The radical filtration of a zigzag algebra

For a finite simple graph without isolated vertices over a field, the Jacobson radical of the zigzag relation quotient is its positive-length part. This file makes that statement intrinsic: TauCeti.zigzagTrivialCoeff reads off the coefficients of the vertex idempotents, and the radical is its kernel. In the vertex-arrow-volume basis this is precisely the span of the arrows and volumes.

The multiplication table then gives the whole filtration. A product of two radical elements is in the span of the volume classes, every volume annihilates the radical, and hence the third radical power vanishes. Conversely every volume is a product of a dart with its reverse, so the second radical layer is exactly the volume span.

Main definitions

• TauCeti.zigzagTrivialCoeff: the algebra homomorphism from the zigzag quotient to DoubledQuiver G → k that retains the vertex coefficients.
• TauCeti.zigzagPositiveSpan: the span of the arrow and volume basis classes.
• TauCeti.zigzagVolumeSpan: the span of the volume basis classes, with the description TauCeti.zigzagVolumeSpan_eq_span and the elimination lemma TauCeti.zigzagVolumeSpan_le.

Main results

• TauCeti.jacobson_nonisolatedZigzagQuotient_eq_ker: the Jacobson radical is the kernel of TauCeti.zigzagTrivialCoeff, equivalently the positive-length span.
• TauCeti.restrictScalars_jacobson_sq_nonisolatedZigzagQuotient_eq_zigzagVolumeSpan: its square is the volume span.
• TauCeti.jacobson_pow_three_nonisolatedZigzagQuotient_eq_bot: its third power is zero.
• TauCeti.mem_zigzagPositiveSpan_of_zigzagVolume_mul_eq_zero: an element of e_u Z killed by the volume class x_u lies in the positive-length span.

References

See Huerfano--Khovanov, A category for the adjoint representation, Section 3, and Ehrig--Tubbenhauer, Algebraic properties of zigzag algebras, Section 2.

The vertex-coefficient homomorphism

noncomputable def TauCeti.zigzagTrivialCoeff {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] : nonisolatedZigzagQuotient k G →ₐ[k] DoubledQuiver G → k

The vertex-coefficient homomorphism of the zigzag relation quotient. It sends a vertex idempotent to its indicator function and kills every arrow and volume class.

@[simp]

theorem TauCeti.zigzagTrivialCoeff_zigzagMk {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] (x : pathAlgebra k (DoubledQuiver G)) : (zigzagTrivialCoeff k G) ((zigzagMk k G) x) = (PathAlgebra.trivialCoeff k (DoubledQuiver G)) x

The vertex-coefficient homomorphism after the zigzag quotient map is the path-algebra trivial-coefficient homomorphism.

@[simp]

theorem TauCeti.zigzagTrivialCoeff_vertexIdempotent {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] [DecidableEq V] (i j : V) : (zigzagTrivialCoeff k G) ((zigzagMk k G) (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G i))) (DoubledQuiver.vertex G j) = if i = j then 1 else 0

The vertex idempotent at i has coefficient one at i and zero at every other vertex.

theorem TauCeti.zigzagTrivialCoeff_ofArrow {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] (d : G.Dart) : (zigzagTrivialCoeff k G) ((zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯))) = 0

Every arrow class has zero vertex coefficient. Deliberately not a simp lemma, TauCeti.PathAlgebra.ofArrow_eq_ofPath already rewriting its left-hand side.

@[simp]

theorem TauCeti.zigzagTrivialCoeff_zigzagVolume {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] (i : V) : (zigzagTrivialCoeff k G) (zigzagVolume k G i) = 0

Every volume class has zero vertex coefficient.

theorem TauCeti.zigzagTrivialCoeff_surjective {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] : Function.Surjective ⇑(zigzagTrivialCoeff k G)

Every family of vertex coefficients occurs.

The positive-length and volume spans

noncomputable def TauCeti.zigzagPositiveSpan {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] : Submodule k (nonisolatedZigzagQuotient k G)

The positive-length subspace of a zigzag algebra: the span of its arrow and volume basis classes. The Jacobson radical is identified with this subspace below.

noncomputable def TauCeti.zigzagVolumeSpan {V : Type u} (k : Type w) [CommRing k] (G : SimpleGraph V) [Finite V] : Submodule k (nonisolatedZigzagQuotient k G)

The degree-two subspace of a zigzag algebra: the span of its volume classes. This is the second radical layer.

theorem TauCeti.zigzagMk_ofArrow_mem_zigzagPositiveSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (d : G.Dart) : (zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯)) ∈ zigzagPositiveSpan k G

Every arrow class belongs to the positive-length span.

theorem TauCeti.zigzagVolume_mem_zigzagPositiveSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (i : V) : zigzagVolume k G i ∈ zigzagPositiveSpan k G

Every volume class belongs to the positive-length span.

theorem TauCeti.zigzagVolume_mem_zigzagVolumeSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (i : V) : zigzagVolume k G i ∈ zigzagVolumeSpan k G

Every volume class belongs to the volume span.

theorem TauCeti.zigzagVolumeSpan_eq_span {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] : zigzagVolumeSpan k G = Submodule.span k (Set.range (zigzagVolume k G))

The volume span is the span of the volume classes.

theorem TauCeti.zigzagVolumeSpan_le {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] {N : Submodule k (nonisolatedZigzagQuotient k G)} (h : ∀ (i : V), zigzagVolume k G i ∈ N) : zigzagVolumeSpan k G ≤ N

A subspace containing every volume class contains the volume span.

theorem TauCeti.zigzagVolumeSpan_le_zigzagPositiveSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] : zigzagVolumeSpan k G ≤ zigzagPositiveSpan k G

The volume span is contained in the positive-length span.

theorem TauCeti.mem_zigzagPositiveSpan_iff {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {x : nonisolatedZigzagQuotient k G} : x ∈ zigzagPositiveSpan k G ↔ ∀ (i : V), ((zigzagBasis k G hns).repr x) (Sum.inl i) = 0

The positive-length span is characterized by vanishing vertex coordinates.

theorem TauCeti.mem_zigzagVolumeSpan_iff {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {x : nonisolatedZigzagQuotient k G} : x ∈ zigzagVolumeSpan k G ↔ (∀ (i : V), ((zigzagBasis k G hns).repr x) (Sum.inl i) = 0) ∧ ∀ (d : G.Dart), ((zigzagBasis k G hns).repr x) (Sum.inr (Sum.inl d)) = 0

The volume span is characterized by vanishing vertex and arrow coordinates.

theorem TauCeti.zigzagTrivialCoeff_apply_eq_repr {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (x : nonisolatedZigzagQuotient k G) (i : V) : (zigzagTrivialCoeff k G) x (DoubledQuiver.vertex G i) = ((zigzagBasis k G hns).repr x) (Sum.inl i)

Evaluating the vertex-coefficient homomorphism at i reads the i-th vertex coordinate in the vertex-arrow-volume basis.

theorem TauCeti.ker_zigzagTrivialCoeff_eq_zigzagPositiveSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : Submodule.restrictScalars k (RingHom.ker (zigzagTrivialCoeff k G).toRingHom) = zigzagPositiveSpan k G

The kernel of the vertex-coefficient map is exactly the positive-length span.

Products in the positive-length filtration

theorem TauCeti.mul_mem_zigzagVolumeSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] {x y : nonisolatedZigzagQuotient k G} (hx : x ∈ zigzagPositiveSpan k G) (hy : y ∈ zigzagPositiveSpan k G) : x * y ∈ zigzagVolumeSpan k G

Multiplying two positive-length elements gives a linear combination of volume classes.

theorem TauCeti.mul_eq_zero_of_mem_zigzagVolumeSpan_of_mem_zigzagPositiveSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] {x y : nonisolatedZigzagQuotient k G} (hx : x ∈ zigzagVolumeSpan k G) (hy : y ∈ zigzagPositiveSpan k G) : x * y = 0

A volume combination annihilates every positive-length element on the left.

theorem TauCeti.mul_mul_eq_zero_of_mem_zigzagPositiveSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] {x y z : nonisolatedZigzagQuotient k G} (hx : x ∈ zigzagPositiveSpan k G) (hy : y ∈ zigzagPositiveSpan k G) (hz : z ∈ zigzagPositiveSpan k G) : x * y * z = 0

Three positive-length elements have zero product.

theorem TauCeti.mul_zigzagMk_ofArrow_mem_zigzagPositiveSpan {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (y : nonisolatedZigzagQuotient k G) (d : G.Dart) : y * (zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯)) ∈ zigzagPositiveSpan k G

Every left multiple of an arrow class has positive length.

theorem TauCeti.zigzagMk_ofArrow_mem_jacobson {V : Type u} {k : Type w} [CommRing k] {G : SimpleGraph V} [Finite V] (d : G.Dart) : (zigzagMk k G) (PathAlgebra.ofArrow (DoubledQuiver.arrow G ⋯)) ∈ Ring.jacobson (nonisolatedZigzagQuotient k G)

An arrow class lies in the Jacobson radical, with no hypothesis on isolated vertices: every left multiple of it is nilpotent, since three positive-length elements multiply to zero.

Identification with the Jacobson radical

theorem TauCeti.jacobson_nonisolatedZigzagQuotient_eq_ker {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : Ring.jacobson (nonisolatedZigzagQuotient k G) = RingHom.ker (zigzagTrivialCoeff k G).toRingHom

The Jacobson radical of a zigzag relation quotient is its positive-length part.

theorem TauCeti.restrictScalars_jacobson_nonisolatedZigzagQuotient_eq_zigzagPositiveSpan {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : Submodule.restrictScalars k (Ring.jacobson (nonisolatedZigzagQuotient k G)) = zigzagPositiveSpan k G

The Jacobson radical, viewed as a k-submodule, is the span of the arrow and volume basis classes.

theorem TauCeti.zigzagVolume_mem_jacobson_sq {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) (i : V) : zigzagVolume k G i ∈ Ring.jacobson (nonisolatedZigzagQuotient k G) ^ 2

Every volume class belongs to the square of the Jacobson radical: it is the product of an arrow with its reverse.

theorem TauCeti.restrictScalars_jacobson_sq_nonisolatedZigzagQuotient_eq_zigzagVolumeSpan {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : Submodule.restrictScalars k (Ring.jacobson (nonisolatedZigzagQuotient k G) ^ 2) = zigzagVolumeSpan k G

The square of the Jacobson radical is the volume span.

theorem TauCeti.jacobson_pow_three_nonisolatedZigzagQuotient_eq_bot {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) : Ring.jacobson (nonisolatedZigzagQuotient k G) ^ 3 = ⊥

The third power of the Jacobson radical of a zigzag algebra vanishes.

theorem TauCeti.mem_zigzagPositiveSpan_of_zigzagVolume_mul_eq_zero {V : Type u} {k : Type w} [Field k] {G : SimpleGraph V} [Finite V] (hns : ∀ (i : V), ∃ (j : V), G.Adj i j) {u : V} {c : nonisolatedZigzagQuotient k G} (hc : (zigzagMk k G) (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G u)) * c = c) (h : zigzagVolume k G u * c = 0) : c ∈ zigzagPositiveSpan k G

An element of e_u Z annihilated by the volume class x_u lies in the radical. Its coefficient at e_u vanishes because x_u c is that coefficient times x_u ≠ 0, and its other vertex coefficients vanish because c = e_u c.