Koszul resolutions of zigzag simples for non-Dynkin graphs and the Koszulity dichotomy

Let G be a finite connected simple graph which is not a finite Dynkin diagram. The Etingof--Eu theorem gives injective left Koszul differentials for the preprojective algebra of every orientation of G, including when G is an affine or wild tree. Transport across the signed--signless comparison for bipartite graphs, together with the direct construction for graphs containing a cycle, then gives a linear graded projective resolution of every vertex simple of the zigzag algebra of G.

Combined with the finite-ADE linear-resolution obstruction of TauCeti.RepresentationTheory.Quiver.Zigzag.Projective.NonKoszul and the finite/infinite dichotomy of TauCeti.RepresentationTheory.Quiver.Preprojective.ADE.FiniteDimensional, this establishes the complete resolution-theoretic Koszulity classification for connected finite simple graphs on at least two vertices: every graded vertex simple of ZigzagAlgebra k G admits a linear graded projective resolution if and only if G is not isomorphic to the diagram of any valid finite Dynkin type, and if and only if the preprojective (or signless preprojective) algebra of G is infinite-dimensional.

Main results

• exists_isLinear_gradedProjectiveResolution_zigzagGradedSimple_of_not_exists_dynkinType_iso: every graded vertex simple of a connected non-Dynkin zigzag algebra has a linear graded projective resolution.
• forall_exists_isLinear_zigzagGradedSimple_iff: the Koszulity dichotomy for zigzag algebras: for a connected finite simple graph on at least two vertices, every graded vertex simple is linearly resolved if and only if the graph is not a Dynkin diagram.
• forall_exists_isLinear_zigzagGradedSimple_iff_forall_preprojectiveKoszulLeft_injective, forall_exists_isLinear_zigzagGradedSimple_iff_not_finiteDimensional_preprojective, and forall_exists_isLinear_zigzagGradedSimple_iff_not_finiteDimensional_signlessPreprojective: equivalence of zigzag Koszulity with exactness of the preprojective Koszul complex and with infinite-dimensionality of the preprojective and signless preprojective algebras.

References

• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), Sections 2 and 3.
• A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), Section 2.
• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), Section 3.

theorem TauCeti.exists_isLinear_gradedProjectiveResolution_zigzagGradedSimple_of_not_exists_dynkinType_iso (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] (hG : G.Connected) (hD : ¬∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)) (v : V) : ∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear

Every vertex simple of a connected non-Dynkin zigzag algebra has a linear resolution. If G is not isomorphic to the diagram of any valid finite Dynkin type, then the graded simple at every vertex has a linear graded projective resolution. This includes all connected non-Dynkin trees as well as connected graphs containing a cycle.

theorem TauCeti.forall_exists_isLinear_zigzagGradedSimple_iff (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [Nontrivial V] (hG : G.Connected) : (∀ (v : V), ∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear) ↔ ¬∃ (t : DynkinType), t.Valid ∧ Nonempty (G ≃g diagramGraph t.cartanMatrix)

The Koszulity dichotomy for zigzag algebras: for a connected finite simple graph G on at least two vertices, every graded vertex simple of ZigzagAlgebra k G admits a linear graded projective resolution if and only if G is not isomorphic to the diagram of any valid Dynkin type.

theorem TauCeti.forall_exists_isLinear_zigzagGradedSimple_iff_forall_preprojectiveKoszulLeft_injective (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [Nontrivial V] (hG : G.Connected) (o : DoubledQuiver.Orientation G) : (∀ (v : V), ∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear) ↔ ∀ (v : DoubledQuiver.OrientedQuiver G o), Function.Injective ⇑(preprojectiveKoszulLeft k v)

For a connected finite simple graph G on at least two vertices and any orientation o of G, every graded vertex simple of ZigzagAlgebra k G admits a linear graded projective resolution if and only if the left Koszul differential of Π_k(OrientedQuiver G o) is injective at every vertex.

theorem TauCeti.forall_exists_isLinear_zigzagGradedSimple_iff_not_finiteDimensional_preprojective (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [Nontrivial V] (hG : G.Connected) (o : DoubledQuiver.Orientation G) : (∀ (v : V), ∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear) ↔ ¬FiniteDimensional k (preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o))

For a connected finite simple graph G on at least two vertices and any orientation o of G, every graded vertex simple of ZigzagAlgebra k G admits a linear graded projective resolution if and only if the preprojective algebra Π_k(OrientedQuiver G o) is infinite-dimensional.

theorem TauCeti.forall_exists_isLinear_zigzagGradedSimple_iff_not_finiteDimensional_signlessPreprojective (k : Type w) [Field k] {V : Type u} {G : SimpleGraph V} [Finite V] [Nontrivial V] [(i : V) → Fintype ↑(G.neighborSet i)] (hG : G.Connected) : (∀ (v : V), ∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear) ↔ ¬FiniteDimensional k (signlessPreprojectiveAlgebra k (DoubledQuiver G))

For a connected finite simple graph G on at least two vertices, every graded vertex simple of ZigzagAlgebra k G admits a linear graded projective resolution if and only if the signless preprojective algebra of DoubledQuiver G is infinite-dimensional.