the image of the degree-n piece of the path-length grading of the doubled path algebra under the quotient map. By TauCeti.isInternal_skewZigzagGrade these pieces form an internal direct sum decomposition of the quotient itself.

theorem TauCeti.mem_skewZigzagGrade_iff (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) {n : ℕ} {x : skewZigzagQuotient k G c} : x ∈ skewZigzagGrade k G c n ↔ ∃ y ∈ PathAlgebra.grade k (DoubledQuiver G) n, (skewZigzagMk k G c) y = x

Membership in a graded piece is being the class of a homogeneous element of that degree.

@[simp]

theorem TauCeti.skewZigzagMk_mem_skewZigzagGrade (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) {n : ℕ} {y : pathAlgebra k (DoubledQuiver G)} (hy : y ∈ PathAlgebra.grade k (DoubledQuiver G) n) : (skewZigzagMk k G c) y ∈ skewZigzagGrade k G c n

A homogeneous element lands in the piece its degree names.

theorem TauCeti.isInternal_skewZigzagGrade (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) : DirectSum.IsInternal (skewZigzagGrade k G c)

The skew-zigzag quotient is the internal direct sum of its graded pieces.

theorem TauCeti.mul_mem_skewZigzagGrade (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) {m n : ℕ} {x y : skewZigzagQuotient k G c} (hx : x ∈ skewZigzagGrade k G c m) (hy : y ∈ skewZigzagGrade k G c n) : x * y ∈ skewZigzagGrade k G c (m + n)

Multiplication adds degrees in the induced grading.

@[instance_reducible]

noncomputable def TauCeti.skewZigzagGradedAlgebra (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) : GradedAlgebra (skewZigzagGrade k G c)

The skew-zigzag relation quotient is a graded algebra for the induced path-length grading. This is kept as a definition rather than an instance so that callers choose when to introduce it locally; see TauCeti.GradedAlgebra.gradedAlgebraQuotientPiece.

The graded pieces

theorem TauCeti.skewZigzagGrade_eq_span_range (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) (n : ℕ) : skewZigzagGrade k G c n = Submodule.span k (Set.range fun (p : { p : Quiver.TotalPath (DoubledQuiver G) // p.snd.snd.length = n }) => (skewZigzagMk k G c) (PathAlgebra.ofPath ↑p))

The degree-n piece is spanned by the classes of paths of length n.

theorem TauCeti.skewZigzagGrade_zero_eq_span_range_vertexIdempotent (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) : skewZigzagGrade k G c 0 = Submodule.span k (Set.range fun (i : V) => (skewZigzagMk k G c) (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G i)))

Degree zero is spanned by the vertex idempotent classes.

theorem TauCeti.skewZigzagGrade_one_eq_span_range_ofArrow (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) : skewZigzagGrade k G c 1 = Submodule.span k (Set.range fun (d : G.Dart) => (skewZigzagMk k G c) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯)))

Degree one is spanned by the arrow classes, one for each dart of the graph.

theorem TauCeti.skewZigzagGrade_two_eq_span_range_backtrackElem (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) : skewZigzagGrade k G c 2 = Submodule.span k (Set.range fun (d : G.Dart) => (skewZigzagMk k G c) (DoubledQuiver.backtrackElem G k ⋯))

Degree two is spanned by the backtrack classes, one for each dart of the graph: every length-two path either does not return, and dies, or returns to its source and is a backtrack. The backtracks at one vertex agree only up to the unit ratios of the parameter, so the family is indexed by darts rather than by one volume class per vertex.

theorem TauCeti.skewZigzagGrade_eq_bot_of_three_le (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c : SkewZigzagParameter k G) {n : ℕ} (hn : 3 ≤ n) : skewZigzagGrade k G c n = ⊥

Every piece of degree at least three vanishes: all long paths are relations.

The comparison isomorphisms are graded

@[simp]

theorem TauCeti.skewZigzagQuotientGaugeEquiv_mem_skewZigzagGrade_iff (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] (c c' : SkewZigzagParameter k G) (u : ⦃x y : DoubledQuiver G⦄ → (x ⟶ y) → kˣ) (hc : c' = c.gauge u) {n : ℕ} {x : skewZigzagQuotient k G c} : (skewZigzagQuotientGaugeEquiv k G c c' u hc) x ∈ skewZigzagGrade k G c' n ↔ x ∈ skewZigzagGrade k G c n

The gauge isomorphism is graded: it and its inverse are arrow rescalings, which preserve path length.

@[simp]

theorem TauCeti.skewZigzagQuotientOneEquiv_mem_zigzagGrade_iff (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] {n : ℕ} {x : skewZigzagQuotient k G 1} : (skewZigzagQuotientOneEquiv k G) x ∈ zigzagGrade k G n ↔ x ∈ skewZigzagGrade k G 1 n

The comparison of the constant parameter with the ordinary zigzag relation quotient is graded: it matches the skew-zigzag grading at the constant parameter with the ordinary zigzag grading.

@[simp]

theorem TauCeti.skewZigzagQuotientEquiv_mem_skewZigzagGrade_iff (k : Type w) [CommRing k] {V : Type u} (G : SimpleGraph V) [Finite V] {W : Type v} [Finite W] {H : SimpleGraph W} (e : G ≃g H) (c : SkewZigzagParameter k G) {n : ℕ} {x : skewZigzagQuotient k G c} : (skewZigzagQuotientEquiv k e c) x ∈ skewZigzagGrade k H (SkewZigzagParameter.relabel e c) n ↔ x ∈ skewZigzagGrade k G c n

The relabelling isomorphism along a graph isomorphism is graded: it and its inverse are induced by relabellings of the doubled path algebras, which preserve path length.

Vertex-fixing graded classification extending Couture's result

theorem TauCeti.SkewZigzagParameter.isGaugeEquivalent_iff_exists_graded_vertexFixing_algEquiv {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [Finite V] {c c' : SkewZigzagParameter k G} : c.IsGaugeEquivalent c' ↔ ∃ (φ : skewZigzagQuotient k G c ≃ₐ[k] skewZigzagQuotient k G c'), (∀ (n : ℕ) (x : skewZigzagQuotient k G c), φ x ∈ skewZigzagGrade k G c' n ↔ x ∈ skewZigzagGrade k G c n) ∧ ∀ (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))

Gauge classes are vertex-fixing graded isomorphism classes. Two skew-zigzag parameters over a commutative ring are gauge equivalent exactly when their relation quotients are isomorphic by a graded algebra isomorphism fixing every vertex idempotent. The gauge isomorphism is graded, and conversely every vertex-fixing isomorphism, graded or not, forces gauge equivalence (AlgEquiv.isGaugeEquivalent_of_vertexFixing).

theorem TauCeti.SkewZigzagParameter.cohomologyClass_eq_iff_exists_graded_vertexFixing_algEquiv {k : Type w} [CommRing k] {V : Type u} {G : SimpleGraph V} [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'), (∀ (n : ℕ) (x : skewZigzagQuotient k G c), φ x ∈ skewZigzagGrade k G c' n ↔ x ∈ skewZigzagGrade k G c n) ∧ ∀ (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))

Vertex-fixing graded classification extending Couture's result: 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 a graded algebra isomorphism fixing every vertex idempotent. Couture's Theorem 4.8 treats connected graphs over fields containing square roots.