Koszul resolutions of zigzag simples from the preprojective Koszul complex

Let G be a finite simple graph and Q an orientation of G. The zigzag Koszul resolution TauCeti.zigzagKoszulResolution of the graded simple modules S_v of the zigzag algebra Z(G) exists as soon as the Koszul complexes 0 → e_w Π → ⨁_{j ∼ w} e_j Π of the signless preprojective algebra Π of the doubled graph, the quadratic dual of Z(G), are exact at their left ends. The Etingof--Eu Koszulity theorem is instead a statement about the signed preprojective algebra Π_k(Q): its left Koszul differentials y ↦ (b* y)_b are injective.

For a bipartite graph the two algebras agree: a colouring identifies the doubled graph with the symmetrification of Q, and then rescales each arrow of Q by the sign of the colour of its head (TauCeti.symmetrifySignlessPreprojectiveAlgebraEquiv). This identification fixes the vertex idempotents and carries each doubled arrow to a unit multiple of the corresponding arrow, so it transports left-end exactness from Π_k(Q) to the signless algebra. Hence, for a bipartite graph, injectivity of the left Koszul differentials of Π_k(Q) gives linear resolutions of all the zigzag simples. Every acyclic graph is bipartite, and every connected graph which is not a tree already has such resolutions (TauCeti.exists_isLinear_gradedProjectiveResolution_zigzagGradedSimple_of_not_isAcyclic), so for a connected graph Koszulity of Π_k(Q) for one orientation implies Koszulity of Z(G).

Main results

• TauCeti.forall_signlessPreprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_coloring: for a two-coloured graph, left-end exactness of the preprojective Koszul complex of any orientation at w gives left-end exactness of the signless Koszul complex of the doubled graph at w.
• TauCeti.exists_isLinear_gradedProjectiveResolution_zigzagGradedSimple_of_coloring: for a two-coloured graph, if the left Koszul differentials of Π_k(Q) are injective, every graded zigzag simple has a linear resolution. The same result for every connected graph is exists_isLinear_gradedProjectiveResolution_zigzagGradedSimple_of_preprojectiveKoszulLeft_injective

References

• R. S. Huerfano and M. Khovanov, A category for the adjoint representation, J. Algebra 246 (2001), Section 3, for the quadratic dual of the zigzag algebra and its comparison with the preprojective algebra of a bipartite graph.
• P. Etingof and C.-H. Eu, Koszulity and the Hilbert series of preprojective algebras, Math. Res. Lett. 14 (2007), Sections 2 and 3, for the vertex Koszul complexes of Π_k(Q).
• A. Beilinson, V. Ginzburg and W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (1996), Section 2, for Koszulity in terms of linear resolutions.

Transporting left-end exactness

theorem TauCeti.forall_signlessPreprojectiveMk_ofArrow_reverse_mul_eq_zero_iff_of_coloring (k : Type w) {V : Type u} {G : SimpleGraph V} [CommRing k] [Finite V] [(v : V) → Fintype ↑(G.neighborSet v)] (C : G.Coloring Bool) (o : DoubledQuiver.Orientation G) (w : V) (hw : ∀ (y : preprojectiveAlgebra k (DoubledQuiver.OrientedQuiver G o)), (preprojectiveMk k (DoubledQuiver.OrientedQuiver G o)) (doubledVertexIdempotent k (DoubledQuiver.OrientedQuiver.vertex G o w)) * y = y → (∀ (i : Quiver.Symmetrify (DoubledQuiver.OrientedQuiver G o)) (b : i ⟶ Quiver.Symmetrify.of.obj (DoubledQuiver.OrientedQuiver.vertex G o w)), (preprojectiveMk k (DoubledQuiver.OrientedQuiver G o)) (PathAlgebra.ofArrow (Quiver.reverse b)) * y = 0) → y = 0) {y : signlessPreprojectiveAlgebra k (DoubledQuiver G)} (hy : (signlessPreprojectiveMk k (DoubledQuiver G)) (PathAlgebra.vertexIdempotent k (DoubledQuiver.vertex G w)) * y = y) : (∀ (i : DoubledQuiver G) (b : i ⟶ DoubledQuiver.vertex G w), (signlessPreprojectiveMk k (DoubledQuiver G)) (PathAlgebra.ofArrow (Quiver.reverse b)) * y = 0) ↔ y = 0

Exactness of the signless Koszul complex at its left end, for a bipartite graph. Let C be a two-colouring of G and o any orientation of G. If the Koszul complex of the preprojective algebra Π_k(o) is exact at its left end at w, then so is the Koszul complex of the signless algebra Π of the doubled graph: for y ∈ e_w Π, b* y = 0 for every arrow b into w exactly when y = 0.

Linear resolutions of zigzag simples

theorem TauCeti.exists_isLinear_gradedProjectiveResolution_zigzagGradedSimple_of_coloring (k : Type w) {V : Type u} {G : SimpleGraph V} [Field k] [Finite V] (C : G.Coloring Bool) (o : DoubledQuiver.Orientation G) (ho : ∀ (v : DoubledQuiver.OrientedQuiver G o), Function.Injective ⇑(preprojectiveKoszulLeft k v)) (v : V) : ∃ (r : GradedProjectiveResolution (zigzagIntegerGrade k G) (zigzagGradedSimple k G v).grading), r.IsLinear

The zigzag simples of a bipartite graph have linear resolutions when the preprojective algebra is Koszul. Let C be a two-colouring of a finite simple graph G and o an orientation of G whose preprojective algebra has injective left Koszul differentials y ↦ (b* y)_b at every vertex. Then every graded simple module S_v of the zigzag algebra, concentrated in degree zero, has a linear graded projective resolution, namely TauCeti.zigzagKoszulResolution.

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

Koszulity of the preprojective algebra gives Koszulity of the zigzag algebra. Let G be a finite preconnected simple graph and o an orientation of G whose preprojective algebra has injective left Koszul differentials y ↦ (b* y)_b at every vertex. Then every graded simple module S_v of the zigzag algebra, concentrated in degree zero, has a linear graded projective resolution. An acyclic graph is two-coloured, and a graph which is not acyclic needs no hypothesis on o.