Quartic Hochschild classes and non-formality of the affine Dₙ zigzag algebras

In the numbering of TauCeti.AffineDynkinType.graph (D n), for n ≥ 5, the nodes 1 and n - 3 are the two branch nodes of the affine Dₙ graph: the node 1 has the neighbours 0, 2 and n - 1, the node n - 3 has the neighbours n - 4, n - 2 and n, and they are joined by the path 1 - 2 - ⋯ - (n - 3). The quartic Hochschild cohomology HH^{4, -2} of the affine Dₙ zigzag algebra is nonzero over every nontrivial commutative ring, as a case of the result for every graph containing an affine Dₙ (TauCeti.not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebra_of_isContained). When 2 is invertible the affine Dₙ zigzag algebra is moreover not intrinsically formal, by the two-branch-vertex criterion TauCeti.not_isIntrinsicallyFormal_zigzagStrictAInfinityAlgebra_of_three_le_degree of TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.NonFormal. The affine D₄ graph, whose only branch node has four neighbours, is treated in TauCeti.RepresentationTheory.Quiver.Zigzag.AInfinity.Affine.D.Four.

Main results

• TauCeti.not_hochschildVanishesAt_four_affineD: HH^{4, -2} of the affine Dₙ zigzag algebra is nonzero for n ≥ 5.
• TauCeti.not_isIntrinsicallyFormal_affineD: the affine Dₙ zigzag algebra is not intrinsically formal for n ≥ 5, when 2 is invertible.

theorem TauCeti.not_hochschildVanishesAt_four_affineD {k : Type w} [CommRing k] [Nontrivial k] {n : ℕ} (hn : 5 ≤ n) : ¬(zigzagStrictAInfinityAlgebra k ⋯).HochschildVanishesAt ⟨4, not_hochschildVanishesAt_four_affineD._proof_3⟩

The quartic Hochschild cohomology HH^{4, -2} of the affine Dₙ zigzag algebra is nonzero for n ≥ 5, over every nontrivial commutative ring: this is the case G = Dₙ of TauCeti.not_hochschildVanishesAt_four_zigzagStrictAInfinityAlgebra_of_isContained. For n = 4 see TauCeti.not_hochschildVanishesAt_four_affineD4.

theorem TauCeti.not_isIntrinsicallyFormal_affineD {k : Type w} [CommRing k] [Nontrivial k] [Invertible 2] {n : ℕ} (hn : 5 ≤ n) : ¬(zigzagStrictAInfinityAlgebra k ⋯).IsIntrinsicallyFormal

The affine Dₙ zigzag algebra is not intrinsically formal for n ≥ 5, over every nontrivial commutative ring in which 2 is invertible: some A∞ algebra whose cohomology algebra is the path-graded zigzag algebra is not formal. The graph is connected, and its two branch nodes 1 and n - 3 have the neighbours 0, 2, n - 1 and n - 4, n - 2, n. For n = 4 see TauCeti.not_isIntrinsicallyFormal_affineD4.